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Mapping and Advancing the Scalability-Accuracy Frontier of Nonlinear Causal Discovery
Hendrik Suhr, Sascha Xu, Jilles Vreeken
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Abstract
Abstract:Scalable nonlinear causal discovery requires methods that combine flexible mechanism estimators with efficient search over large graph spaces. Several algorithmic families have been proposed to address this challenge, yet their accuracy-runtime trade-offs remain poorly understood. We empirically compare the four major approaches: differentiable structure learning, amortized structure learning, score-matching, and combinatorial search. Our results reveal complementary bottlenecks: differentiable and amortized methods scale well but exhibit an accuracy gap, score-matching methods can be accurate in low dimensions but degrade quickly for increasing feature sizes, and combinatorial methods remain accurate but are slowed by repeated and redundant local scoring. Motivated by this bottleneck, we develop SPADE, a spline-based score-evaluation scheme that compiles sufficient statistics once and reuses them throughout combinatorial search. Under bounded indegree, its Gaussian variant reduces algorithmic complexity from O(nd^3) to O(nd^2+d^3). Empirically, SPADE shifts the observed scalability-accuracy frontier by orders of magnitude: it solves 100-variable problems with 160K samples in seconds and 1600-variable problems with 2.5K samples in minutes, while retaining high structural accuracy across synthetic and real-world benchmarks. These results reveal a substantial shift in the practical scale of combinatorial search and highlight the importance of evaluating scalable causal-discovery methods along the full accuracy-runtime frontier.
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- Source: https://arxiv.org/abs/2610.03258v1
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Mapping and Advancing the Scalability–Accuracy Frontier of Nonlinear Causal Discovery Hendrik Suhr Sascha Xu Jilles Vreeken CISPA Helmholtz Center for Information Security hendrik.suhr,sascha.xu,jv@cispa.de Abstract Scalable nonlinear causal discovery requires methods that combine flexible mechanism estimators with efficient search over large graph spaces. Several algorithmic families have been proposed to address this challenge, yet their accuracy–runtime trade-offs remain poorly understood. We empirically compare the four major approaches: differentiable structure learning, amortized structure learning, score-matching, and combinatorial search. Our results reveal complementary bottlenecks: differentiable and amortized methods scale well but exhibit an accuracy gap, score-matching methods can be accurate in low dimensions but degrade quickly for increasing feature sizes, and combinatorial methods remain accurate but are slowed by repeated and redundant local scoring. Motivated by this bottleneck, we develop Spade, a spline-based score-evaluation scheme that compiles sufficient statistics once and reuses them throughout combinatorial search. Under bounded indegree, its Gaussian variant reduces algorithmic complexity from O(nd3)O(nd^3) to O(nd2+d3)O(nd^2+d^3). Empirically, Spade shifts the observed scalability–accuracy frontier by orders of magnitude: it solves 100-variable problems with 160K samples in seconds and 1600-variable problems with 2.5K samples in minutes, while retaining high structural accuracy across synthetic and real-world benchmarks. These results reveal a substantial shift in the practical scale of combinatorial search and highlight the importance of evaluating scalable causal-discovery methods along the full accuracy–runtime frontier. 1 Introduction (a) Scalability–accuracy frontier on synthetic data. (b) Scalability–accuracy frontier on real-world data. Figure 1: Scalability–accuracy Pareto comparison on the synthetic qualitative benchmark, with F1 scores and runtimes averaged over 480 datasets (left), and on the real-world Causal Chamber benchmark (right). Learning causal structure from observational data is a central problem in science and machine learning. In many applications, interventions are costly, infeasible, or unethical, whereas large observational datasets are readily available. Causal discovery can support scientific hypothesis generation, improve robustness under distribution shift, and guide downstream decisions with causal rather than predictive relationships. In this work, we focus on causally sufficient nonlinear additive-noise models, in which the directed acyclic graph (DAG) is generically identifiable from observational data rather than only up to a Markov equivalence class [6, 22]. Yet identifiability does not make high-dimensional discovery easy: the space of DAGs grows super-exponentially, finite samples make structure selection uncertain, and nonlinear mechanisms typically require flexible conditional models rather than simple closed-form scores. Four main algorithmic families have been proposed to tackle this challenge. Combinatorial score-based methods assign DAGs or causal orders a likelihood score and search for high-scoring solutions [3, 2, 15, 31]. Score-matching methods exploit differential properties of the observational density, replacing explicit graph search by estimation of the Hessian of the log density [24, 17, 26, 8]. Differentiable structure-learning methods relax the discrete adjacency matrix to continuous parameters and impose acyclicity through smooth constraints or penalties [37, 10, 33, 19]. Amortized causal discovery uses transformers that are pretrained on synthetic causal models, which act as a prior, enabling causal discovery with a single forward pass [12, 11, 9, 21]. Despite this progress, it remains unclear how these approaches compare along the scalability–accuracy frontier, and which bottlenecks prevent accurate methods from scaling further. Our contributions are threefold: First, we empirically map the scalability–accuracy frontier of four major families of nonlinear causal-discovery methods as the number of variables and samples increases. Second, we identify repeated nonlinear local-score evaluation as a key bottleneck limiting the scalability of otherwise accurate combinatorial methods, motivating a reusable-score approach to graph search. Third, we instantiate this approach in Spade, which combines a spline-accelerated local score with the Topic search algorithm [31]. By compiling reusable sufficient statistics for additive-regression scores, Spade reduces repeated nonlinear scoring during sparse order search and substantially improves the observed scalability–accuracy frontier (Fig. 1). 2 Preliminaries We observe an i.i.d. sample =(x(1),…,x(n))⊤∈ℝn×dX=(x^(1),…,x^(n)) ^n× d from a distribution PXP_X over random variables X1,…,XdX_1,…,X_d. We assume that PXP_X is generated by a causally sufficient structural causal model (SCM) with directed acyclic graph (DAG) =(V,E)G=(V,E), where V=1,…,dV=\1,…,d\. An edge i→j∈Ei→ j∈ E indicates that XiX_i is a direct cause of XjX_j. We denote the causal parents of XjX_j by Paj:=i∈V:i→j∈E, _j:=\i∈ V:i→ j∈ E\, and write XPaj:=(Xi)i∈PajX_ _j:=(X_i)_i∈ _j. A DAG induces the usual recursive factorization p(x)=∏j=1dp(xj∣xPaj).p(x)= _j=1^dp\! (x_j x_ _j ). (1) Thus, score-based causal discovery can be viewed as searching over acyclic parent sets (S1,…,Sd)(S_1,…,S_d), where Sj⊆V∖jS_j V \j\ and the directed graph with edges i→ji→ j for i∈Sji∈ S_j is acyclic. A causal order is a permutation π of V such that all edges are oriented from earlier to later variables: if i→ji→ j, then π−1(i)<π−1(j)π^-1(i)<π^-1(j). Orders are therefore a convenient parameterization of acyclic search spaces: conditioning on π reduces DAG search to selecting, for each node j, a parent set from its predecessors. Additive-noise models. We focus on additive-noise models (ANMs), in which each variable is generated as Xj=fj(XPaj)+Nj,j=1,…,d,X_j=f_j(X_ _j)+N_j, j=1,…,d, (2) where fjf_j is an unknown structural function and N1,…,NdN_1,…,N_d are jointly independent exogenous noise variables. Under standard regularity conditions, nonlinear ANMs are generically identifiable from the observational distribution: except for known exceptional cases, most notably the linear Gaussian case, an incorrect causal direction does not admit an ANM representation with independent noise [6]. Hence, under these assumptions, the DAG is identifiable from PXP_X. In linear Gaussian SEMs, identifiability can be recovered under the equal-error-variance assumption, which requires all noise variables to have the same variance [22]. This assumption is fairly restrictive for heterogeneous real-world data, where quantities are typically recorded on human-made, non-comparable scales. Thus, we focus only on scale-invariant criteria. More general identifiable functional causal models have also been studied [35, 7]. They broaden the class of identifiable data-generating processes, but also introduce additional modeling and estimation challenges. We focus on ANMs because scalability already remains difficult in this simpler fully identifiable nonlinear setting. For a broader overview of causal-discovery methods, we refer to Zanga et al. [34]. 3 Scalability–Accuracy Trade-Offs We first examine the main algorithmic families proposed for scalable nonlinear causal discovery and inspect their trade-offs in terms of structural accuracy and runtime. The goal is to identify the bottleneck that currently limits this trade-off, and thereby to determine which aspect can and should be accelerated. To this end, we focus on four families of nonlinear causal-discovery methods: combinatorial score optimization, differentiable structure learning, amortized structure learning, and score matching. We briefly describe each family here, and provide the exact functioning of each tested method in Appx. A.1. Combinatorial score optimization. Combinatorial methods use discrete optimization to maximize a decomposable score S over candidate DAGs. For a graph G, with parent set Paj _j^G for node j, a score function commonly takes the form (,X)=∑j=1dsj(Xj∣XPaj),S(G;X)= _j=1^ds_j(X_j X_ _j^G)\;, (3) where each local term evaluates how well the proposed parents explain the corresponding child. In nonlinear additive-noise models, this typically amounts to fitting a local model Xj=fj(XPaj)+NjX_j=f_j(X_ _j^G)+N_j (4) and scoring the resulting residuals, optionally including a complexity penalty [13]. Under suitable model specification, such scores can identify the true DAG in the population limit [14]. We evaluate the order-based methods CAM [2] and Topic [31], as well as the greedy DAG-search method Globe [15]. These methods share related local score structures while exploring the graph space differently. Differentiable structure learning. Differentiable methods replace discrete graph search by continuous optimization over a relaxed adjacency matrix while jointly fitting causal mechanisms. Here, the objective that is optimized is typically of the form minW,θℒ(X,W,θ)s.t.h(W)=0, _W,θ\;L(X;W,θ) .t. h(W)=0\;, (5) where W∈ℝd×dW ^d× d is a continuous adjacency matrix, θ parameterizes the nonlinear mechanisms, and h(W)=0h(W)=0 enforces acyclicity. This formulation turns structure learning into a gradient-based optimization problem and is therefore naturally compatible with flexible neural estimators as well as fast runtime through highly parallelized GPU implementation. We evaluate nonlinear variants of NoTears [38], Dagma [1], and SDCD [19], using non-equal-variance objectives where required to match our benchmark setting. The resulting methods differ in both their mechanism parameterizations and acyclicity formulations. Score matching. Score-matching methods avoid explicit graph search by estimating derivatives of the observational log-density. Writing ψ(x)=∇xlogp(x)ψ(x)= _x p(x) (6) for the score function of the data distribution, these methods use score-derived quantities to recover a causal order under suitable assumptions [24]. This replaces optimization over DAGs by density-derivative estimation, and can provide identifiability under comparatively weak assumptions on the functional mechanisms. Some variants also allow more general noise distributions [17]. We evaluate DAS [18], DiffAN [26], and SciNO [8], focusing on recent score-matching variants that are explicitly designed for scalable nonlinear causal discovery. Since DiffAN and SciNO return an order rather than an edge set, we apply standard CAM pruning to obtain a DAG estimate. Amortized causal discovery. Amortized causal-discovery methods replace dataset-specific graph search with a learned predictor that maps an entire dataset directly to a distribution over causal graphs. They are pretrained on simulated SCMs: ϕ⋆=argminϕ(,X)∼Psim[ℒtrain(fϕ(X),)].φ = _φ\;E_(G,X) P_sim [L_train (f_φ(X),G ) ]. (7) After pretraining, they enable inference with a single forward pass ^=fϕ⋆(X) G=f_φ (X) at test time, shifting computation from per-dataset optimization to offline training [12, 11, 9, 21]. Their inductive biases are therefore determined in part by the simulator distribution, including assumptions on graph structure, mechanisms, noise, and interventions. We evaluate Avici [11] and CauScale [21], which provide public pretrained checkpoints and explicitly report generalization to graph sizes beyond their training domain. (a) Feature complexity. (b) Sample complexity. Figure 2: Scalability–accuracy trade-offs. Runtime and F1 score are shown in separate panels with matched dimensions. Dotted lines indicate the 24h timeout. Results are averaged across 5 seeds, shaded regions correspond to standard errors. 3.1 Experimental Setup Our goal is to evaluate how structural accuracy and runtime change as the number of variables and samples increases. To isolate this scalability–accuracy trade-off, we use a controlled nonlinear additive-noise setting that matches the assumptions of the compared method families as closely as possible. For the feature-scaling experiment, we vary the number of variables as d=25⋅2kd=25· 2^k and fix the sample size to n=2 500n=2\,500. For each dimension, we generate five random Erdős–R’enyi (ER) DAGs with expected degree three and sample one dataset from each DAG. We increase k until all methods, or all but one method, exceed the time limit of 2424 hours. GPU runs that exceed the 40 GB device-memory capacity of the available accelerators are recorded as OOM. For the sample-scaling experiment, we fix d=100d=100 and increase the sample size as n=2 500⋅2kn=2\,500· 2^k. We generate nonlinear mechanisms by sampling Gaussian-process functions and use Gaussian exogenous noise. To avoid exploiting equal-variance or scale cues, we standardize each variable before applying the methods [20]. We provide further details of the data-generating process in Appx. A.2, hardware in Appx. A.3, and implementation details for all methods in Appx. A.7. 3.2 Results We present the results in Fig. 2. For each family, we highlight the representative that achieved the most completed runs: SDCD for differentiable structure learning, DAS for score matching, CauScale for amortized inference, and Topic for combinatorial search. Full metric tables, including precision, recall, SHD, topological divergence, and runtime, are reported in Appx. A.6. Feature complexity. Fig. 2(a) shows a clear separation between runtime scalability and structural accuracy as the number of variables increases. SDCD and CauScale scale to the largest feature dimensions, reflecting the efficient continuous optimization of SDCD and the single-forward-pass inference of CauScale. However, this computational advantage comes at a substantial cost in structural accuracy: both methods remain well below the best F1 scores achieved in the benchmark, especially at the largest feature sizes. The highest F1F_1 scores are achieved by the combinatorial method Topic. Its accuracy degrades only mildly as d grows, indicating that its nonlinear local score retains a strong finite-sample signal even in larger graphs. The limitation is computational: Topic requires a cubic number of local score evaluations in the number of variables, and each evaluation involves fitting a nonlinear model. This makes the method substantially more accurate than the differentiable alternatives, but increasingly expensive as the graph grows. Among the score-matching methods, DAS gives the strongest feature-scaling performance, due to its efficient pruning. It is competitive at d=25d=25, but its structural accuracy deteriorates rapidly when the number of variables increases. This suggests that the bottleneck is the estimation of high-dimensional score-derived quantities, which becomes increasingly difficult as the number of variables grows. The remaining methods either achieve lower accuracy or become substantially slower, timing out between 5050 and 400400 variables. Sample complexity. The sample-scaling experiment in Fig. 2(b) sheds further light on the trade-off between sample size and structural accuracy. SDCD remains computationally efficient, but only partially closes the accuracy gap to Topic. This suggests that its main limitation might not be sample size alone, but the difficulty of extracting a reliable structural signal from the coupled graph–mechanism objective. We investigate this hypothesis further in Appx. A.4, and the results support this interpretation: Across the considered differentiable methods and regimes, the learned mechanisms do not reliably rank the true DAG above randomly sampled alternatives. CauScale worsens with increasing sample sizes and runs OOM already at 20K samples. This points to a potential mismatch between the pretraining distribution and the benchmark regime, which is a known limitation of amortized approaches [16]. Nonetheless, CauScale remains the most efficient method in its applicable regime. DAS improves as the sample size increases, consistent with the benefit of more accurate estimation of the score-derived quantities used by the method. However, this improvement comes with substantial runtime cost: the employed Stein estimator has cubic runtime in the sample size, and DAS reaches the time limit at 1010k samples. Thus, DAS can benefit from additional data but does not attain the same scalability–accuracy trade-off as combinatorial search in the present regime. Topic remains the strongest structural performer as n increases. This supports the conclusion that nonlinear local scores provide a reliable causal signal when evaluated accurately. At the same time, larger samples increase the cost of each nonlinear local fit, reinforcing that the bottleneck is not the search signal itself but the repeated evaluation of closely related local models. Implications. The benchmark reveals complementary bottlenecks. Differentiable and amortized methods offer the best runtime scaling, but lose too much structural accuracy. Score-matching methods avoid explicit graph search, but their performance is limited by high-dimensional derivative estimation. Combinatorial score-based search provides the strongest structural signal, but pays for it through repeated nonlinear local model fits. The latter bottleneck presents a concrete opportunity to improve scalability without sacrificing structural accuracy. The limitation of Topic is not that the local score lacks signal, but that evaluating it repeatedly is too expensive. In the next section, we show that for additive spline scores, much of the sample-dependent computation can be cached and reused across parent-set evaluations, enabling significantly faster search. 4 Scalable Combinatorial Causal Discovery Motivated by the benchmark in the previous section, we target the repeated local-score evaluations that limit accurate combinatorial methods. The key idea is to use a nonlinear additive score whose sample-dependent algebra can be compiled once and reused across the score queries issued during order search. We instantiate this idea with fixed-basis splines and combine the resulting score with the order-search framework of Topic. This yields Spade, an efficient scoring framework that retains likelihood-based nonlinear scoring while making repeated local-score evaluations computationally feasible. 4.1 Spline-Accelerated Score Computation We consider a decomposable BIC-type score for candidate parent sets [27]. During structure search, Paj _j is unknown, so the combinatorial search queries local scores for candidate pairs (j,S)(j,S), with S⊆V∖jS V \j\. For each queried pair, the score fits the conditional mean of YjY_j given XSX_S and evaluates the resulting residuals by a penalized negative log-likelihood. For a residual-likelihood choice ℓ , the local score is j(Yj∣S;ℓ)=ℒjℓ(Yj∣S)+penBIC(S),S_j(Y_j S; )=L _j(Y_j S)+pen_BIC(S), (8) where ℒjℓ(Yj∣S)L _j(Y_j S) is the residual negative log-likelihood and penBIC(S)pen_BIC(S) is the BIC complexity penalty. Decomposable BIC-type scores are a standard choice in score-based causal discovery. Our contribution is an efficient scheme to evaluate related local scores repeatedly during combinatorial search, thereby reducing the overall computational complexity. Fixed-basis splines. We now describe the spline model that exposes the reusable algebra. We instantiate the conditional-mean model with a fixed-basis generalized additive spline regression [5, 30]. The basis and smoothing penalty are fixed across score queries, so each queried parent set is scored by a single penalized least-squares solve rather than by an iterative GAM fit. This preserves a flexible nonlinear additive model while giving all local regressions a shared feature representation. For each variable XkX_k, we construct a centered spline design matrix Bk∈ℝn×KB_k ^n× K, where K is fixed. Let Yj∈ℝnY_j ^n denote the centered response for node j. For a candidate parent set S, define the concatenated design matrix BS=[Bk]k∈SB_S=[B_k]_k∈ S. We represent the conditional mean on the observed sample as fj,S(XS)=BSβj,S.f_j,S(X_S)=B_S _j,S. The spline coefficients are estimated by penalized least squares, β^j,S=argminβ‖Yj−BSβ‖22+β⊤PSβ, β_j,S= _β \ \|Y_j-B_Sβ \|_2^2+β P_Sβ \, (9) where PS=λΩS+εIP_S=λ _S+ I. ΩS _S is the spline roughness-penalty matrix, λ≥0λ≥ 0 controls the degree of smoothing, and ε>0 >0 is a small ridge parameter used for numerical stability. Hence, we obtain the fitted residuals as per N^j,S=Yj−BSβ^j,S. N_j,S=Y_j-B_S β_j,S\;. (10) Compiling repeated local queries. Using combinatorial search, the direct, repeated evaluation of Equation (9) is the main computational bottleneck. The same spline bases, response variables, and overlapping parent sets recur across many local-score queries, yet a naive implementation repeatedly forms local Gram matrices and feature–response cross-products from the observations. Because the spline basis and penalty rule are fixed, this sample-dependent algebra can instead be computed once and reused. Let B=[B1,…,Bd]∈ℝn×dKB=[B_1,…,B_d] ^n× dK denote the global spline design matrix and let Y=[Y1,…,Yd]Y=[Y_1,…,Y_d] denote the matrix of centered responses. Before search, we compute G=B⊤B,H=B⊤Y,cj=Yj⊤Yj,j∈1,…,d. G=B B,\,H=B Y,\,c_j=Y_j Y_j,\,\,\,\,j∈\1,…,d\. For a candidate parent set S, let I(S)I(S) denote the columns of B corresponding to the spline blocks of variables in S. The normal equations for Equation (9) become [GI(S),I(S)+PS]β^j,S=HI(S),j. [G_I(S),I(S)+P_S ] β_j,S=H_I(S),j. (11) Thus, each queried local regression reduces to extracting the cached blocks GI(S),I(S)G_I(S),I(S) and HI(S),jH_I(S),j and solving the corresponding penalized linear system of dimension |S|K|S|K. A direct fit would instead have to form BS⊤BSB_S B_S and BS⊤YjB_S Y_j from the observations for every query, with sample-dependent costs proportional to n(|S|K)2n(|S|K)^2 and n|S|Kn|S|K, respectively. For fixed bases and penalties, the cached formulation is equivalent to the direct fit in Equation (9), up to numerical precision, but removes this repeated pass over the data. Residual likelihoods. After fitting the spline, the remaining step is to evaluate the residual term in Equation (8). We use two variants. The Gaussian likelihood is computationally cheapest because it can be evaluated entirely from cached quadratic quantities. The histogram likelihood is more flexible because it estimates a univariate residual density and can better accommodate non-Gaussian noise. Both variants use the same additive spline mean model and the same BIC penalty; they differ only in the residual likelihood. Gaussian likelihood. For the Gaussian variant, the cache suffices to compute the residual sum of squares: RSSj,S=cj−2β^j,S⊤HI(S),j+β^j,S⊤GI(S),I(S)β^j,S.RSS_j,S=c_j-2 β_j,S H_I(S),j+ β_j,S G_I(S),I(S) β_j,S. (12) This is exactly equal to ‖Yj−BSβ^j,S‖22 \|Y_j-B_S β_j,S \|_2^2, but does not require materializing the residual vector. With the plug-in variance estimate σ^j,S2=RSSj,Sn σ_j,S^2= RSS_j,Sn, we obtain ℒjGauss(S)=n2[log(2π)+1+log(RSSj,Sn)]L^Gauss_j(S)= n2 [ (2π)+1+ ( RSS_j,Sn ) ] (13) for the Gaussian negative log-likelihood. Thus, after preprocessing, Gaussian local-score evaluation depends on the parent-set size but not on n. Histogram likelihood. To allow more flexible residual distributions, we also use a histogram-based negative log-likelihood. The histogram variant uses the same cached spline mean fit, but evaluates a smoothed leave-one-out histogram density on the materialized residuals. This gives a more flexible residual model at the cost of a sample-dependent residual-scoring step. We give the exact binning, smoothing, and scoring rule in Appx. A.7. This variant uses the same cached spline mean fit from Equation (11), but must materialize N^j,S N_j,S to estimate and evaluate the empirical residual density. Algorithm 1 in Appx. A.5 summarizes the procedure, and we provide implementation details in Appx. A.7. The preprocessing stage is executed once per dataset to enable fast queries during the combinatorial search. 4.2 Search The compiled score can be paired with any combinatorial search procedure that queries decomposable local scores. We adopt the topological-order search procedure introduced by Topic, the strongest combinatorial baseline in Section 3. At each step, the procedure selects the next node in the order, adds score-improving outgoing edges, and prunes redundant incoming edges. It therefore repeatedly evaluates small candidate parent sets, for which our cached spline score is particularly effective. The published Topic instantiation evaluated in Section 3 fits a separate cubic-spline model for each queried parent set, selecting its smoothing parameter during fitting. It also uses a two-part MDL score with a constant coding cost per parameter and edge. We instead use a shared fixed spline basis and smoothing penalty together with a BIC-type criterion. The shared representation enables the efficient score evaluation through sufficient-statistic compilation. The BIC penalty, kSlognk_S n, provides increasingly strong complexity control as the sample size grows, which is particularly relevant to the large-sample regimes. Pairing the Topic search procedure with our compiled spline score yields Spade 11 1 Spline-Accelerated DAG Estimation. We refer to the Gaussian and histogram-based variants as Spade-Gauss and Spade-Hist, respectively. In Section 5, we evaluate both variants extensively on a range of synthetic and real-world causal discovery benchmarks. (a) Feature complexity (b) Sample complexity Figure 3: Scalability in features and samples on clean data. Runtime and F1 score of the obtained DAG are shown in separate panels with matched dimensions. Results are averaged across 5 seeds, shaded regions correspond to standard errors. 4.3 Complexity We analyze the cost of score evaluation and assume a fixed number K of spline basis functions per variable. The global spline design has dKdK columns, so constructing the cache costs O(n(dK)2)O(n(dK)^2) time and O((dK)2)O((dK)^2) storage, which we write as O(nd2)O(nd^2) time and O(d2)O(d^2) storage for fixed K. Spade-Hist additionally retains the spline design matrix in order to construct residuals, requiring O(ndK)=O(nd)O(ndK)=O(nd) storage. Let p=|S|p=|S| be the size of a candidate parent set. Fixed-basis spline evaluations from scratch require O(np2+p3)O(np^2+p^3) time: O(np2)O(np^2) to form local Gram and feature–response products and O(p3)O(p^3) to solve the penalized system. Spade-Gauss instead evaluates a local score in O(p2+p3)=O(p3)O(p^2+p^3)=O(p^3) time by using only extracted cache blocks and quadratic forms. Spade-Hist uses the same cached mean fit but additionally constructs and scores the residual vector, adding O(np+p3)O(np+p^3) time per query. Thus, under bounded in-degree, Spade-Gauss requires constant time and Spade-Hist requires O(n)O(n) time per local-score query. Since Topic performs O(d3)O(d^3) local-score queries under bounded degree [31], the total score-evaluation cost of Spade-Gauss is O(nd2+d3)O(nd^2+d^3), compared with O(nd3)O(nd^3) for repeated direct spline evaluation. For Spade-Hist, the total asymptotic cost remains O(nd3)O(nd^3) because each query still scores residuals over the sample. However, caching removes the repeated mean-fit computations, enabling substantial speed-ups for both variants. 5 Experiments We evaluate Spade along three complementary axes. First, we evaluate Spade on the scalability benchmark from Sec. 3. Second, at moderate sample and feature sizes, we compare structural accuracy among a richer set of competitors across a broad synthetic grid. Finally, we test whether the observed gains transfer to real-world benchmark data. Additional experiments, including no-cache ablations of Spade-Gauss and Spade-Hist, a scalability benchmark with heterogeneous mechanisms and noise distributions, and results on the Sachs dataset [25], are provided in Appx. A.4. 5.1 Scalability We evaluate scalability in the setting from Sec. 3. Fig. 3 shows that both Spade variants substantially advance the scalability–accuracy frontier. In the feature-scaling experiment, they achieve higher F1 scores than Topic while being orders of magnitud– faster. The accuracy gain is not a consequence of caching, which only changes evaluation cost; rather, it suggests that the BIC-penalized fixed-basis spline score gives a more favorable parent-set criterion in this regime. At the largest feature size for which Topic terminates, d=800d=800, Topic requires 12.5 hours, whereas Spade-Gauss finishes in 99.3 seconds (456×456× speed-up) and Spade-Hist in 386.9 seconds (117×117× speed-up). At d=1600d=1600, Topic no longer completes within 24 hours, while Spade-Gauss and Spade-Hist terminate well within the limit. SDCD and CauScale are faster at the largest feature sizes, but achieve substantially lower F1 scores. The advantage is even more pronounced in the sample-scaling experiment. Spade-Gauss and Spade-Hist continue to improve with additional samples, whereas Topic plateaus. At the largest sample size for which Topic terminates, Spade-Gauss and Spade-Hist achieve speed-ups of 2708×2708× and 291×291×, respectively. At n=160 000n=160\,000, both variants still complete, while maintaining the strongest structural accuracy. To verify that these gains persist under more heterogenous mechanisms and noise distributions we repeated the experiment with a more diverse data-generating process in Appx. A.4. The results show that the runtime advantage persists, and that Spade-Hist is more robust to non-Gaussian noise than Spade-Gauss. Overall, these results show that Spade advances the empirical scalability–accuracy frontier on synthetic data. It improves the structural accuracy of the most accurate baseline while being orders of magnitude faster, handles both very large feature and sample sizes, and achieves substantially higher F1 scores than the most scalable baselines. 5.2 Robustness Across Data-Generating Regimes Previously, we have focused on runtime and F1 score as a function of feature and sample size. We now assess robustness across graph density, mechanism class, and noise family by reporting conditional F1 scores that marginalize over the remaining factors; full regime-wise results are in Appx. A.6. The 96 settings cover ER DAGs with expected degrees in 2,3,4\2,3,4\, additive and non-additive mechanisms, and Gaussian, unimodal symmetric, unimodal asymmetric, or multimodal additive noise; details are given in Appx. A.2. To isolate robustness across data-generating regimes, and to compare to a broader set of competitors, we consider moderate problem sizes with n∈1 000,2 000n∈\1\,000,2\,000\, and d∈20,50d∈\20,50\. We extend the competitor set with Score [24], NoGAM [17], Caps [32], and kernel-based PC and FCI [28, 29, 36]. Since PC and FCI return Markov equivalence classes, we evaluate only their directed edges and treat ambiguous edges as abstentions. We present the results for n=1 000n=1\,000 in Fig. 4, and defer the results for n=2 000n=2\,000 to Appx. A.4. We present Spade and the three strongest competitors by average F1 per problem size. Overall, Spade-Hist achieves the strongest performance and is most robust under non-Gaussian noise, whereas Spade-Gauss performs best in the Gaussian-noise regime and remains competitive elsewhere. Across all marginalized settings, one of the two variants obtained the best F1 score. We therefore recommend Spade-Hist as the default unless scalability requirements are extreme or the residual noise is known to be close to Gaussian. Finally, we aggregate F1 scores and runtimes across all synthetic settings to assess the moderate-size scalability–accuracy frontier. We include all methods except PC and FCI, which did not complete all n=2000,d=50n=2000,d=50 runs within 24 hours. In Fig. 1(a), ”Ours” refers to Spade-Hist. Averaged across all settings, Spade-Hist achieves the best F1 score, 0.748, compared to 0.616 for the strongest non-Spade competitor, Topic, while reducing mean runtime from 75.35 seconds to 1.24 seconds, a speed-up of roughly 61×61×. Even compared to the fastest external method, CauScale, which runs in 6.23 seconds on average but reaches only 0.542 F1, Spade-Hist is both more accurate and about 5×5× faster. 5.3 Real-World Data We finally evaluate whether the improvements on synthetic data transfer to real-world datasets. We focus on the Causal Chamber benchmark [4], a controlled physical benchmark with known causal structure. As shown in Fig. 1(b), Spade-Hist achieves the strongest runtime–accuracy trade-off. It obtains an F1 score of 0.50 in 0.97 seconds on average. The closest competitor in F1 is SDCD, with 0.49 F1, but it requires 520 seconds. The fastest competitor, CauScale, runs in 10 seconds but reaches only 0.27 F1. We provide results on the Sachs protein-signaling dataset [25] in Appx. A.4, where Spade remains among the strongest and fastest methods. Overall, the real-world results support the synthetic findings: Spade is the only method in this comparison that combines competitive accuracy with consistently low runtime, advancing the empirical scalability–accuracy frontier beyond simulated benchmarks. Figure 4: Qualitative performance on synthetic datasets with n=1000n=1000 in F1 score. Each spoke isolates the respective condition while marginalizing over remaining conditions. 6 Conclusion We compared the scalability–accuracy trade-offs of four major families of nonlinear causal-discovery methods. Across the considered benchmarks, combinatorial methods provided the strongest signal, but were limited by repeated nonlinear local-score evaluations. We introduced Spade, a nonlinear additive score whose fixed-basis spline representation enables shared sample-dependent computations to be compiled and reused throughout sparse order search. This substantially advances the empirical scalability–accuracy frontier, retaining strong structural accuracy at problem sizes where existing nonlinear combinatorial methods become impractical. More broadly, our results highlight that scalability in nonlinear causal discovery should be evaluated along the full scalability–accuracy frontier and across all major algorithmic families. Such comparisons help localize where improvements occur and make the corresponding trade-offs explicit. We hope that our findings can guide future work both by providing a strong reference point and by revealing practical bottlenecks of existing approaches. 6.1 Limitations & Future Work Our comparison is empirical. The observed strengths and bottlenecks of the evaluated method families should therefore be interpreted as patterns in the studied benchmarks rather than universal characterizations of each approach. Reproducing these findings across further simulators and application domains is an important direction for future work. We focus on causally sufficient, fully identifiable additive-noise models. Extending compiled nonlinear local scores to partially identifiable CPDAG settings, heteroscedastic mechanisms, latent confounding, and broader functional causal models remains open. Finally, combinatorial search is not the only promising route toward scalable causal discovery. Improvements in score-derivative estimation, differentiable optimization, and amortized causal discovery could provide complementary solutions. AI use statement In this work, we used generative AI tools to assist with conceptual and methodological discussion, technical formulation, and the interpretation and presentation of experimental results. We additionally used generative AI tools to draft and edit text, improve readability, and assist with coding. 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Advances in neural information processing systems 31. Cited by: §1. [38] X. Zheng, C. Dan, B. Aragam, P. Ravikumar, and E. Xing (2020) Learning sparse nonparametric dags. In International conference on artificial intelligence and statistics, p. 3414–3425. Cited by: §A.1, §3. Checklist 1. For all models and algorithms presented, check if you include: (a) A clear description of the mathematical setting, assumptions, algorithm, and/or model. [Yes/No/Not Applicable] Yes. Sections 2 and 4 describe the mathematical setting, assumptions, local score, compiled sufficient statistics, and search procedure. (b) An analysis of the properties and complexity (time, space, sample size) of any algorithm. [Yes/No/Not Applicable] Yes. Section 4.3 analyzes the time and storage complexity of direct and compiled score evaluation and the resulting search procedure. (c) (Optional) Anonymized source code, with specification of all dependencies, including external libraries. [Yes/No/Not Applicable] Yes. The supplementary material includes anonymized source code, a Dockerfile specifying the execution environment and dependencies, and commands for reproducing the experiments. 2. For any theoretical claim, check if you include: (a) Statements of the full set of assumptions of all theoretical results. [Yes/No/Not Applicable] Yes. The assumptions used for the complexity analysis, including fixed spline-basis size and bounded parent-set size, are stated in Section 4.3. (b) Complete proofs of all theoretical results. [Yes/No/Not Applicable] Not Applicable. The paper does not present novel theorem or recovery-guarantee claims; its formal claims are direct computational-complexity derivations given in Section 4.3. (c) Clear explanations of any assumptions. [Yes/No/Not Applicable] Yes. The modeling assumptions are explained in Section 2, and the assumptions used in the complexity analysis are explained in Section 4.3. 3. For all figures and tables that present empirical results, check if you include: (a) The code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL). [Yes/No/Not Applicable] Yes. The supplementary reproduction package includes the experimental code, a containerized environment, exact paper presets, seeds, data-loading instructions, and commands for reproducing the main results. (b) All the training details (e.g., data splits, hyperparameters, how they were chosen). [Yes/No/Not Applicable] Yes. Appendix A.2 gives the data-generation and sampling procedures, and Appendix A.7 reports the hyperparameters and how they were selected. (c) A clear definition of the specific measure or statistics and error bars (e.g., with respect to the random seed after running experiments multiple times). [Yes/No/Not Applicable] Yes. Appendix A.6 defines the evaluation metrics and aggregation, while figure captions and the appendix report the number of seeds and identify shaded regions and tabulated uncertainties as standard errors. (d) A description of the computing infrastructure used. (e.g., type of GPUs, internal cluster, or cloud provider). [Yes/No/Not Applicable] Yes. Appendix A.3 reports the compute-cluster setting, CPU model and allocation, and GPU model and memory. 4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets, check if you include: (a) Citations of the creator If your work uses existing assets. [Yes/No/Not Applicable] Yes. The paper cites the creators of the existing datasets and baseline methods used in the evaluation. (b) The license information of the assets, if applicable. [Yes/No/Not Applicable] No. The current manuscript does not list license information for the external datasets, code, and pretrained models used in the evaluation. (c) New assets either in the supplemental material or as a URL, if applicable. [Yes/No/Not Applicable] Yes. The implementation introduced in this work is included in the supplementary reproduction package together with build and execution instructions. (d) Information about consent from data providers/curators. [Yes/No/Not Applicable] Not Applicable. We use previously published benchmark datasets and do not collect data directly from participants or providers. (e) Discussion of sensible content if applicable, e.g., personally identifiable information or offensive content. [Yes/No/Not Applicable] Not Applicable. The synthetic, physical-system, and protein-signaling benchmark data used here do not contain personally identifiable or offensive content. 5. If you used crowdsourcing or conducted research with human subjects, check if you include: (a) The full text of instructions given to participants and screenshots. [Yes/No/Not Applicable] Not Applicable. This work did not use crowdsourcing or conduct research with human participants. (b) Descriptions of potential participant risks, with links to Institutional Review Board (IRB) approvals if applicable. [Yes/No/Not Applicable] Not Applicable. This work did not use crowdsourcing or conduct research with human participants. (c) The estimated hourly wage paid to participants and the total amount spent on participant compensation. [Yes/No/Not Applicable] Not Applicable. This work did not use crowdsourcing or conduct research with human participants. Appendix A Appendix A.1 Causal Discovery Methods This appendix describes the causal-discovery methods considered in the empirical comparison. The methods span combinatorial score optimization, differentiable structure learning, score-matching and ordering-based discovery, amortized causal discovery, and classical constraint-based references. Combinatorial additive-score methods. CAM estimates causal additive models by separating the problem into order search and edge selection [2]. It assumes an additive structural equation model in which each variable is a sum of nonlinear effects of its parents plus independent noise. The method first searches for a topological order using restricted likelihood scores for additive regressions and then prunes the dense DAG induced by that order with sparse additive regression. This decoupling makes the search more manageable than enumerating all DAGs, but each candidate still requires nonlinear regression fits. Topic is an information-theoretic causal-discovery method that also works in a topological-order framework [31]. It scores candidate parent sets with local code-length or likelihood criteria for continuous data and interleaves ordering, edge insertion, and pruning rather than treating these as fully separate stages. The resulting search maintains a sparse graph throughout the optimization and repeatedly compares nonlinear local mechanism scores, which makes Topic a strong but computationally demanding representative of combinatorial nonlinear search. Globe searches directly over fully oriented causal networks using an MDL-inspired score for functional causal models [15]. It constructs a directed graph by greedily modifying parent sets and evaluating whether the resulting local mechanisms yield a shorter description of the data. Compared with order-only procedures, Globe performs a more direct graph search, but it relies on repeated local model evaluation and therefore faces the same basic scalability pressure in high dimensions. Differentiable structure learning. NoTears replaces the discrete DAG constraint with a smooth acyclicity equality, enabling gradient optimization over a continuous weighted adjacency matrix. The nonlinear formulation parameterizes node mechanisms by neural networks and uses sparsity of input derivatives to induce graph sparsity [38]. Thus, graph structure and nonlinear mechanisms are learned jointly by continuous optimization, with acyclicity enforced through the differentiable constraint. Dagma is another continuous DAG-learning method, but replaces the matrix exponential acyclicity function by a log-determinant characterization defined over an M-matrix domain [1]. The method optimizes a sequence of unconstrained objectives along a barrier-like central path, with sparsity regularization on the adjacency. Its nonlinear variant combines this acyclicity formulation with neural network mechanisms, yielding a continuous graph–mechanism objective that is designed to have better numerical behavior than earlier matrix-exponential relaxations. SDCD is designed to stabilize differentiable causal discovery at larger dimensions [19]. It uses an acyclicity constraint with quadratic computational structure and a training procedure tailored to sparse graphs, avoiding the cubic matrix operations that dominate several earlier continuous relaxations. The method separates training into stages that first obtain a stable sparse structural representation and then refine the graph and mechanisms, making it a scalable representative of differentiable DAG learning. Score-matching and ordering methods. Score recovers a causal order in nonlinear additive Gaussian-noise models from derivatives of the observational log density [24]. The key signal is based on the score function ∇xlogp(x) _x p(x) and its derivatives: leaf nodes can be identified from properties of the Hessian of the log density, removed, and the procedure iterated. After an order has been estimated, sparse edge selection can be performed within the DAGs compatible with that order. NoGAM extends the score-matching route beyond Gaussian noise by using additive-model assumptions with arbitrary noise distributions [17]. It retains the idea of estimating density-score-derived quantities to infer an order, but avoids relying on the Gaussian-noise form of the original Score criterion. This makes NoGAM a score-matching method for nonlinear additive models with weaker restrictions on the exogenous noise distribution. DAS uses score matching to recover the whole graph in nonlinear additive Gaussian-noise models while reducing the cost of the pruning stage substantially [18]. Instead of first estimating a full order and then applying an expensive dense pruning pass, it exploits second-derivative information of the log density to remove many candidate edges more directly. The method therefore targets the same score-derived identifiability signal as Score, but couples order estimation with a more efficient graph-recovery procedure. DiffAN is an order-estimation method that replaces kernel or Stein estimates of the required Hessian quantities by a diffusion-model estimator [26]. It learns a neural score model and uses the estimated Hessian information to identify leaves recursively, yielding a topological order for an additive-noise model. The central idea is to replace explicit high-dimensional density-derivative estimation by a learned score model whose derivatives can be used for causal ordering. SciNO also targets ordering-based causal discovery from score-derived Hessian information, but uses a score-informed neural operator to approximate the Hessian diagonal more stably than directly differentiating a score network [8]. It represents score-related functions in a smooth operator-learning framework, aiming to preserve the ordering information needed for additive-noise causal discovery while improving numerical stability over direct Hessian estimation. Caps is an ordering-based method for data containing both linear and nonlinear causal relations [32]. It estimates an order from score-derived Hessian information and then uses parent scores to guide post-processing: weak candidate parents are pruned before additional high-score edges are considered subject to acyclicity. The method is designed for mixed linear and nonlinear mechanisms, where a single purely linear or purely nonlinear criterion may be poorly matched to the data-generating process. Amortized causal discovery. Avici performs amortized variational inference for causal structure learning [11]. A neural inference model is pretrained on simulated SCMs and then maps a new dataset directly to edge probabilities, so test-time inference is a forward pass rather than dataset-specific graph search. Its architecture is designed to respect permutation symmetries of the causal structure learning problem, while its statistical behavior is governed by the synthetic simulator distribution used during pretraining. CauScale is a neural causal-discovery foundation model designed for large graphs [21]. It improves the memory and runtime profile of transformer-style amortized discovery by compressing data embeddings and using tied attention weights, while combining a data stream with a graph-prior stream. As with other amortized methods, the computational cost of learning is shifted to pretraining, after which structure prediction for a new dataset is performed by neural inference. Constraint-based references. PC and FCI are included as classical constraint-based references in the qualitative synthetic comparison [28, 29]. PC assumes causal sufficiency and recovers a Markov equivalence class by iteratively testing conditional independences, whereas FCI allows latent confounding and selection effects and returns a partial ancestral graph. Kernel conditional independence tests extend these procedures to nonlinear relationships [36]. Unlike the additive-noise methods above, constraint-based methods identify an equivalence class under Markov and faithfulness assumptions rather than a single fully oriented DAG in general. A.2 Data Synthetic data All synthetic datasets are generated from causally sufficient structural causal models with independent additive noise. Unless stated otherwise, we first draw a random topological order π on d variables and then sample an Erdős–R’enyi DAG by adding each order-compatible edge independently with probability p=mink¯/(d−1),1,p= \ k/(d-1),1\, where k¯ k is the target expected degree. If the resulting graph is empty, we add the edge from the first to the last variable in the sampled order. Each present edge receives a signed weight whose magnitude is sampled uniformly from [0.5,2][0.5,2]. Variables are generated in topological order. Root variables are exogenous noise variables. For a non-root node j, with parent values XPa(j)X_ (j), we sample Xj=fj(XPa(j))+Nj.X_j=f_j(X_ (j))+N_j. The noise variables NjN_j are independent across nodes and samples. After generation, every observed variable is standardized to zero empirical mean and unit empirical variance. The generic simulator supports four mechanism classes. The linear class uses the sampled edge weights directly. The additive class generates a sum of univariate nonlinear parent effects. For each non-root node, the nonlinear effect family is sampled from basis functions, polynomials, random Fourier features approximating a Gaussian process, and one-hidden-layer neural networks. Basis effects use random transformations from sin,cos,tanh,sigmoid,x3,sign(x)|x|,exp(−|x|)\ , , ,sigmoid,x^3,sign(x) |x|, (-|x|)\ with random scale and shift; polynomial effects use clipped powers of degree 22 to 44; Gaussian-process effects use 1616 to 4848 random Fourier features with lengthscale in [0.4,2.5][0.4,2.5]; neural mechanisms use hidden width 44 to 1212 and an activation sampled from tanh,ReLU,sigmoid,sin\ ,ReLU,sigmoid, \. The nonadditive class adds parent-parent interaction terms to the additive effects; for Gaussian-process and neural-network generators it also includes a multivariate parent effect. The never-additive class removes the additive component whenever a node has at least two parents, leaving only interaction effects; nodes with a single parent fall back to a univariate nonlinear effect. In heterogeneous settings, the mechanism class is sampled independently for each non-root node from the available classes. Noise distributions are sampled from four families. The Gaussian family is standard normal. The unimodal symmetric family contains Laplace noise scaled to unit variance, Student-t noise with three degrees of freedom scaled by 3 3, and uniform noise on [−3,3][- 3, 3]. The unimodal asymmetric family contains centered exponential noise, standardized lognormal noise, standardized Gamma noise, and a standardized skew-normal-like mixture formed by adding an absolute Gaussian component. The multimodal family contains two- and three-component Gaussian mixtures, three-component Laplace and Student mixtures, and a four-component mixed Gaussian/Laplace/Student/ exponential mixture; all multimodal noises are standardized. In homoscedastic settings, each non-root node receives a constant noise scale sampled uniformly from [0.4,1][0.4,1], except in the controlled scalability experiments described below. Controlled scalability. The controlled scalability experiments use a deliberately clean additive-noise setting. The graph is sampled with exactly round(k¯d/2)round( kd/2) order-compatible edges rather than only in expectation, with k¯=3 k=3. All non-root structural mechanisms are Gaussian-process functions represented by random Fourier features with lengthscale in [0.4,2.5][0.4,2.5], and all noise variables are Gaussian. Non-root noise scales are fixed to 11, so the only scale variation is introduced by the final column standardization. The feature-scaling panel fixes n=2,500n=2,500 and varies d∈25,50,100,200,400,800,1 600,d∈\25,50,100,200,400,800,1\,600\, whereas the sample-scaling panel fixes d=100d=100 and varies n∈2 500,5 000,10 000,20 000,40 000,80 000,160 000.n∈\2\,500,5\,000,10\,000,20\,000,40\,000,80\,000,160\,000\. Each setting is averaged over five independently generated datasets. The runtime-ablation experiment uses the same controlled data-generating process and the same feature- and sample-scaling grids. Heterogeneous scalability. The heterogeneous scalability experiment keeps the same graph-density and problem-size grids as the controlled scalability experiment, but replaces the clean GP/Gaussian mechanisms by node-wise heterogeneous mechanisms and noises. The graph is an order-based Erdős–R’enyi DAG with expected degree k¯=3 k=3. For each non-root node, the mechanism class is sampled at random, and nonlinear mechanisms further sample their concrete generator from the basis, polynomial, Gaussian-process, and neural-network families described above. The noise distribution is also sampled node-wise by first choosing one of the Gaussian, unimodal symmetric, unimodal asymmetric, or multimodal noise families and then sampling a concrete distribution from that family. The data are homoscedastic, with node-wise constant noise scales sampled from [0.4,1][0.4,1], and are standardized column-wise after generation. We use five independent seeds per setting. Qualitative synthetic comparison. The qualitative comparison uses the same generic SCM simulator, but varies the data-generating factors in a full moderate-size grid. We use (n,d)∈(1000,20),(2000,20),(1000,50),(2000,50),k¯∈2,3,4,(n,d)∈\(1000,20),(2000,20),(1000,50),(2000,50)\, k∈\2,3,4\, with either additive or never-additive mechanisms. The noise family is one of Gaussian, unimodal symmetric, unimodal asymmetric, and multimodal noise. This yields 4×3×2×4=964× 3× 2× 4=96 data-generating settings. For each setting we generate five independent datasets, giving 480480 synthetic datasets in total. The noise family is resolved independently for each node to one of the concrete distributions in that family, and all datasets are homoscedastic and column-standardized. Differentiable-mechanism diagnostic. The diagnostic experiment in Appx. A.4 uses a separate non-additive neural SCM designed to be favorable to neural mechanism learners. For each d∈10,25,50,100d∈\10,25,50,100\ and seed s∈0,…,4s∈\0,…,4\, we sample n=1000n=1000 observations from an order-based Erdős–R’enyi DAG with expected degree 44, forcing one edge if the sampled graph is empty. Root variables are independent standard Gaussian noises. For each non-root variable, the main effect is a random one-hidden-layer neural network of its parents with hidden width sampled uniformly from 6,…,16\6,…,16\, Gaussian weights scaled by the input width, Gaussian biases, and activation sampled from tanh,ReLU,sigmoid,sin\ ,ReLU,sigmoid, \. If a node has at least two parents, every parent pair contributes an additional random neural interaction, multiplied by a coefficient sampled uniformly from [0.5,1.5][0.5,1.5]. The deterministic effect is standardized within the generated sample and then combined with Gaussian noise scaled by a coefficient sampled uniformly from [0.4,1][0.4,1]. The final observed variables are standardized column-wise. Real-world data The real-world experiments use two observational tabular benchmarks with reference graphs. For Causal Chamber [4], we use the light-tunnel dataset lt_interventions_standard_v1 and its uniform_reference experiment, which provides non-image observations from the standard light-tunnel configuration together with a chamber-level ground-truth graph. We keep the numeric measurement variables, remove rows with missing or non-finite values, discard constant columns, and standardize each variable. The reference graph is restricted to the retained variables. The reported Causal Chamber setting contains 10,00010,000 observations of 2323 variables and 3939 reference edges; the table reports five bootstrap resamples of this observational table. For Sachs [25], we use the single-condition observational Benchpress [23] version of the protein-signaling data. The reference graph is the standard 11-node consensus DAG over the measured signaling proteins. After removing one incomplete row, the reported Sachs setting contains five bootstrap samples of 853853 observations, 1111 variables, and 1717 reference edges. A.3 Hardware Experiments were run on a compute cluster. GPU-based methods were allocated one NVIDIA A100 40 GB GPU and four CPU threads per run. CPU-based methods were run on nodes equipped with AMD EPYC 7773X processors; each CPU run was allocated four CPU threads and no GPU. Reported runtimes are wall-clock runtimes of the benchmark invocation. A.4 Experiments In this section we present additional experimental details for the experiments in the main paper and provide additional experiments. (a) Feature complexity. (b) Sample complexity. Figure 5: Scalability in features and samples on heterogeneous data. Runtime and F1 score of the obtained DAG are shown in separate panels with matched dimensions. Results are averaged across 5 seeds, shaded regions correspond to standard errors. Heterogeneous benchmark. To verify that the scalability–accuracy gains observed in Sec. 5.1 persist under more heterogeneous mechanisms and noise distributions, we run a variant of the controlled scalability experiment with node-wise heterogeneous mechanisms and noises, we repeat the experiments from Sec. 5.1 with the same feature- and sample-scaling grids, but with node-wise heterogeneous mechanisms and noise distributions. The diverse simulator samples node-wise mechanism classes and noise families, including linear, additive, non-additive, and interaction-based nonlinear mechanisms, as well as Gaussian, unimodal, asymmetric, and multimodal additive noise. We describe the data generation process in detail in Appx. A.2. Fig. 5 shows that the gains persist in the diverse setting. In the feature-scaling experiment, both variants again outperform Topic in F1 score. Spade-Hist achieves the strongest accuracy, consistent with its more flexible residual likelihood. At d=1600d=1600, Topic does not complete within 24 hours, while Spade-Gauss and Spade-Hist finish in 918 and 2416 seconds. As before, SDCD and CauScale remain faster at the largest feature sizes, but are substantially less accurate. The sample-scaling experiment further highlights the robustness of the histogram score. Spade-Hist reaches an F1 score of 0.829, compared to 0.724 for Spade-Gauss and 0.622 for Topic. In terms of runtime, Spade-Gauss is even faster than SDCD and CauScale, which runs out of memory after n=10 000n=10\,000 samples. Differentiable Structure Learning Experiment To investigate the lower structural quality of the differentiable methods compared to Topic observed in Sec. 3, we test whether the learned mechanisms are capable of guiding the search towards the true DAG. While these mechanisms are typically more expressive than the additive spline models used by combinatorial methods, the joint optimization of a DAG parametrization and nonlinear mechanisms can get stuck in local optima: the mechanisms may adapt to the currently preferred candidate DAG and fail to assign better scores to better structures. To test this hypothesis, we consider NoTears, Dagma and SDCD. We use a data-generating setup that is favorable to neural mechanism learners. For each dimension d∈10,25,50,100d∈\10,25,50,100\ and seed s∈0,…,4s∈\0,…,4\, we sample n=1000n=1000 observations from a random nonlinear additive-noise SCM. The graph is an Erdos–Renyi DAG generated by first sampling a random topological order and then adding each forward edge independently with probability 4/(d−1)4/(d-1), corresponding to expected degree 44; if this produces an empty graph, one forward edge is added. Root variables are independent standard Gaussian noise variables. For non-root variables, the structural function is a random one-hidden-layer neural network of its parents, with hidden width sampled uniformly from 6,…,16\6,…,16\, Gaussian weights scaled by the input width, Gaussian biases, and an activation sampled from tanh,ReLU,sigmoid,sin\ ,ReLU,sigmoid, \. To make the mechanisms non-additive, whenever a node has at least two parents we additionally add random neural effects of every parent pair, each multiplied by a coefficient sampled uniformly from [0.5,1.5][0.5,1.5]. The deterministic effect is standardized within the generated sample and then combined with Gaussian noise scaled by a coefficient sampled uniformly from [0.4,1.0][0.4,1.0]. Finally, all variables are standardized column-wise before fitting the methods. During the training of each method, we freeze the learned mechanisms at checkpoints and test how well they rank the true DAG compared to 10001000 dense DAGs induced by independent random orders. Each candidate DAG is scored by applying its mask to the frozen learned mechanism and evaluating the resulting reconstruction or likelihood score on the validation split. We compare these rankings to an additive regression spline score, which is misspecified for the non-additive MLP mechanisms but refits a separate regression for each candidate DAG. We illustrate the results in Fig. 6. No differentiable method reliably ranks the true DAG highest among only 10001000 random DAGs. No differentiable method achieves an ideal rank. In contrast, the spline score ranks the true DAG highest consistently despite the generating mechanisms being non-additive. This is possible because ranking the true DAG highest among only 10001000 random-order DAGs is a weak requirement: most random orders are far from the true one. The fact that the learned mechanisms fail this task suggests that the benchmarked differentiable learners get stuck in poor local optima of the joint DAG–mechanism space. Bridging the gap to the structural quality of combinatorial methods therefore likely requires optimization strategies that produce more reliable structural ranking signals and escape these local optima. Figure 6: The mechanisms learned by differentiable structure learning methods do not rank the true DAG the highest reliably. Runtime Ablation To isolate the gains of compiling the Gram matrices and feature–response cross-products, we run Spade-Gauss and Spade-Hist without compilation on setting with Gaussian processes and Gaussian noise. We refer to these variants as Spade-Gauss-NoCache and Spade-Hist-NoCache. We present the results in Fig. 7. As expected we observe significant speed-up through caching, especially for Spade-Gauss. (a) Feature complexity. (b) Sample complexity. Figure 7: Runtime ablation of the cached sufficient-statistics implementation. Each cached variant is compared to an otherwise identical variant that refits the additive-spline score from scratch. Results are averaged across 5 seeds, and shaded regions show standard errors. Qualitative results at n=2000n=2000 We present the qualitative synthetic comparison at n=2000n=2000 in Fig. 8. While the set of the three strongest competitors differs, the patterns of the Spade-variants are similar to the n=1000n=1000-setting, which is presented in the main paper. Figure 8: Qualitative performance on synthetic datasets with n=2000n=2000 in F1 score. Each spoke isolates the respective condition while marginalizing over remaining conditions. Sachs We evaluate the two variants of Spade and all competitors on the real-world Sachs dataset [25]. Because of the skewedness of the distributions, we applied a log and arcsinharcsinh transformations to the data. We report the results for the raw data and the two transformations in Tab. 42. While the Spade-variants do not achieve the best F1 score, they are among the 3 highest F1 scores in all 3 settings, and consistently achieve the lowest SHD, while also achieving the lowest runtimes. A.5 Cached Sufficient Statistics We present the algorithm that summarizes the algebraic compilation algorithm underlying Spade in Algorithm 1. Algorithm 1 Compiled local-score oracle used by Spade 0: Dataset X∈ℝn×dX ^n× d; residual likelihood ℓ ; spline basis size K; spline-penalty rule S↦PSS P_S 0: Gaussian cache Gauss=(G,H,c)C_Gauss=(G,H,c) or histogram cache Hist=(B,G,H,c)C_Hist=(B,G,H,c) 1: Construct centered spline blocks B1,…,BdB_1,…,B_d and centered responses Y1,…,YdY_1,…,Y_d 2: B←[B1,…,Bd]B←[B_1,…,B_d] 3: G←B⊤BG← B B, H←B⊤YH← B Y 4: cj←Yj⊤Yjc_j← Y_j Y_j for all j∈Vj∈ V 5: if ℓ=Gaussian = Gaussian then 6: Store Gauss←(G,H,c)C_Gauss←(G,H,c) 7: else 8: Store Hist←(B,G,H,c)C_Hist←(B,G,H,c) 9: end if 10: LocalScore(j,S,ℓ,)(j,S, ,C) 11: I←I(S)I← I(S) 12: AS←GI,I+PSA_S← G_I,I+P_S, hj,S←HI,jh_j,S← H_I,j 13: Solve ASβ^j,S=hj,SA_S β_j,S=h_j,S 14: if ℓ=Gaussian = Gaussian then 15: RSSj,S←cj−2β^j,S⊤hj,S+β^j,S⊤GI,Iβ^j,SRSS_j,S← c_j-2 β_j,S h_j,S+ β_j,S G_I,I β_j,S 16: 17: return ℒjGauss(S)+penBIC(j,S)L^Gauss_j(S)+pen_BIC(j,S) 18: else 19: N^j,S←Yj−BSβ^j,S N_j,S← Y_j-B_S β_j,S 20: 21: return HistogramNLL(N^j,S)+penBIC(j,S)HistogramNLL( N_j,S)+pen_BIC(j,S) 22: end if A.6 Additional Metrics In this appendix, we present detailed results for the experiments in the main paper, including additional graph-recovery and runtime metrics. The controlled tables report directed edge precision, recall, and F1 by comparing the ordered edge sets of the learned graph and the ground-truth DAG. Skeleton metrics use the same precision, recall, and F1 definitions after replacing each directed graph by its undirected skeleton. Empty predicted graphs have precision 11 only when the true graph is also empty, and recall 11 only when both graphs are empty. We report structural Hamming distance (SHD) with a reversed edge counting as one operation: if M is the number of missing directed edges, E the number of extra directed edges, and R the number of reversed true edges, then SHD=M+E−RSHD=M+E-R. The normalized topological divergence is the fraction of true directed edges whose endpoints appear in the wrong order in the estimated topological order. Runtime is measured in seconds for each method run. The table column n reports the number of successful rows entering each mean, and failed, timed-out, or OOM runs are excluded from the corresponding means. Entries are reported as means with standard errors over these successful rows. The controlled feature-scaling results are shown in Tables 1–7, and the controlled sample-scaling results are shown in Tables 8–14. The corresponding heterogeneous scalability results are reported in Tables 15 and 16. The Causal Chamber results are shown in Table 41. Table 1: Controlled scalability feature-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_features=25). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.785 ± 0.023 0.993 ± 0.007 0.653 ± 0.033 0.761 ± 0.015 0.964 ± 0.019 0.632 ± 0.025 14.2 ± 0.9 0.068 ± 0.018 1.95 ± 0.06 25.0 ± 1.4 Spade-Gauss 5 0.785 ± 0.020 1.000 0.647 ± 0.027 0.785 ± 0.020 1.000 0.647 ± 0.027 13.4 ± 1.0 0.047 ± 0.015 0.16 ± 0.01 24.6 ± 1.0 Spade-Hist-NoCache 5 0.785 ± 0.023 0.993 ± 0.007 0.653 ± 0.033 0.761 ± 0.015 0.964 ± 0.019 0.632 ± 0.025 14.2 ± 0.9 0.068 ± 0.018 3.54 ± 0.22 25.0 ± 1.4 Spade-Gauss-NoCache 5 0.785 ± 0.020 1.000 0.647 ± 0.027 0.785 ± 0.020 1.000 0.647 ± 0.027 13.4 ± 1.0 0.047 ± 0.015 3.75 ± 0.15 24.6 ± 1.0 Topic 5 0.791 ± 0.024 0.864 ± 0.023 0.732 ± 0.028 0.729 ± 0.027 0.795 ± 0.024 0.674 ± 0.031 16.8 ± 1.6 0.174 ± 0.040 39.21 ± 1.81 32.2 ± 1.0 Cam 5 0.809 ± 0.017 0.698 ± 0.021 0.963 ± 0.016 0.791 ± 0.023 0.682 ± 0.023 0.942 ± 0.027 18.2 ± 2.0 0.021 ± 0.015 918.03 ± 3.79 52.6 ± 1.5 CauScale 5 0.804 ± 0.030 0.957 ± 0.030 0.695 ± 0.035 0.504 ± 0.062 0.597 ± 0.067 0.437 ± 0.058 22.6 ± 2.5 0.379 ± 0.076 10.00 ± 1.31 27.6 ± 1.2 SDCD 5 0.737 ± 0.028 0.956 ± 0.025 0.600 ± 0.028 0.568 ± 0.056 0.735 ± 0.066 0.463 ± 0.048 21.4 ± 2.3 0.332 ± 0.060 161.66 ± 4.10 23.8 ± 0.6 Globe 5 0.585 ± 0.022 0.870 ± 0.039 0.442 ± 0.019 0.536 ± 0.018 0.795 ± 0.023 0.405 ± 0.018 25.2 ± 1.0 0.305 ± 0.046 546.22 ± 28.03 19.4 ± 0.9 Das 5 0.782 ± 0.020 0.776 ± 0.029 0.795 ± 0.032 0.674 ± 0.031 0.669 ± 0.037 0.684 ± 0.036 21.0 ± 2.0 0.163 ± 0.023 128.98 ± 0.34 39.2 ± 2.3 Avici 5 0.478 ± 0.029 1.000 0.316 ± 0.025 0.454 ± 0.029 0.951 ± 0.020 0.300 ± 0.024 26.6 ± 0.9 0.363 ± 0.037 30.99 ± 1.30 12.0 ± 0.9 NOTEARS 5 0.772 ± 0.017 0.853 ± 0.016 0.705 ± 0.017 0.634 ± 0.031 0.700 ± 0.032 0.579 ± 0.030 20.6 ± 1.6 0.242 ± 0.028 516.07 ± 49.59 31.4 ± 0.4 DAGMA 5 0.638 ± 0.036 0.813 ± 0.040 0.532 ± 0.042 0.259 ± 0.022 0.329 ± 0.023 0.216 ± 0.023 34.6 ± 1.4 0.632 ± 0.024 2059.20 ± 67.92 25.0 ± 2.2 DiffAN-NoRes 5 0.881 ± 0.024 0.884 ± 0.024 0.879 ± 0.024 0.728 ± 0.047 0.730 ± 0.047 0.726 ± 0.048 14.8 ± 2.6 0.216 ± 0.042 175.15 ± 7.56 37.8 ± 0.5 SciNO 5 0.799 ± 0.022 0.753 ± 0.031 0.853 ± 0.013 0.480 ± 0.044 0.453 ± 0.045 0.511 ± 0.043 29.4 ± 2.7 0.458 ± 0.051 977.20 ± 24.87 43.2 ± 1.2 Table 2: Controlled scalability feature-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_features=50). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.796 ± 0.022 0.992 ± 0.008 0.667 ± 0.029 0.777 ± 0.032 0.966 ± 0.023 0.651 ± 0.036 26.6 ± 3.0 0.080 ± 0.025 3.24 ± 0.10 50.4 ± 2.0 Spade-Gauss 5 0.810 ± 0.024 0.996 ± 0.004 0.685 ± 0.033 0.807 ± 0.026 0.991 ± 0.009 0.683 ± 0.034 24.0 ± 2.7 0.021 ± 0.008 0.55 ± 0.01 51.6 ± 2.4 Spade-Hist-NoCache 5 0.797 ± 0.021 0.996 ± 0.004 0.667 ± 0.029 0.778 ± 0.026 0.971 ± 0.011 0.651 ± 0.033 26.4 ± 2.5 0.072 ± 0.015 10.42 ± 0.45 50.2 ± 2.1 Spade-Gauss-NoCache 5 0.810 ± 0.024 0.996 ± 0.004 0.685 ± 0.033 0.807 ± 0.026 0.991 ± 0.009 0.683 ± 0.034 24.0 ± 2.7 0.021 ± 0.008 12.52 ± 0.40 51.6 ± 2.4 Topic 5 0.771 ± 0.021 0.824 ± 0.018 0.725 ± 0.028 0.740 ± 0.017 0.792 ± 0.018 0.696 ± 0.023 34.4 ± 2.5 0.072 ± 0.018 162.32 ± 4.47 66.0 ± 2.1 Cam 5 0.682 ± 0.022 0.533 ± 0.025 0.949 ± 0.011 0.661 ± 0.025 0.517 ± 0.027 0.920 ± 0.019 69.4 ± 6.5 0.045 ± 0.013 14909.67 ± 175.55 134.6 ± 5.6 CauScale 5 0.880 ± 0.011 0.975 ± 0.006 0.803 ± 0.021 0.653 ± 0.037 0.722 ± 0.033 0.597 ± 0.040 31.8 ± 2.8 0.283 ± 0.031 9.12 ± 1.29 61.8 ± 1.9 SDCD 5 0.790 ± 0.019 0.974 ± 0.007 0.667 ± 0.028 0.679 ± 0.029 0.836 ± 0.023 0.573 ± 0.034 33.4 ± 2.2 0.301 ± 0.014 162.83 ± 2.86 51.4 ± 2.4 Globe 5 0.560 ± 0.016 0.838 ± 0.023 0.421 ± 0.016 0.532 ± 0.015 0.796 ± 0.025 0.400 ± 0.014 51.2 ± 1.6 0.341 ± 0.015 1680.44 ± 215.85 37.8 ± 1.5 Das 5 0.700 ± 0.025 0.687 ± 0.020 0.715 ± 0.033 0.566 ± 0.029 0.555 ± 0.024 0.579 ± 0.035 56.0 ± 3.4 0.213 ± 0.022 348.80 ± 12.17 78.0 ± 2.5 Avici 5 0.362 ± 0.007 1.000 0.221 ± 0.005 0.345 ± 0.016 0.950 ± 0.031 0.211 ± 0.011 59.2 ± 0.8 0.483 ± 0.037 53.45 ± 2.33 16.6 ± 0.4 NOTEARS 5 0.795 ± 0.019 0.857 ± 0.016 0.747 ± 0.037 0.660 ± 0.029 0.712 ± 0.037 0.619 ± 0.034 38.2 ± 2.7 0.264 ± 0.032 2394.08 ± 190.52 65.6 ± 3.9 DAGMA 5 0.700 ± 0.016 0.884 ± 0.013 0.581 ± 0.023 0.383 ± 0.031 0.484 ± 0.039 0.317 ± 0.026 57.0 ± 2.4 0.536 ± 0.024 6584.66 ± 819.73 49.4 ± 2.3 DiffAN-NoRes 5 0.863 ± 0.017 0.850 ± 0.019 0.880 ± 0.027 0.691 ± 0.020 0.680 ± 0.019 0.704 ± 0.028 34.0 ± 2.2 0.224 ± 0.025 95.77 ± 1.23 77.8 ± 3.0 SciNO 5 0.770 ± 0.023 0.712 ± 0.028 0.840 ± 0.024 0.419 ± 0.030 0.388 ± 0.032 0.456 ± 0.028 66.6 ± 5.0 0.493 ± 0.017 1022.13 ± 30.39 88.8 ± 3.1 Table 3: Controlled scalability feature-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_features=100). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.786 ± 0.011 0.988 ± 0.004 0.653 ± 0.016 0.746 ± 0.012 0.939 ± 0.017 0.620 ± 0.014 58.2 ± 1.9 0.140 ± 0.013 8.56 ± 0.28 99.2 ± 2.6 Spade-Gauss 5 0.783 ± 0.014 0.996 ± 0.003 0.645 ± 0.019 0.776 ± 0.012 0.988 ± 0.002 0.640 ± 0.017 54.4 ± 2.7 0.052 ± 0.011 1.90 ± 0.06 97.2 ± 2.7 Spade-Hist-NoCache 5 0.788 ± 0.013 0.990 ± 0.003 0.656 ± 0.019 0.755 ± 0.014 0.948 ± 0.012 0.628 ± 0.018 56.8 ± 2.6 0.132 ± 0.010 35.79 ± 1.24 99.4 ± 2.9 Spade-Gauss-NoCache 5 0.786 ± 0.010 0.996 ± 0.003 0.649 ± 0.013 0.776 ± 0.011 0.984 ± 0.005 0.641 ± 0.013 54.2 ± 2.2 0.065 ± 0.011 54.82 ± 1.61 97.8 ± 1.8 Topic 5 0.757 ± 0.010 0.811 ± 0.009 0.709 ± 0.013 0.728 ± 0.010 0.780 ± 0.008 0.683 ± 0.012 72.4 ± 2.5 0.088 ± 0.021 630.21 ± 16.82 131.2 ± 1.9 CauScale 5 0.860 ± 0.007 0.949 ± 0.007 0.787 ± 0.014 0.657 ± 0.018 0.726 ± 0.021 0.601 ± 0.018 66.2 ± 2.7 0.253 ± 0.013 6.99 ± 0.01 124.4 ± 2.9 SDCD 5 0.727 ± 0.009 0.926 ± 0.010 0.599 ± 0.012 0.619 ± 0.011 0.788 ± 0.018 0.509 ± 0.010 80.8 ± 1.8 0.331 ± 0.024 170.91 ± 6.06 97.0 ± 2.0 Globe 5 0.452 ± 0.052 0.841 ± 0.032 0.320 ± 0.046 0.425 ± 0.044 0.797 ± 0.040 0.300 ± 0.040 115.2 ± 4.0 0.401 ± 0.014 5442.12 ± 659.79 58.2 ± 9.0 Das 5 0.663 ± 0.020 0.663 ± 0.021 0.665 ± 0.025 0.507 ± 0.019 0.507 ± 0.018 0.509 ± 0.024 124.6 ± 5.2 0.223 ± 0.011 1050.34 ± 28.86 150.8 ± 5.1 Avici 5 0.198 ± 0.016 0.794 ± 0.127 0.115 ± 0.008 0.173 ± 0.025 0.712 ± 0.154 0.100 ± 0.013 142.2 ± 6.4 0.508 ± 0.008 82.76 ± 1.22 24.4 ± 4.4 NOTEARS 5 0.738 ± 0.018 0.842 ± 0.018 0.659 ± 0.023 0.568 ± 0.026 0.649 ± 0.032 0.507 ± 0.025 92.6 ± 5.4 0.355 ± 0.033 10407.45 ± 760.20 117.4 ± 3.9 DAGMA 5 0.643 ± 0.017 0.863 ± 0.012 0.515 ± 0.023 0.321 ± 0.034 0.427 ± 0.035 0.259 ± 0.031 123.6 ± 3.2 0.543 ± 0.029 13490.10 ± 884.38 89.6 ± 4.8 DiffAN-NoRes 5 0.169 ± 0.002 0.093 ± 0.001 0.916 ± 0.018 0.125 ± 0.003 0.069 ± 0.002 0.679 ± 0.024 1387.0 ± 34.9 0.272 ± 0.025 53158.73 ± 1684.60 1476.2 ± 39.9 SciNO 5 0.155 ± 0.001 0.085 ± 0.001 0.867 ± 0.012 0.090 ± 0.001 0.049 ± 0.001 0.500 ± 0.011 1470.0 ± 16.9 0.452 ± 0.016 54972.90 ± 1683.25 1525.0 ± 19.9 Table 4: Controlled scalability feature-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_features=200). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.776 ± 0.011 0.988 ± 0.004 0.639 ± 0.014 0.742 ± 0.015 0.946 ± 0.012 0.611 ± 0.016 118.8 ± 5.5 0.151 ± 0.010 27.85 ± 0.38 193.8 ± 3.7 Spade-Gauss 5 0.787 ± 0.010 1.000 0.649 ± 0.014 0.779 ± 0.009 0.990 ± 0.006 0.642 ± 0.013 107.4 ± 3.8 0.089 ± 0.015 7.25 ± 0.08 194.6 ± 4.2 Spade-Hist-NoCache 5 0.775 ± 0.010 0.989 ± 0.004 0.637 ± 0.013 0.740 ± 0.013 0.944 ± 0.012 0.609 ± 0.015 119.6 ± 4.9 0.152 ± 0.008 136.26 ± 1.99 193.4 ± 3.8 Spade-Gauss-NoCache 5 0.787 ± 0.010 1.000 0.649 ± 0.014 0.779 ± 0.009 0.990 ± 0.006 0.642 ± 0.013 107.4 ± 3.8 0.089 ± 0.015 297.55 ± 4.22 194.6 ± 4.2 Topic 5 0.728 ± 0.007 0.776 ± 0.009 0.685 ± 0.007 0.699 ± 0.008 0.746 ± 0.011 0.658 ± 0.006 161.8 ± 4.5 0.105 ± 0.012 2615.19 ± 26.09 264.6 ± 2.7 CauScale 5 0.841 ± 0.013 0.951 ± 0.013 0.755 ± 0.016 0.640 ± 0.018 0.723 ± 0.015 0.575 ± 0.019 139.4 ± 7.5 0.295 ± 0.019 10.46 ± 1.61 238.2 ± 4.9 SDCD 5 0.738 ± 0.011 0.922 ± 0.005 0.616 ± 0.013 0.592 ± 0.019 0.739 ± 0.016 0.494 ± 0.019 167.4 ± 6.3 0.336 ± 0.021 249.18 ± 70.08 200.4 ± 3.3 Globe 5 0.513 ± 0.011 0.837 ± 0.021 0.371 ± 0.012 0.473 ± 0.009 0.773 ± 0.027 0.341 ± 0.007 219.6 ± 3.9 0.389 ± 0.006 19924.54 ± 784.13 133.2 ± 5.5 Das 5 0.567 ± 0.016 0.599 ± 0.022 0.539 ± 0.014 0.415 ± 0.016 0.439 ± 0.022 0.395 ± 0.012 290.6 ± 11.4 0.258 ± 0.015 3592.97 ± 252.14 270.8 ± 8.5 Avici 5 0.087 ± 0.017 1.000 0.046 ± 0.010 0.086 ± 0.017 0.986 ± 0.014 0.045 ± 0.010 286.4 ± 2.9 0.503 ± 0.010 160.07 ± 2.35 13.8 ± 2.9 NOTEARS 5 0.714 ± 0.008 0.862 ± 0.015 0.610 ± 0.008 0.500 ± 0.021 0.604 ± 0.029 0.427 ± 0.017 201.6 ± 8.1 0.421 ± 0.018 58971.43 ± 2828.33 212.6 ± 4.2 DAGMA 5 0.627 ± 0.014 0.942 ± 0.008 0.471 ± 0.016 0.386 ± 0.011 0.581 ± 0.018 0.290 ± 0.010 221.8 ± 3.2 0.517 ± 0.015 49687.30 ± 2114.38 150.2 ± 5.8 Table 5: Controlled scalability feature-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_features=400). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.782 ± 0.008 0.989 ± 0.001 0.646 ± 0.011 0.739 ± 0.007 0.936 ± 0.002 0.611 ± 0.010 237.4 ± 5.5 0.166 ± 0.010 105.26 ± 1.40 392.0 ± 6.7 Spade-Gauss 5 0.793 ± 0.006 0.997 ± 0.001 0.658 ± 0.008 0.787 ± 0.005 0.989 ± 0.001 0.653 ± 0.008 209.2 ± 4.4 0.073 ± 0.006 26.62 ± 0.29 396.0 ± 4.8 Spade-Hist-NoCache 5 0.780 ± 0.007 0.990 ± 0.001 0.644 ± 0.009 0.738 ± 0.007 0.936 ± 0.003 0.609 ± 0.009 238.4 ± 5.6 0.161 ± 0.009 533.05 ± 12.78 390.2 ± 5.6 Spade-Gauss-NoCache 5 0.793 ± 0.006 0.997 ± 0.001 0.658 ± 0.008 0.787 ± 0.005 0.989 ± 0.001 0.653 ± 0.008 209.2 ± 4.4 0.073 ± 0.006 1838.97 ± 19.86 396.0 ± 4.8 Topic 5 0.728 ± 0.008 0.769 ± 0.009 0.691 ± 0.010 0.714 ± 0.007 0.755 ± 0.008 0.678 ± 0.009 317.8 ± 7.7 0.083 ± 0.005 10821.17 ± 248.61 539.0 ± 8.4 CauScale 5 0.840 ± 0.014 0.986 ± 0.002 0.732 ± 0.020 0.639 ± 0.014 0.751 ± 0.013 0.557 ± 0.016 271.8 ± 9.9 0.302 ± 0.015 10.23 ± 1.01 445.4 ± 11.7 SDCD 5 0.730 ± 0.008 0.843 ± 0.010 0.645 ± 0.009 0.602 ± 0.013 0.694 ± 0.017 0.531 ± 0.010 353.8 ± 11.0 0.284 ± 0.007 137.47 ± 2.52 459.2 ± 6.1 Globe 5 0.436 ± 0.009 0.853 ± 0.015 0.294 ± 0.009 0.395 ± 0.007 0.772 ± 0.015 0.266 ± 0.008 471.6 ± 2.0 0.453 ± 0.013 63710.79 ± 2201.24 207.2 ± 9.5 Das 5 0.546 ± 0.007 0.578 ± 0.011 0.517 ± 0.006 0.398 ± 0.009 0.421 ± 0.008 0.377 ± 0.011 600.8 ± 7.8 0.272 ± 0.017 12492.48 ± 803.63 537.4 ± 8.4 Avici 5 0.024 ± 0.004 1.000 0.012 ± 0.002 0.024 ± 0.004 1.000 0.012 ± 0.002 592.6 ± 1.3 0.510 ± 0.013 360.44 ± 21.74 7.4 ± 1.3 Table 6: Controlled scalability feature-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_features=800). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.760 ± 0.004 0.989 ± 0.002 0.618 ± 0.006 0.718 ± 0.006 0.934 ± 0.004 0.584 ± 0.007 508.0 ± 8.1 0.174 ± 0.009 386.91 ± 5.20 749.8 ± 7.1 Spade-Gauss 5 0.770 ± 0.005 0.998 ± 0.001 0.626 ± 0.007 0.761 ± 0.004 0.988 ± 0.002 0.620 ± 0.006 457.8 ± 7.2 0.089 ± 0.004 99.31 ± 1.38 753.0 ± 8.9 Spade-Hist-NoCache 5 0.761 ± 0.003 0.990 ± 0.002 0.618 ± 0.005 0.721 ± 0.004 0.938 ± 0.005 0.586 ± 0.005 504.6 ± 5.3 0.169 ± 0.007 2089.16 ± 42.92 749.8 ± 6.9 Spade-Gauss-NoCache 5 0.770 ± 0.005 0.998 ± 0.001 0.626 ± 0.007 0.761 ± 0.004 0.988 ± 0.002 0.620 ± 0.006 457.8 ± 7.2 0.089 ± 0.004 12680.18 ± 136.39 753.0 ± 8.9 Topic 5 0.701 ± 0.006 0.748 ± 0.007 0.659 ± 0.005 0.687 ± 0.005 0.733 ± 0.006 0.646 ± 0.005 691.4 ± 12.6 0.100 ± 0.005 45302.69 ± 534.96 1056.8 ± 3.4 CauScale 5 0.696 ± 0.009 0.875 ± 0.035 0.581 ± 0.013 0.479 ± 0.008 0.602 ± 0.026 0.400 ± 0.009 827.4 ± 30.2 0.391 ± 0.008 11.75 ± 0.33 804.6 ± 47.9 SDCD 5 0.700 ± 0.004 0.816 ± 0.009 0.614 ± 0.008 0.589 ± 0.006 0.687 ± 0.005 0.517 ± 0.010 746.8 ± 7.7 0.281 ± 0.011 137.28 ± 2.06 903.6 ± 20.3 Das 5 0.482 ± 0.011 0.592 ± 0.013 0.406 ± 0.010 0.342 ± 0.015 0.420 ± 0.018 0.288 ± 0.013 1189.4 ± 24.9 0.268 ± 0.010 47316.79 ± 3686.07 823.2 ± 5.0 Avici 5 0.006 ± 0.001 0.971 ± 0.029 0.003 ± 0.001 0.006 ± 0.001 0.931 ± 0.043 0.003 ± 0.001 1196.8 ± 0.6 0.491 ± 0.006 953.19 ± 124.35 3.8 ± 1.0 Table 7: Controlled scalability feature-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_features=1600). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.762 ± 0.004 0.990 ± 0.002 0.620 ± 0.005 0.722 ± 0.007 0.939 ± 0.004 0.587 ± 0.007 1005.2 ± 17.1 0.174 ± 0.009 1587.04 ± 24.79 1501.8 ± 13.3 Spade-Gauss 5 0.769 ± 0.004 0.998 ± 0.001 0.625 ± 0.005 0.760 ± 0.005 0.986 ± 0.002 0.618 ± 0.006 919.6 ± 13.7 0.094 ± 0.005 427.20 ± 7.73 1503.6 ± 11.0 Spade-Hist-NoCache 5 0.760 ± 0.003 0.989 ± 0.002 0.618 ± 0.005 0.721 ± 0.006 0.938 ± 0.004 0.586 ± 0.007 1011.0 ± 15.7 0.176 ± 0.007 8415.97 ± 105.52 1498.6 ± 11.9 CauScale 5 0.611 ± 0.016 0.684 ± 0.049 0.560 ± 0.010 0.399 ± 0.014 0.447 ± 0.036 0.365 ± 0.006 2193.4 ± 149.0 0.438 ± 0.005 25.79 ± 0.98 2012.6 ± 170.4 SDCD 5 0.691 ± 0.003 0.827 ± 0.002 0.594 ± 0.005 0.588 ± 0.005 0.704 ± 0.004 0.505 ± 0.006 1485.8 ± 12.4 0.282 ± 0.008 231.34 ± 3.21 1721.6 ± 12.7 Avici 2 0.000 ± 0.000 0.500 ± 0.500 0.000 ± 0.000 0.000 ± 0.000 0.500 ± 0.500 0.000 ± 0.000 2399.5 ± 0.5 0.492 ± 0.007 2016.08 ± 34.10 0.5 ± 0.5 Table 8: Controlled scalability sample-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_samples=2500). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.786 ± 0.011 0.988 ± 0.004 0.653 ± 0.016 0.746 ± 0.012 0.939 ± 0.017 0.620 ± 0.014 58.2 ± 1.9 0.140 ± 0.013 8.56 ± 0.28 99.2 ± 2.6 Spade-Gauss 5 0.783 ± 0.014 0.996 ± 0.003 0.645 ± 0.019 0.776 ± 0.012 0.988 ± 0.002 0.640 ± 0.017 54.4 ± 2.7 0.052 ± 0.011 1.90 ± 0.06 97.2 ± 2.7 Spade-Hist-NoCache 5 0.788 ± 0.013 0.990 ± 0.003 0.656 ± 0.019 0.755 ± 0.014 0.948 ± 0.012 0.628 ± 0.018 56.8 ± 2.6 0.132 ± 0.010 35.79 ± 1.24 99.4 ± 2.9 Spade-Gauss-NoCache 5 0.786 ± 0.010 0.996 ± 0.003 0.649 ± 0.013 0.776 ± 0.011 0.984 ± 0.005 0.641 ± 0.013 54.2 ± 2.2 0.065 ± 0.011 54.82 ± 1.61 97.8 ± 1.8 Topic 5 0.757 ± 0.010 0.811 ± 0.009 0.709 ± 0.013 0.728 ± 0.010 0.780 ± 0.008 0.683 ± 0.012 72.4 ± 2.5 0.088 ± 0.021 630.21 ± 16.82 131.2 ± 1.9 CauScale 5 0.860 ± 0.007 0.949 ± 0.007 0.787 ± 0.014 0.657 ± 0.018 0.726 ± 0.021 0.601 ± 0.018 66.2 ± 2.7 0.253 ± 0.013 6.99 ± 0.01 124.4 ± 2.9 SDCD 5 0.727 ± 0.009 0.926 ± 0.010 0.599 ± 0.012 0.619 ± 0.011 0.788 ± 0.018 0.509 ± 0.010 80.8 ± 1.8 0.331 ± 0.024 170.91 ± 6.06 97.0 ± 2.0 Globe 5 0.452 ± 0.052 0.841 ± 0.032 0.320 ± 0.046 0.425 ± 0.044 0.797 ± 0.040 0.300 ± 0.040 115.2 ± 4.0 0.401 ± 0.014 5442.12 ± 659.79 58.2 ± 9.0 Das 5 0.663 ± 0.020 0.663 ± 0.021 0.665 ± 0.025 0.507 ± 0.019 0.507 ± 0.018 0.509 ± 0.024 124.6 ± 5.2 0.223 ± 0.011 1050.34 ± 28.86 150.8 ± 5.1 Avici 5 0.198 ± 0.016 0.794 ± 0.127 0.115 ± 0.008 0.173 ± 0.025 0.712 ± 0.154 0.100 ± 0.013 142.2 ± 6.4 0.508 ± 0.008 82.76 ± 1.22 24.4 ± 4.4 NOTEARS 5 0.738 ± 0.018 0.842 ± 0.018 0.659 ± 0.023 0.568 ± 0.026 0.649 ± 0.032 0.507 ± 0.025 92.6 ± 5.4 0.355 ± 0.033 10407.45 ± 760.20 117.4 ± 3.9 DAGMA 5 0.643 ± 0.017 0.863 ± 0.012 0.515 ± 0.023 0.321 ± 0.034 0.427 ± 0.035 0.259 ± 0.031 123.6 ± 3.2 0.543 ± 0.029 13490.10 ± 884.38 89.6 ± 4.8 DiffAN-NoRes 5 0.169 ± 0.002 0.093 ± 0.001 0.916 ± 0.018 0.125 ± 0.003 0.069 ± 0.002 0.679 ± 0.024 1387.0 ± 34.9 0.272 ± 0.025 53158.73 ± 1684.60 1476.2 ± 39.9 SciNO 5 0.155 ± 0.001 0.085 ± 0.001 0.867 ± 0.012 0.090 ± 0.001 0.049 ± 0.001 0.500 ± 0.011 1470.0 ± 16.9 0.452 ± 0.016 54972.90 ± 1683.25 1525.0 ± 19.9 Table 9: Controlled scalability sample-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_samples=5000). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.862 ± 0.010 0.993 ± 0.003 0.761 ± 0.015 0.828 ± 0.013 0.955 ± 0.010 0.732 ± 0.016 41.0 ± 2.6 0.088 ± 0.010 11.45 ± 0.17 115.0 ± 2.1 Spade-Gauss 5 0.858 ± 0.008 0.996 ± 0.002 0.753 ± 0.012 0.846 ± 0.008 0.982 ± 0.004 0.743 ± 0.011 39.0 ± 1.8 0.057 ± 0.013 2.67 ± 0.09 113.4 ± 1.7 Spade-Hist-NoCache 5 0.859 ± 0.009 0.993 ± 0.003 0.757 ± 0.013 0.827 ± 0.013 0.956 ± 0.011 0.729 ± 0.016 41.4 ± 2.6 0.088 ± 0.010 63.87 ± 2.41 114.4 ± 1.9 Spade-Gauss-NoCache 5 0.858 ± 0.008 0.996 ± 0.002 0.753 ± 0.012 0.846 ± 0.008 0.982 ± 0.004 0.743 ± 0.011 39.0 ± 1.8 0.057 ± 0.013 81.01 ± 2.65 113.4 ± 1.7 Topic 5 0.811 ± 0.010 0.816 ± 0.009 0.807 ± 0.014 0.783 ± 0.014 0.788 ± 0.014 0.779 ± 0.016 60.4 ± 3.5 0.071 ± 0.009 1199.87 ± 23.24 148.2 ± 2.1 CauScale 5 0.831 ± 0.007 0.874 ± 0.007 0.792 ± 0.015 0.586 ± 0.021 0.616 ± 0.020 0.559 ± 0.023 83.4 ± 2.9 0.332 ± 0.022 7.36 ± 0.10 136.0 ± 3.3 SDCD 5 0.734 ± 0.014 0.944 ± 0.009 0.601 ± 0.017 0.652 ± 0.015 0.838 ± 0.023 0.533 ± 0.013 75.4 ± 2.7 0.305 ± 0.026 323.41 ± 7.16 95.6 ± 2.5 Globe 5 0.604 ± 0.019 0.906 ± 0.016 0.456 ± 0.023 0.576 ± 0.023 0.862 ± 0.021 0.435 ± 0.025 92.2 ± 3.0 0.308 ± 0.020 10869.50 ± 859.99 75.8 ± 4.7 Das 5 0.667 ± 0.011 0.629 ± 0.010 0.709 ± 0.014 0.525 ± 0.016 0.495 ± 0.016 0.559 ± 0.018 129.0 ± 4.2 0.200 ± 0.013 4660.60 ± 12.87 169.2 ± 2.2 Avici 5 0.187 ± 0.024 0.822 ± 0.143 0.107 ± 0.012 0.161 ± 0.029 0.716 ± 0.155 0.092 ± 0.016 143.2 ± 7.8 0.491 ± 0.009 297.51 ± 14.40 23.0 ± 5.2 NOTEARS 5 0.768 ± 0.013 0.860 ± 0.012 0.695 ± 0.018 0.627 ± 0.018 0.703 ± 0.026 0.567 ± 0.016 82.0 ± 3.6 0.289 ± 0.026 18090.72 ± 1431.21 121.2 ± 3.3 DAGMA 5 0.668 ± 0.017 0.860 ± 0.006 0.548 ± 0.023 0.362 ± 0.020 0.465 ± 0.014 0.297 ± 0.021 118.8 ± 2.5 0.527 ± 0.013 36365.67 ± 2073.24 95.6 ± 4.3 Table 10: Controlled scalability sample-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_samples=10000). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.910 ± 0.007 0.992 ± 0.002 0.840 ± 0.013 0.881 ± 0.008 0.961 ± 0.009 0.813 ± 0.012 29.0 ± 1.7 0.069 ± 0.006 19.30 ± 0.39 127.0 ± 2.3 Spade-Gauss 5 0.914 ± 0.009 0.995 ± 0.003 0.845 ± 0.017 0.892 ± 0.014 0.972 ± 0.010 0.825 ± 0.020 26.8 ± 2.8 0.049 ± 0.009 4.21 ± 0.50 127.4 ± 2.7 Spade-Hist-NoCache 5 0.908 ± 0.008 0.991 ± 0.003 0.839 ± 0.014 0.876 ± 0.011 0.956 ± 0.011 0.809 ± 0.014 29.8 ± 2.2 0.072 ± 0.004 117.39 ± 1.72 127.0 ± 2.3 Spade-Gauss-NoCache 5 0.914 ± 0.009 0.995 ± 0.003 0.845 ± 0.017 0.892 ± 0.014 0.972 ± 0.010 0.825 ± 0.020 26.8 ± 2.8 0.049 ± 0.009 130.96 ± 3.57 127.4 ± 2.7 Topic 5 0.847 ± 0.008 0.810 ± 0.007 0.888 ± 0.013 0.819 ± 0.010 0.784 ± 0.010 0.859 ± 0.011 52.4 ± 2.5 0.064 ± 0.011 2769.23 ± 108.37 164.4 ± 2.0 CauScale 5 0.812 ± 0.011 0.827 ± 0.015 0.800 ± 0.018 0.567 ± 0.016 0.577 ± 0.015 0.559 ± 0.020 91.6 ± 3.2 0.319 ± 0.022 11.78 ± 1.29 145.4 ± 4.5 SDCD 5 0.737 ± 0.011 0.940 ± 0.007 0.607 ± 0.014 0.656 ± 0.010 0.838 ± 0.015 0.540 ± 0.010 74.8 ± 1.7 0.279 ± 0.015 689.38 ± 11.19 96.8 ± 2.5 Globe 5 0.797 ± 0.017 0.976 ± 0.007 0.676 ± 0.026 0.769 ± 0.020 0.941 ± 0.009 0.652 ± 0.028 54.8 ± 3.7 0.165 ± 0.013 35421.71 ± 3940.90 104.0 ± 4.5 Das 5 0.676 ± 0.017 0.611 ± 0.014 0.759 ± 0.027 0.546 ± 0.018 0.495 ± 0.019 0.612 ± 0.019 130.6 ± 5.7 0.191 ± 0.018 35257.70 ± 436.56 186.2 ± 5.7 Avici 3 0.157 ± 0.019 0.716 ± 0.284 0.098 ± 0.002 0.148 ± 0.023 0.686 ± 0.279 0.091 ± 0.004 165.0 ± 29.0 0.480 ± 0.007 1300.88 ± 123.24 43.3 ± 28.8 NOTEARS 5 0.772 ± 0.019 0.869 ± 0.014 0.696 ± 0.025 0.645 ± 0.024 0.726 ± 0.027 0.581 ± 0.025 78.6 ± 5.2 0.284 ± 0.026 26463.03 ± 1908.11 120.2 ± 4.0 DAGMA 5 0.690 ± 0.014 0.878 ± 0.016 0.571 ± 0.022 0.415 ± 0.022 0.528 ± 0.023 0.344 ± 0.023 110.6 ± 2.7 0.469 ± 0.029 80096.65 ± 2215.01 97.8 ± 5.1 Table 11: Controlled scalability sample-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_samples=20000). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.935 ± 0.009 0.986 ± 0.006 0.889 ± 0.017 0.897 ± 0.008 0.946 ± 0.010 0.853 ± 0.015 24.0 ± 2.1 0.071 ± 0.008 34.66 ± 0.67 135.4 ± 3.1 Spade-Gauss 5 0.942 ± 0.009 0.990 ± 0.005 0.900 ± 0.016 0.916 ± 0.013 0.962 ± 0.010 0.875 ± 0.018 20.2 ± 2.8 0.051 ± 0.010 5.66 ± 0.15 136.4 ± 2.7 Spade-Hist-NoCache 5 0.934 ± 0.009 0.986 ± 0.006 0.888 ± 0.017 0.896 ± 0.008 0.946 ± 0.010 0.852 ± 0.015 24.2 ± 2.1 0.072 ± 0.009 278.99 ± 7.07 135.2 ± 3.0 Spade-Gauss-NoCache 5 0.942 ± 0.009 0.990 ± 0.005 0.900 ± 0.016 0.916 ± 0.013 0.962 ± 0.010 0.875 ± 0.018 20.2 ± 2.8 0.051 ± 0.010 259.90 ± 8.44 136.4 ± 2.7 Topic 5 0.845 ± 0.009 0.778 ± 0.009 0.927 ± 0.018 0.814 ± 0.012 0.749 ± 0.015 0.892 ± 0.016 56.0 ± 3.4 0.059 ± 0.008 6428.36 ± 432.32 178.8 ± 4.1 SDCD 5 0.749 ± 0.015 0.942 ± 0.013 0.623 ± 0.019 0.663 ± 0.013 0.833 ± 0.014 0.551 ± 0.016 73.2 ± 2.8 0.288 ± 0.021 1175.25 ± 37.55 99.2 ± 3.1 NOTEARS 3 0.764 ± 0.020 0.881 ± 0.015 0.676 ± 0.026 0.649 ± 0.027 0.747 ± 0.023 0.573 ± 0.031 77.7 ± 5.5 0.278 ± 0.049 34282.95 ± 1981.63 115.0 ± 3.8 Table 12: Controlled scalability sample-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_samples=40000). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.950 ± 0.008 0.977 ± 0.007 0.927 ± 0.017 0.915 ± 0.007 0.941 ± 0.012 0.892 ± 0.013 19.6 ± 1.9 0.067 ± 0.009 64.16 ± 1.68 142.4 ± 3.2 Spade-Gauss 5 0.952 ± 0.009 0.978 ± 0.006 0.929 ± 0.018 0.924 ± 0.013 0.949 ± 0.016 0.901 ± 0.018 18.0 ± 2.9 0.052 ± 0.013 8.04 ± 0.15 142.6 ± 3.3 Spade-Hist-NoCache 5 0.950 ± 0.008 0.977 ± 0.007 0.925 ± 0.017 0.913 ± 0.007 0.939 ± 0.011 0.889 ± 0.014 20.0 ± 2.0 0.068 ± 0.010 572.68 ± 14.92 142.2 ± 3.2 Spade-Gauss-NoCache 5 0.952 ± 0.009 0.978 ± 0.006 0.929 ± 0.018 0.924 ± 0.013 0.949 ± 0.016 0.901 ± 0.018 18.0 ± 2.9 0.052 ± 0.013 546.27 ± 26.11 142.6 ± 3.3 Topic 5 0.845 ± 0.012 0.763 ± 0.012 0.947 ± 0.016 0.814 ± 0.015 0.735 ± 0.016 0.912 ± 0.015 57.4 ± 4.6 0.057 ± 0.010 15747.31 ± 973.76 186.2 ± 3.0 SDCD 5 0.751 ± 0.012 0.943 ± 0.009 0.625 ± 0.019 0.660 ± 0.008 0.829 ± 0.017 0.549 ± 0.011 73.4 ± 1.8 0.295 ± 0.023 2164.41 ± 13.30 99.6 ± 3.5 NOTEARS 3 0.785 ± 0.026 0.887 ± 0.012 0.704 ± 0.036 0.663 ± 0.038 0.749 ± 0.026 0.596 ± 0.044 74.0 ± 7.1 0.253 ± 0.014 66641.65 ± 4163.24 119.0 ± 5.0 Table 13: Controlled scalability sample-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_samples=80000). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.958 ± 0.007 0.973 ± 0.010 0.945 ± 0.018 0.915 ± 0.004 0.930 ± 0.013 0.903 ± 0.013 18.6 ± 1.3 0.067 ± 0.008 148.94 ± 6.08 145.8 ± 3.9 Spade-Gauss 5 0.958 ± 0.008 0.967 ± 0.009 0.951 ± 0.015 0.926 ± 0.014 0.935 ± 0.019 0.919 ± 0.015 17.2 ± 2.9 0.048 ± 0.010 16.02 ± 2.78 147.6 ± 3.1 Spade-Hist-NoCache 5 0.958 ± 0.007 0.973 ± 0.010 0.945 ± 0.018 0.915 ± 0.004 0.930 ± 0.013 0.903 ± 0.013 18.6 ± 1.3 0.067 ± 0.008 1353.85 ± 140.56 145.8 ± 3.9 Spade-Gauss-NoCache 5 0.958 ± 0.008 0.967 ± 0.009 0.951 ± 0.015 0.926 ± 0.014 0.935 ± 0.019 0.919 ± 0.015 17.2 ± 2.9 0.048 ± 0.010 1328.62 ± 68.57 147.6 ± 3.1 Topic 5 0.825 ± 0.014 0.723 ± 0.016 0.963 ± 0.016 0.797 ± 0.016 0.698 ± 0.019 0.929 ± 0.014 66.2 ± 5.7 0.049 ± 0.007 43376.76 ± 2382.63 200.0 ± 4.5 SDCD 5 0.751 ± 0.014 0.948 ± 0.009 0.623 ± 0.019 0.669 ± 0.014 0.844 ± 0.015 0.555 ± 0.017 72.0 ± 2.8 0.283 ± 0.025 4163.32 ± 67.10 98.6 ± 3.2 Table 14: Controlled scalability sample-size experiment. Graph metrics are computed from stored DAG outputs where available; plot-only aggregate rows retain only reported metrics (n_samples=160000). method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges Spade-Hist 5 0.958 ± 0.007 0.956 ± 0.010 0.961 ± 0.019 0.912 ± 0.005 0.910 ± 0.015 0.915 ± 0.012 19.6 ± 1.5 0.067 ± 0.006 575.00 ± 42.56 151.0 ± 4.2 Spade-Gauss 5 0.964 ± 0.007 0.958 ± 0.009 0.971 ± 0.015 0.925 ± 0.013 0.920 ± 0.019 0.931 ± 0.013 16.8 ± 2.8 0.049 ± 0.009 25.72 ± 0.86 152.0 ± 3.2 Spade-Hist-NoCache 5 0.959 ± 0.008 0.956 ± 0.010 0.963 ± 0.019 0.911 ± 0.005 0.909 ± 0.015 0.915 ± 0.012 19.6 ± 1.5 0.067 ± 0.006 3639.34 ± 267.12 151.2 ± 4.2 Spade-Gauss-NoCache 5 0.964 ± 0.007 0.958 ± 0.009 0.971 ± 0.015 0.925 ± 0.013 0.920 ± 0.019 0.931 ± 0.013 16.8 ± 2.8 0.049 ± 0.009 3274.59 ± 236.83 152.0 ± 3.2 SDCD 5 0.759 ± 0.014 0.950 ± 0.009 0.632 ± 0.017 0.679 ± 0.015 0.850 ± 0.014 0.565 ± 0.016 70.2 ± 3.0 0.280 ± 0.016 8078.58 ± 132.57 99.8 ± 2.5 Table 15: Heterogeneous scalability feature-size experiment. Entries are means with standard errors over successful seeds. n_features method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges 25 Spade-Hist 5 0.775 ± 0.034 0.912 ± 0.036 0.676 ± 0.035 0.746 ± 0.052 0.876 ± 0.057 0.651 ± 0.051 14.8 ± 3.7 0.103 ± 0.057 0.97 ± 0.06 25.0 ± 2.1 25 Spade-Gauss 5 0.787 ± 0.028 0.860 ± 0.025 0.731 ± 0.042 0.658 ± 0.033 0.720 ± 0.039 0.611 ± 0.041 17.8 ± 3.1 0.161 ± 0.019 0.23 ± 0.03 28.6 ± 2.5 25 Topic 5 0.732 ± 0.018 0.705 ± 0.027 0.771 ± 0.043 0.559 ± 0.029 0.540 ± 0.041 0.587 ± 0.033 25.4 ± 3.3 0.224 ± 0.013 54.38 ± 8.34 36.8 ± 3.0 25 CauScale 5 0.733 ± 0.049 0.796 ± 0.041 0.689 ± 0.066 0.475 ± 0.060 0.512 ± 0.058 0.450 ± 0.065 25.4 ± 4.6 0.359 ± 0.029 5.99 ± 0.13 29.0 ± 3.0 25 SDCD 5 0.693 ± 0.046 0.766 ± 0.048 0.646 ± 0.064 0.479 ± 0.078 0.528 ± 0.086 0.447 ± 0.078 25.4 ± 4.1 0.355 ± 0.036 159.94 ± 2.69 29.2 ± 4.3 25 Das 5 0.601 ± 0.039 0.528 ± 0.043 0.716 ± 0.055 0.456 ± 0.029 0.397 ± 0.020 0.551 ± 0.058 39.0 ± 6.4 0.270 ± 0.045 133.03 ± 1.28 47.2 ± 7.3 50 Spade-Hist 5 0.857 ± 0.015 0.958 ± 0.022 0.775 ± 0.011 0.837 ± 0.022 0.937 ± 0.030 0.757 ± 0.017 19.8 ± 2.3 0.102 ± 0.020 2.64 ± 0.20 58.0 ± 3.3 50 Spade-Gauss 5 0.810 ± 0.020 0.859 ± 0.022 0.767 ± 0.020 0.680 ± 0.030 0.721 ± 0.032 0.643 ± 0.029 35.4 ± 6.0 0.213 ± 0.012 0.94 ± 0.13 64.2 ± 4.4 50 Topic 5 0.743 ± 0.015 0.693 ± 0.017 0.801 ± 0.023 0.611 ± 0.019 0.570 ± 0.017 0.660 ± 0.027 50.6 ± 5.8 0.217 ± 0.018 258.72 ± 29.33 83.0 ± 5.3 50 CauScale 5 0.779 ± 0.034 0.771 ± 0.048 0.790 ± 0.025 0.569 ± 0.057 0.566 ± 0.067 0.574 ± 0.049 47.2 ± 6.3 0.286 ± 0.038 5.90 ± 0.03 73.8 ± 3.6 50 SDCD 5 0.694 ± 0.006 0.698 ± 0.017 0.691 ± 0.008 0.456 ± 0.039 0.461 ± 0.046 0.452 ± 0.033 61.0 ± 5.5 0.379 ± 0.024 156.34 ± 3.42 71.2 ± 4.8 50 Das 5 0.560 ± 0.024 0.473 ± 0.032 0.695 ± 0.016 0.352 ± 0.042 0.299 ± 0.041 0.434 ± 0.043 97.6 ± 10.0 0.370 ± 0.036 321.60 ± 5.58 106.8 ± 8.8 100 Spade-Hist 5 0.809 ± 0.006 0.938 ± 0.012 0.712 ± 0.004 0.784 ± 0.016 0.910 ± 0.023 0.690 ± 0.012 53.6 ± 2.0 0.125 ± 0.019 10.06 ± 0.69 114.4 ± 3.8 100 Spade-Gauss 5 0.808 ± 0.014 0.893 ± 0.023 0.737 ± 0.009 0.689 ± 0.027 0.763 ± 0.036 0.629 ± 0.021 69.0 ± 5.4 0.172 ± 0.017 3.66 ± 0.31 124.4 ± 3.3 100 Topic 5 0.700 ± 0.008 0.653 ± 0.014 0.756 ± 0.006 0.579 ± 0.011 0.540 ± 0.014 0.626 ± 0.011 117.2 ± 4.3 0.188 ± 0.009 1035.38 ± 77.52 174.8 ± 5.7 100 CauScale 5 0.845 ± 0.003 0.908 ± 0.017 0.792 ± 0.010 0.667 ± 0.012 0.716 ± 0.014 0.625 ± 0.016 68.8 ± 2.4 0.219 ± 0.011 6.39 ± 0.13 131.2 ± 1.8 100 SDCD 5 0.675 ± 0.015 0.667 ± 0.024 0.685 ± 0.014 0.406 ± 0.026 0.402 ± 0.031 0.411 ± 0.023 141.6 ± 11.9 0.408 ± 0.016 151.59 ± 4.29 155.4 ± 8.0 100 Das 5 0.518 ± 0.012 0.442 ± 0.019 0.630 ± 0.007 0.317 ± 0.020 0.271 ± 0.023 0.384 ± 0.017 215.2 ± 15.8 0.363 ± 0.019 818.49 ± 9.20 217.0 ± 13.9 200 Spade-Hist 5 0.795 ± 0.017 0.942 ± 0.008 0.688 ± 0.021 0.762 ± 0.021 0.903 ± 0.013 0.659 ± 0.025 115.4 ± 7.7 0.152 ± 0.014 38.70 ± 2.31 221.4 ± 10.0 200 Spade-Gauss 5 0.781 ± 0.016 0.874 ± 0.021 0.707 ± 0.017 0.669 ± 0.017 0.749 ± 0.022 0.605 ± 0.017 150.8 ± 10.9 0.197 ± 0.009 13.72 ± 1.12 245.8 ± 12.5 200 Topic 5 0.683 ± 0.015 0.643 ± 0.020 0.729 ± 0.011 0.571 ± 0.019 0.537 ± 0.023 0.609 ± 0.015 241.8 ± 13.9 0.212 ± 0.013 4034.05 ± 230.33 344.8 ± 13.5 200 CauScale 5 0.855 ± 0.006 0.950 ± 0.002 0.777 ± 0.010 0.678 ± 0.017 0.753 ± 0.017 0.616 ± 0.017 128.4 ± 4.7 0.243 ± 0.013 6.57 ± 0.19 248.0 ± 8.3 200 SDCD 5 0.675 ± 0.006 0.668 ± 0.016 0.685 ± 0.013 0.414 ± 0.011 0.409 ± 0.013 0.420 ± 0.014 280.0 ± 11.1 0.411 ± 0.012 135.46 ± 0.88 312.2 ± 17.7 200 Das 5 0.486 ± 0.011 0.429 ± 0.010 0.562 ± 0.015 0.280 ± 0.010 0.247 ± 0.009 0.323 ± 0.011 431.6 ± 11.6 0.375 ± 0.011 3606.43 ± 62.52 397.2 ± 14.2 400 Spade-Hist 5 0.792 ± 0.010 0.945 ± 0.002 0.682 ± 0.014 0.756 ± 0.011 0.902 ± 0.003 0.651 ± 0.014 231.6 ± 15.0 0.180 ± 0.015 145.14 ± 4.54 428.4 ± 4.2 400 Spade-Gauss 5 0.785 ± 0.008 0.881 ± 0.004 0.708 ± 0.011 0.665 ± 0.008 0.747 ± 0.008 0.600 ± 0.009 295.2 ± 14.0 0.212 ± 0.007 51.51 ± 1.63 477.4 ± 7.6 400 Topic 5 0.693 ± 0.007 0.656 ± 0.009 0.735 ± 0.012 0.586 ± 0.006 0.554 ± 0.010 0.621 ± 0.009 455.0 ± 14.0 0.211 ± 0.006 14842.05 ± 340.50 665.8 ± 9.2 400 CauScale 5 0.848 ± 0.006 0.954 ± 0.004 0.763 ± 0.009 0.674 ± 0.004 0.759 ± 0.004 0.607 ± 0.006 256.0 ± 10.4 0.240 ± 0.006 7.19 ± 0.09 474.8 ± 7.7 400 SDCD 5 0.643 ± 0.008 0.629 ± 0.009 0.657 ± 0.009 0.371 ± 0.013 0.363 ± 0.014 0.379 ± 0.012 598.8 ± 10.7 0.455 ± 0.018 127.59 ± 1.22 620.2 ± 11.4 400 Das 5 0.462 ± 0.007 0.445 ± 0.010 0.482 ± 0.011 0.264 ± 0.009 0.254 ± 0.010 0.276 ± 0.011 788.6 ± 10.6 0.406 ± 0.008 10551.73 ± 605.04 644.2 ± 8.6 500 Topic 5 0.693 ± 0.004 0.651 ± 0.005 0.741 ± 0.004 0.581 ± 0.011 0.546 ± 0.011 0.621 ± 0.011 579.0 ± 9.0 0.204 ± 0.009 26004.56 ± 734.33 849.2 ± 4.7 500 SDCD 5 0.632 ± 0.007 0.609 ± 0.012 0.658 ± 0.007 0.358 ± 0.011 0.346 ± 0.014 0.373 ± 0.009 784.0 ± 24.5 0.463 ± 0.014 126.35 ± 0.45 806.4 ± 22.1 500 Das 5 0.440 ± 0.006 0.436 ± 0.005 0.446 ± 0.011 0.253 ± 0.006 0.250 ± 0.006 0.256 ± 0.008 986.6 ± 18.1 0.398 ± 0.010 19335.05 ± 992.65 764.4 ± 26.0 800 Spade-Hist 5 0.786 ± 0.002 0.931 ± 0.007 0.681 ± 0.003 0.742 ± 0.004 0.878 ± 0.009 0.642 ± 0.004 478.8 ± 9.5 0.196 ± 0.005 583.72 ± 10.34 856.6 ± 10.8 800 Spade-Gauss 5 0.781 ± 0.003 0.856 ± 0.007 0.718 ± 0.008 0.656 ± 0.004 0.720 ± 0.008 0.604 ± 0.007 605.4 ± 2.5 0.205 ± 0.003 221.35 ± 7.53 982.4 ± 17.7 800 Topic 5 0.687 ± 0.003 0.639 ± 0.003 0.744 ± 0.009 0.572 ± 0.004 0.531 ± 0.006 0.619 ± 0.007 939.0 ± 9.4 0.211 ± 0.005 62753.40 ± 1896.28 1364.0 ± 20.5 800 CauScale 5 0.718 ± 0.015 0.693 ± 0.020 0.747 ± 0.011 0.541 ± 0.017 0.522 ± 0.021 0.562 ± 0.013 902.6 ± 46.5 0.290 ± 0.011 10.49 ± 0.16 1264.4 ± 23.1 800 SDCD 5 0.621 ± 0.005 0.591 ± 0.007 0.656 ± 0.009 0.341 ± 0.003 0.324 ± 0.006 0.360 ± 0.003 1282.8 ± 22.4 0.464 ± 0.005 131.62 ± 0.94 1301.8 ± 27.9 800 Das 5 0.376 ± 0.007 0.392 ± 0.005 0.361 ± 0.009 0.234 ± 0.008 0.244 ± 0.007 0.225 ± 0.009 1561.6 ± 18.7 0.388 ± 0.010 47139.59 ± 1424.06 1076.4 ± 15.1 1600 Spade-Hist 5 0.768 ± 0.010 0.914 ± 0.009 0.663 ± 0.010 0.712 ± 0.016 0.847 ± 0.017 0.614 ± 0.015 1068.6 ± 49.2 0.224 ± 0.016 2416.47 ± 39.21 1729.4 ± 16.5 1600 Spade-Gauss 5 0.775 ± 0.004 0.850 ± 0.005 0.712 ± 0.004 0.659 ± 0.005 0.723 ± 0.006 0.606 ± 0.005 1240.2 ± 20.8 0.206 ± 0.006 918.17 ± 19.48 2000.0 ± 10.2 1600 CauScale 5 0.501 ± 0.062 0.415 ± 0.065 0.650 ± 0.051 0.350 ± 0.053 0.291 ± 0.053 0.452 ± 0.053 3765.6 ± 630.5 0.358 ± 0.032 26.71 ± 1.18 4007.4 ± 458.0 1600 SDCD 5 0.628 ± 0.003 0.614 ± 0.004 0.643 ± 0.004 0.356 ± 0.004 0.348 ± 0.006 0.364 ± 0.003 2481.4 ± 23.3 0.447 ± 0.006 215.13 ± 3.74 2498.4 ± 24.5 Table 16: Heterogeneous scalability sample-size experiment. Entries are means with standard errors over successful seeds. n_samples method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges 2500 Spade-Hist 5 0.809 ± 0.006 0.938 ± 0.012 0.712 ± 0.004 0.784 ± 0.016 0.910 ± 0.023 0.690 ± 0.012 53.6 ± 2.0 0.125 ± 0.019 10.06 ± 0.69 114.4 ± 3.8 2500 Spade-Gauss 5 0.808 ± 0.014 0.893 ± 0.023 0.737 ± 0.009 0.689 ± 0.027 0.763 ± 0.036 0.629 ± 0.021 69.0 ± 5.4 0.172 ± 0.017 3.66 ± 0.31 124.4 ± 3.3 2500 Topic 5 0.700 ± 0.008 0.653 ± 0.014 0.756 ± 0.006 0.579 ± 0.011 0.540 ± 0.014 0.626 ± 0.011 117.2 ± 4.3 0.188 ± 0.009 1035.38 ± 77.52 174.8 ± 5.7 2500 CauScale 5 0.845 ± 0.003 0.908 ± 0.017 0.792 ± 0.010 0.667 ± 0.012 0.716 ± 0.014 0.625 ± 0.016 68.8 ± 2.4 0.219 ± 0.011 6.39 ± 0.13 131.2 ± 1.8 2500 SDCD 5 0.675 ± 0.015 0.667 ± 0.024 0.685 ± 0.014 0.406 ± 0.026 0.402 ± 0.031 0.411 ± 0.023 141.6 ± 11.9 0.408 ± 0.016 151.59 ± 4.29 155.4 ± 8.0 2500 Das 5 0.518 ± 0.012 0.442 ± 0.019 0.630 ± 0.007 0.317 ± 0.020 0.271 ± 0.023 0.384 ± 0.017 215.2 ± 15.8 0.363 ± 0.019 818.49 ± 9.20 217.0 ± 13.9 5000 Spade-Hist 5 0.827 ± 0.013 0.956 ± 0.013 0.729 ± 0.015 0.800 ± 0.014 0.925 ± 0.014 0.705 ± 0.016 49.4 ± 3.4 0.129 ± 0.013 13.05 ± 0.60 115.0 ± 4.8 5000 Spade-Gauss 5 0.814 ± 0.016 0.870 ± 0.016 0.767 ± 0.023 0.665 ± 0.028 0.711 ± 0.032 0.626 ± 0.027 73.4 ± 5.1 0.189 ± 0.016 4.40 ± 0.35 132.8 ± 3.7 5000 Topic 5 0.716 ± 0.015 0.641 ± 0.019 0.812 ± 0.017 0.583 ± 0.023 0.523 ± 0.027 0.660 ± 0.018 120.0 ± 7.5 0.203 ± 0.022 2103.94 ± 160.50 191.2 ± 6.8 5000 CauScale 5 0.775 ± 0.011 0.825 ± 0.025 0.732 ± 0.007 0.583 ± 0.013 0.621 ± 0.023 0.551 ± 0.011 91.6 ± 4.8 0.271 ± 0.024 6.28 ± 0.11 134.2 ± 6.2 5000 SDCD 5 0.658 ± 0.009 0.700 ± 0.019 0.624 ± 0.018 0.359 ± 0.013 0.383 ± 0.020 0.340 ± 0.011 140.4 ± 6.6 0.470 ± 0.029 324.13 ± 5.44 135.0 ± 8.4 5000 Das 5 0.474 ± 0.003 0.391 ± 0.009 0.608 ± 0.021 0.265 ± 0.010 0.219 ± 0.011 0.339 ± 0.013 243.6 ± 12.2 0.390 ± 0.019 4835.82 ± 154.65 235.4 ± 14.3 10000 Spade-Hist 5 0.862 ± 0.021 0.939 ± 0.007 0.798 ± 0.033 0.829 ± 0.016 0.904 ± 0.012 0.767 ± 0.026 43.4 ± 5.7 0.124 ± 0.017 22.93 ± 0.99 127.6 ± 3.2 10000 Spade-Gauss 5 0.846 ± 0.011 0.856 ± 0.012 0.836 ± 0.018 0.697 ± 0.022 0.706 ± 0.026 0.689 ± 0.022 68.0 ± 4.7 0.197 ± 0.016 6.25 ± 0.34 146.8 ± 1.7 10000 Topic 5 0.740 ± 0.012 0.644 ± 0.022 0.874 ± 0.014 0.608 ± 0.023 0.529 ± 0.027 0.717 ± 0.020 115.8 ± 5.3 0.192 ± 0.015 4240.60 ± 207.83 204.8 ± 4.6 10000 CauScale 5 0.758 ± 0.020 0.780 ± 0.026 0.738 ± 0.022 0.548 ± 0.014 0.564 ± 0.016 0.534 ± 0.018 102.6 ± 8.3 0.290 ± 0.014 6.82 ± 0.05 142.8 ± 6.3 10000 SDCD 5 0.697 ± 0.017 0.699 ± 0.024 0.698 ± 0.025 0.410 ± 0.022 0.414 ± 0.035 0.408 ± 0.013 135.4 ± 9.4 0.410 ± 0.029 625.97 ± 16.81 151.0 ± 8.9 20000 Spade-Hist 5 0.843 ± 0.016 0.878 ± 0.023 0.812 ± 0.011 0.797 ± 0.017 0.830 ± 0.023 0.768 ± 0.011 52.4 ± 5.5 0.138 ± 0.008 43.70 ± 1.96 139.8 ± 5.8 20000 Spade-Gauss 5 0.856 ± 0.016 0.858 ± 0.019 0.856 ± 0.021 0.724 ± 0.026 0.726 ± 0.029 0.724 ± 0.027 62.6 ± 5.4 0.198 ± 0.028 9.07 ± 0.63 150.4 ± 6.0 20000 Topic 5 0.739 ± 0.015 0.628 ± 0.019 0.900 ± 0.011 0.622 ± 0.025 0.529 ± 0.026 0.756 ± 0.023 117.0 ± 7.0 0.192 ± 0.028 12831.99 ± 1014.66 216.2 ± 6.4 20000 SDCD 5 0.677 ± 0.017 0.689 ± 0.023 0.667 ± 0.018 0.405 ± 0.023 0.413 ± 0.026 0.399 ± 0.022 137.4 ± 11.7 0.412 ± 0.013 1252.47 ± 26.83 146.8 ± 9.4 40000 Spade-Hist 5 0.842 ± 0.013 0.852 ± 0.015 0.833 ± 0.020 0.763 ± 0.024 0.772 ± 0.022 0.756 ± 0.030 59.0 ± 6.3 0.152 ± 0.019 106.74 ± 11.01 147.8 ± 8.0 40000 Spade-Gauss 5 0.855 ± 0.015 0.833 ± 0.019 0.877 ± 0.013 0.688 ± 0.035 0.671 ± 0.036 0.706 ± 0.035 70.2 ± 6.2 0.229 ± 0.030 11.95 ± 0.78 158.6 ± 3.5 40000 Topic 5 0.714 ± 0.012 0.587 ± 0.014 0.914 ± 0.012 0.567 ± 0.031 0.466 ± 0.027 0.725 ± 0.037 138.2 ± 6.7 0.233 ± 0.030 32126.00 ± 4203.79 234.8 ± 6.6 40000 SDCD 5 0.676 ± 0.020 0.699 ± 0.032 0.657 ± 0.019 0.407 ± 0.023 0.422 ± 0.030 0.395 ± 0.019 135.8 ± 12.1 0.415 ± 0.024 2391.59 ± 81.21 143.6 ± 11.2 80000 Spade-Hist 5 0.795 ± 0.011 0.757 ± 0.027 0.841 ± 0.017 0.765 ± 0.010 0.729 ± 0.025 0.810 ± 0.018 70.6 ± 5.4 0.109 ± 0.011 329.40 ± 38.03 168.6 ± 10.0 80000 Spade-Gauss 5 0.828 ± 0.010 0.763 ± 0.016 0.908 ± 0.014 0.661 ± 0.011 0.609 ± 0.016 0.725 ± 0.012 84.4 ± 4.5 0.215 ± 0.011 25.31 ± 4.06 179.8 ± 7.9 80000 Topic 2 0.693 ± 0.033 0.563 ± 0.048 0.908 ± 0.012 0.545 ± 0.052 0.443 ± 0.059 0.712 ± 0.024 149.5 ± 13.5 0.228 ± 0.026 78209.11 ± 1503.32 243.0 ± 4.0 80000 SDCD 5 0.670 ± 0.020 0.672 ± 0.021 0.673 ± 0.034 0.394 ± 0.030 0.393 ± 0.024 0.397 ± 0.037 141.2 ± 8.4 0.425 ± 0.036 4317.04 ± 89.90 150.8 ± 6.6 160000 Spade-Hist 5 0.758 ± 0.012 0.660 ± 0.025 0.898 ± 0.020 0.710 ± 0.012 0.618 ± 0.021 0.842 ± 0.030 94.6 ± 4.2 0.101 ± 0.017 1032.69 ± 72.45 205.8 ± 9.3 160000 Spade-Gauss 5 0.824 ± 0.008 0.732 ± 0.018 0.943 ± 0.011 0.667 ± 0.032 0.594 ± 0.033 0.764 ± 0.034 87.6 ± 6.2 0.207 ± 0.035 37.33 ± 2.77 194.4 ± 6.7 160000 SDCD 5 0.687 ± 0.024 0.685 ± 0.032 0.692 ± 0.021 0.412 ± 0.034 0.411 ± 0.038 0.414 ± 0.030 138.2 ± 14.4 0.425 ± 0.017 7944.92 ± 117.60 153.4 ± 8.9 Table 17: Qualitative comparison without marginalization (n=1000n=1000, d=20d=20, degree 2, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.843 ± 0.035 0.986 ± 0.014 0.745 ± 0.052 0.749 ± 0.039 0.880 ± 0.052 0.659 ± 0.046 6.0 ± 0.9 0.118 ± 0.047 0.69 ± 0.06 12.6 ± 1.1 gaussian Spade-Gauss 5 0.867 ± 0.033 0.986 ± 0.014 0.780 ± 0.051 0.835 ± 0.024 0.951 ± 0.020 0.750 ± 0.041 4.6 ± 1.0 0.052 ± 0.026 0.08 ± 0.00 13.2 ± 1.1 gaussian Topic 5 0.763 ± 0.028 0.709 ± 0.060 0.849 ± 0.042 0.651 ± 0.044 0.604 ± 0.057 0.725 ± 0.052 10.8 ± 1.1 0.213 ± 0.048 12.14 ± 0.71 20.2 ± 1.1 gaussian Cam 5 0.725 ± 0.016 0.588 ± 0.018 0.946 ± 0.015 0.718 ± 0.020 0.583 ± 0.020 0.937 ± 0.022 12.4 ± 1.6 0.018 ± 0.018 235.86 ± 1.13 27.2 ± 2.5 gaussian CauScale 5 0.816 ± 0.037 0.965 ± 0.021 0.714 ± 0.053 0.504 ± 0.042 0.592 ± 0.029 0.443 ± 0.050 10.2 ± 1.8 0.390 ± 0.029 5.95 ± 0.07 12.2 ± 0.5 gaussian SDCD 5 0.762 ± 0.047 0.848 ± 0.051 0.704 ± 0.061 0.494 ± 0.054 0.554 ± 0.067 0.454 ± 0.052 12.0 ± 2.6 0.358 ± 0.042 75.33 ± 1.56 14.0 ± 1.6 gaussian Globe 5 0.760 ± 0.039 0.880 ± 0.019 0.674 ± 0.053 0.725 ± 0.029 0.844 ± 0.034 0.641 ± 0.035 7.8 ± 1.4 0.176 ± 0.050 130.72 ± 13.93 12.8 ± 1.3 gaussian Das 5 0.564 ± 0.042 0.511 ± 0.026 0.640 ± 0.077 0.381 ± 0.071 0.343 ± 0.060 0.435 ± 0.092 19.8 ± 2.6 0.335 ± 0.044 14.05 ± 0.64 20.6 ± 1.6 gaussian Score 5 0.664 ± 0.023 0.532 ± 0.029 0.894 ± 0.033 0.483 ± 0.026 0.386 ± 0.025 0.651 ± 0.041 19.2 ± 1.3 0.335 ± 0.044 12.64 ± 0.47 28.2 ± 1.7 gaussian NoGAM 5 0.688 ± 0.029 0.557 ± 0.040 0.917 ± 0.027 0.490 ± 0.032 0.398 ± 0.038 0.648 ± 0.015 18.4 ± 1.7 0.343 ± 0.016 44.08 ± 1.04 27.8 ± 1.7 gaussian Avici 5 0.763 ± 0.045 0.943 ± 0.025 0.655 ± 0.069 0.716 ± 0.046 0.883 ± 0.005 0.616 ± 0.072 7.4 ± 1.4 0.144 ± 0.043 7.52 ± 0.07 11.8 ± 1.7 gaussian NOTEARS 5 0.739 ± 0.043 0.705 ± 0.065 0.807 ± 0.066 0.501 ± 0.092 0.487 ± 0.114 0.537 ± 0.079 14.8 ± 3.6 0.355 ± 0.095 252.81 ± 54.81 20.0 ± 3.2 gaussian DAGMA 5 0.777 ± 0.038 0.800 ± 0.054 0.771 ± 0.052 0.474 ± 0.114 0.510 ± 0.145 0.454 ± 0.094 13.0 ± 3.3 0.398 ± 0.141 702.32 ± 35.10 16.6 ± 2.4 gaussian DiffAN 5 0.771 ± 0.023 0.749 ± 0.037 0.812 ± 0.058 0.484 ± 0.029 0.471 ± 0.040 0.509 ± 0.039 13.2 ± 1.6 0.430 ± 0.032 17.47 ± 0.45 18.0 ± 0.8 gaussian SciNO 5 0.794 ± 0.025 0.760 ± 0.032 0.847 ± 0.059 0.507 ± 0.052 0.493 ± 0.068 0.532 ± 0.040 12.6 ± 1.8 0.403 ± 0.053 408.55 ± 17.28 18.6 ± 1.3 gaussian Caps 5 0.753 ± 0.023 0.774 ± 0.031 0.753 ± 0.067 0.433 ± 0.067 0.447 ± 0.078 0.431 ± 0.065 13.4 ± 1.5 0.435 ± 0.043 8.03 ± 0.25 16.0 ± 0.4 gaussian PC 5 0.582 ± 0.045 0.807 ± 0.066 0.466 ± 0.054 0.439 ± 0.068 0.613 ± 0.111 0.350 ± 0.056 13.4 ± 2.3 0.385 ± 0.059 353.03 ± 135.93 9.6 ± 0.7 gaussian FCI 5 0.400 ± 0.077 1.000 0.260 ± 0.054 0.336 ± 0.075 0.867 ± 0.097 0.218 ± 0.052 13.4 ± 1.9 0.422 ± 0.062 339.19 ± 123.29 4.2 ± 0.9 multimodal Spade-Hist 5 0.885 ± 0.040 0.976 ± 0.015 0.820 ± 0.062 0.885 ± 0.040 0.976 ± 0.015 0.820 ± 0.062 3.8 ± 1.4 0.010 ± 0.010 0.62 ± 0.04 14.0 ± 1.2 multimodal Spade-Gauss 5 0.868 ± 0.034 0.971 ± 0.017 0.794 ± 0.054 0.763 ± 0.055 0.849 ± 0.036 0.700 ± 0.068 5.8 ± 1.5 0.111 ± 0.024 0.09 ± 0.01 13.6 ± 1.0 multimodal Topic 5 0.714 ± 0.018 0.641 ± 0.018 0.820 ± 0.057 0.605 ± 0.026 0.545 ± 0.031 0.693 ± 0.046 13.0 ± 1.0 0.137 ± 0.021 12.10 ± 0.99 21.2 ± 1.1 multimodal Cam 5 0.722 ± 0.031 0.582 ± 0.043 0.968 ± 0.021 0.668 ± 0.034 0.538 ± 0.040 0.897 ± 0.039 13.4 ± 0.7 0.083 ± 0.031 239.93 ± 2.23 28.0 ± 1.3 multimodal CauScale 5 0.783 ± 0.033 0.901 ± 0.043 0.698 ± 0.041 0.622 ± 0.065 0.711 ± 0.064 0.558 ± 0.068 9.4 ± 2.1 0.220 ± 0.046 5.87 ± 0.05 13.0 ± 1.0 multimodal SDCD 5 0.759 ± 0.039 0.812 ± 0.037 0.715 ± 0.043 0.490 ± 0.085 0.518 ± 0.085 0.467 ± 0.084 12.2 ± 2.6 0.369 ± 0.063 76.62 ± 2.32 15.0 ± 1.9 multimodal Globe 5 0.758 ± 0.036 0.880 ± 0.019 0.674 ± 0.053 0.731 ± 0.027 0.851 ± 0.027 0.648 ± 0.041 7.8 ± 1.4 0.168 ± 0.040 142.47 ± 13.41 12.8 ± 1.3 multimodal Das 5 0.677 ± 0.049 0.673 ± 0.052 0.689 ± 0.056 0.563 ± 0.076 0.557 ± 0.076 0.576 ± 0.081 12.4 ± 1.2 0.220 ± 0.043 13.69 ± 0.24 17.2 ± 1.5 multimodal Score 5 0.719 ± 0.018 0.603 ± 0.029 0.897 ± 0.015 0.621 ± 0.046 0.523 ± 0.049 0.770 ± 0.039 13.6 ± 1.0 0.220 ± 0.043 14.12 ± 0.60 25.0 ± 1.6 multimodal NoGAM 5 0.723 ± 0.026 0.600 ± 0.039 0.920 ± 0.021 0.642 ± 0.028 0.532 ± 0.030 0.821 ± 0.047 13.6 ± 1.5 0.156 ± 0.040 44.92 ± 2.25 26.0 ± 2.2 multimodal Avici 5 0.803 ± 0.043 0.948 ± 0.034 0.708 ± 0.060 0.606 ± 0.062 0.705 ± 0.055 0.541 ± 0.066 8.8 ± 1.5 0.329 ± 0.072 7.59 ± 0.09 12.6 ± 1.1 multimodal NOTEARS 5 0.806 ± 0.021 0.799 ± 0.049 0.833 ± 0.042 0.678 ± 0.066 0.677 ± 0.080 0.692 ± 0.056 10.0 ± 3.2 0.187 ± 0.060 314.24 ± 40.25 18.6 ± 3.9 multimodal DAGMA 5 0.818 ± 0.029 0.869 ± 0.039 0.779 ± 0.037 0.585 ± 0.074 0.617 ± 0.072 0.561 ± 0.078 10.0 ± 2.3 0.324 ± 0.065 824.46 ± 35.95 15.6 ± 2.5 multimodal DiffAN 5 0.855 ± 0.015 0.820 ± 0.030 0.897 ± 0.015 0.638 ± 0.056 0.613 ± 0.061 0.667 ± 0.054 9.4 ± 2.0 0.300 ± 0.051 17.50 ± 0.19 18.8 ± 2.5 multimodal SciNO 5 0.777 ± 0.015 0.773 ± 0.050 0.801 ± 0.043 0.482 ± 0.060 0.487 ± 0.080 0.491 ± 0.049 13.2 ± 2.4 0.476 ± 0.065 383.32 ± 13.01 18.6 ± 3.5 multimodal Caps 5 0.820 ± 0.018 0.894 ± 0.033 0.762 ± 0.027 0.619 ± 0.062 0.669 ± 0.056 0.580 ± 0.068 8.8 ± 1.2 0.278 ± 0.036 7.59 ± 0.25 14.4 ± 1.3 multimodal PC 5 0.568 ± 0.034 0.821 ± 0.035 0.437 ± 0.032 0.449 ± 0.067 0.641 ± 0.083 0.347 ± 0.055 12.6 ± 1.4 0.336 ± 0.029 404.52 ± 180.74 9.2 ± 1.3 multimodal FCI 5 0.408 ± 0.066 1.000 0.266 ± 0.056 0.408 ± 0.066 1.000 0.266 ± 0.056 12.6 ± 1.9 0.250 ± 0.050 405.99 ± 156.54 4.4 ± 0.8 unimodal_asymmetric Spade-Hist 5 0.908 ± 0.038 0.987 ± 0.013 0.847 ± 0.058 0.908 ± 0.038 0.987 ± 0.013 0.847 ± 0.058 3.2 ± 1.4 0.018 ± 0.018 0.52 ± 0.01 14.2 ± 0.7 unimodal_asymmetric Spade-Gauss 5 0.861 ± 0.026 0.988 ± 0.012 0.765 ± 0.039 0.765 ± 0.045 0.877 ± 0.036 0.681 ± 0.052 5.8 ± 1.4 0.142 ± 0.027 0.08 ± 0.01 13.0 ± 1.1 unimodal_asymmetric Topic 5 0.760 ± 0.012 0.729 ± 0.023 0.798 ± 0.026 0.654 ± 0.030 0.627 ± 0.027 0.689 ± 0.045 10.4 ± 1.2 0.145 ± 0.032 11.59 ± 0.83 18.4 ± 1.4 unimodal_asymmetric Cam 5 0.658 ± 0.034 0.515 ± 0.046 0.938 ± 0.029 0.575 ± 0.033 0.451 ± 0.044 0.820 ± 0.032 19.2 ± 3.2 0.117 ± 0.028 234.79 ± 1.19 31.8 ± 4.1 unimodal_asymmetric CauScale 5 0.795 ± 0.044 0.878 ± 0.047 0.728 ± 0.044 0.506 ± 0.018 0.558 ± 0.021 0.463 ± 0.018 11.0 ± 1.8 0.417 ± 0.038 5.92 ± 0.05 14.0 ± 1.3 unimodal_asymmetric SDCD 5 0.809 ± 0.023 0.864 ± 0.019 0.771 ± 0.052 0.520 ± 0.029 0.563 ± 0.053 0.490 ± 0.019 10.6 ± 0.7 0.351 ± 0.053 77.99 ± 0.81 14.8 ± 0.9 unimodal_asymmetric Globe 5 0.761 ± 0.052 0.891 ± 0.075 0.674 ± 0.052 0.622 ± 0.031 0.727 ± 0.049 0.552 ± 0.038 9.4 ± 1.4 0.245 ± 0.032 125.90 ± 6.20 12.8 ± 1.0 unimodal_asymmetric Das 5 0.677 ± 0.030 0.603 ± 0.043 0.781 ± 0.022 0.509 ± 0.027 0.452 ± 0.032 0.588 ± 0.032 15.8 ± 1.4 0.293 ± 0.043 14.89 ± 0.41 22.0 ± 1.8 unimodal_asymmetric Score 5 0.703 ± 0.028 0.578 ± 0.042 0.918 ± 0.034 0.512 ± 0.040 0.421 ± 0.040 0.666 ± 0.047 17.6 ± 2.5 0.293 ± 0.043 13.59 ± 0.33 27.2 ± 2.8 unimodal_asymmetric NoGAM 5 0.733 ± 0.034 0.614 ± 0.058 0.937 ± 0.026 0.593 ± 0.029 0.497 ± 0.046 0.759 ± 0.028 15.0 ± 2.5 0.200 ± 0.014 46.17 ± 0.25 26.6 ± 3.2 unimodal_asymmetric Avici 5 0.833 ± 0.024 1.000 0.716 ± 0.035 0.696 ± 0.035 0.834 ± 0.024 0.600 ± 0.040 7.0 ± 1.3 0.228 ± 0.057 7.60 ± 0.10 12.0 ± 0.9 unimodal_asymmetric NOTEARS 5 0.794 ± 0.018 0.731 ± 0.035 0.879 ± 0.037 0.615 ± 0.070 0.573 ± 0.084 0.673 ± 0.060 11.6 ± 2.7 0.253 ± 0.069 282.91 ± 42.45 20.6 ± 2.5 unimodal_asymmetric DAGMA 5 0.827 ± 0.056 0.813 ± 0.080 0.853 ± 0.026 0.443 ± 0.103 0.441 ± 0.109 0.451 ± 0.095 14.4 ± 5.1 0.452 ± 0.092 717.62 ± 29.28 19.2 ± 4.0 unimodal_asymmetric DiffAN 5 0.768 ± 0.030 0.705 ± 0.056 0.858 ± 0.007 0.496 ± 0.037 0.456 ± 0.049 0.554 ± 0.034 14.6 ± 2.5 0.381 ± 0.023 17.74 ± 0.26 21.6 ± 3.2 unimodal_asymmetric SciNO 5 0.823 ± 0.023 0.774 ± 0.044 0.892 ± 0.031 0.486 ± 0.043 0.457 ± 0.047 0.525 ± 0.043 12.6 ± 1.2 0.389 ± 0.062 375.92 ± 13.04 19.8 ± 2.2 unimodal_asymmetric Caps 5 0.787 ± 0.041 0.769 ± 0.070 0.820 ± 0.027 0.524 ± 0.016 0.509 ± 0.032 0.548 ± 0.021 12.6 ± 1.9 0.352 ± 0.032 8.03 ± 0.38 18.8 ± 2.5 unimodal_asymmetric PC 5 0.588 ± 0.062 0.840 ± 0.037 0.458 ± 0.061 0.552 ± 0.065 0.791 ± 0.058 0.430 ± 0.061 11.0 ± 1.8 0.264 ± 0.030 353.07 ± 133.90 9.4 ± 1.6 unimodal_asymmetric FCI 5 0.400 ± 0.050 1.000 0.255 ± 0.041 0.365 ± 0.037 0.927 ± 0.045 0.232 ± 0.031 13.0 ± 1.3 0.264 ± 0.030 368.78 ± 135.86 4.4 ± 0.8 unimodal_symmetric Spade-Hist 5 0.875 ± 0.052 1.000 0.794 ± 0.086 0.875 ± 0.052 1.000 0.794 ± 0.086 4.0 ± 1.7 0.010 ± 0.010 0.52 ± 0.02 13.0 ± 0.8 unimodal_symmetric Spade-Gauss 5 0.858 ± 0.043 0.963 ± 0.025 0.790 ± 0.076 0.822 ± 0.051 0.922 ± 0.036 0.757 ± 0.079 5.2 ± 1.6 0.093 ± 0.034 0.09 ± 0.00 13.6 ± 1.2 unimodal_symmetric Topic 5 0.755 ± 0.023 0.731 ± 0.043 0.811 ± 0.071 0.664 ± 0.046 0.638 ± 0.041 0.720 ± 0.091 10.2 ± 1.1 0.139 ± 0.048 11.44 ± 0.75 18.4 ± 1.0 unimodal_symmetric Cam 5 0.609 ± 0.027 0.471 ± 0.033 0.899 ± 0.065 0.568 ± 0.028 0.438 ± 0.030 0.843 ± 0.077 20.0 ± 1.2 0.066 ± 0.035 239.65 ± 1.87 32.0 ± 2.3 unimodal_symmetric CauScale 5 0.844 ± 0.038 0.955 ± 0.018 0.763 ± 0.058 0.534 ± 0.074 0.600 ± 0.078 0.486 ± 0.073 9.6 ± 2.1 0.395 ± 0.057 5.93 ± 0.11 13.2 ± 0.7 unimodal_symmetric SDCD 5 0.801 ± 0.036 0.921 ± 0.025 0.716 ± 0.051 0.492 ± 0.072 0.559 ± 0.067 0.443 ± 0.076 10.8 ± 2.0 0.430 ± 0.080 78.96 ± 0.89 13.2 ± 1.7 unimodal_symmetric Globe 5 0.747 ± 0.048 0.841 ± 0.030 0.689 ± 0.084 0.719 ± 0.057 0.808 ± 0.036 0.666 ± 0.092 8.4 ± 2.0 0.136 ± 0.058 138.30 ± 6.68 13.4 ± 0.9 unimodal_symmetric Das 5 0.572 ± 0.023 0.528 ± 0.027 0.627 ± 0.028 0.450 ± 0.044 0.416 ± 0.042 0.493 ± 0.050 18.0 ± 1.8 0.281 ± 0.060 14.21 ± 0.43 20.0 ± 1.6 unimodal_symmetric Score 5 0.630 ± 0.023 0.507 ± 0.024 0.859 ± 0.074 0.480 ± 0.052 0.386 ± 0.043 0.658 ± 0.089 19.8 ± 1.2 0.281 ± 0.060 13.49 ± 0.64 28.0 ± 1.7 unimodal_symmetric NoGAM 5 0.638 ± 0.042 0.513 ± 0.043 0.880 ± 0.074 0.501 ± 0.055 0.401 ± 0.046 0.697 ± 0.093 19.8 ± 2.4 0.251 ± 0.071 46.72 ± 0.39 29.0 ± 2.9 unimodal_symmetric Avici 5 0.842 ± 0.036 0.983 ± 0.017 0.742 ± 0.052 0.760 ± 0.042 0.885 ± 0.022 0.670 ± 0.055 6.0 ± 1.3 0.159 ± 0.029 8.13 ± 0.38 12.6 ± 1.1 unimodal_symmetric NOTEARS 5 0.766 ± 0.037 0.740 ± 0.028 0.796 ± 0.050 0.595 ± 0.048 0.575 ± 0.043 0.619 ± 0.056 11.4 ± 2.1 0.278 ± 0.047 319.63 ± 30.05 18.2 ± 1.8 unimodal_symmetric DAGMA 5 0.793 ± 0.044 0.812 ± 0.044 0.789 ± 0.065 0.447 ± 0.065 0.457 ± 0.064 0.447 ± 0.077 13.2 ± 2.8 0.432 ± 0.044 829.40 ± 63.61 16.4 ± 1.8 unimodal_symmetric DiffAN 5 0.730 ± 0.043 0.694 ± 0.043 0.800 ± 0.087 0.549 ± 0.052 0.522 ± 0.057 0.602 ± 0.080 13.4 ± 2.4 0.312 ± 0.049 17.57 ± 0.49 19.4 ± 2.3 unimodal_symmetric SciNO 5 0.734 ± 0.046 0.716 ± 0.046 0.776 ± 0.081 0.468 ± 0.074 0.452 ± 0.068 0.502 ± 0.095 14.2 ± 2.3 0.424 ± 0.058 396.63 ± 15.82 18.2 ± 2.0 unimodal_symmetric Caps 5 0.764 ± 0.054 0.782 ± 0.038 0.755 ± 0.077 0.547 ± 0.041 0.562 ± 0.039 0.539 ± 0.051 11.6 ± 1.9 0.347 ± 0.046 7.77 ± 0.23 16.0 ± 1.3 unimodal_symmetric PC 5 0.569 ± 0.087 0.915 ± 0.038 0.438 ± 0.084 0.430 ± 0.087 0.725 ± 0.118 0.328 ± 0.078 12.6 ± 2.2 0.360 ± 0.042 353.55 ± 134.35 7.8 ± 1.3 unimodal_symmetric FCI 5 0.446 ± 0.084 0.975 ± 0.025 0.307 ± 0.070 0.376 ± 0.078 0.833 ± 0.080 0.259 ± 0.062 13.0 ± 1.9 0.386 ± 0.052 343.25 ± 119.76 5.2 ± 1.2 Table 18: Qualitative comparison without marginalization (n=1000n=1000, d=20d=20, degree 2, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.726 ± 0.070 1.000 0.587 ± 0.083 0.636 ± 0.045 0.893 ± 0.056 0.509 ± 0.052 8.4 ± 1.2 0.310 ± 0.061 0.48 ± 0.01 9.8 ± 1.4 gaussian Spade-Gauss 5 0.736 ± 0.067 0.935 ± 0.048 0.613 ± 0.074 0.718 ± 0.057 0.915 ± 0.046 0.597 ± 0.062 7.6 ± 1.6 0.112 ± 0.027 0.08 ± 0.00 10.8 ± 1.0 gaussian Topic 5 0.608 ± 0.028 0.582 ± 0.042 0.650 ± 0.046 0.546 ± 0.042 0.527 ± 0.061 0.579 ± 0.038 15.0 ± 1.2 0.183 ± 0.065 11.03 ± 0.37 18.6 ± 0.5 gaussian Cam 5 0.650 ± 0.021 0.534 ± 0.033 0.850 ± 0.041 0.600 ± 0.029 0.493 ± 0.040 0.782 ± 0.039 16.2 ± 0.7 0.096 ± 0.024 242.98 ± 1.97 26.6 ± 1.0 gaussian CauScale 5 0.702 ± 0.038 0.905 ± 0.046 0.582 ± 0.051 0.473 ± 0.038 0.610 ± 0.053 0.392 ± 0.040 11.4 ± 1.2 0.365 ± 0.064 5.93 ± 0.04 10.8 ± 1.1 gaussian SDCD 5 0.765 ± 0.050 0.882 ± 0.040 0.682 ± 0.059 0.551 ± 0.045 0.631 ± 0.031 0.492 ± 0.051 10.2 ± 1.4 0.334 ± 0.039 73.22 ± 6.59 13.2 ± 1.8 gaussian Globe 5 0.633 ± 0.047 0.829 ± 0.027 0.526 ± 0.060 0.584 ± 0.038 0.773 ± 0.046 0.483 ± 0.047 10.6 ± 0.9 0.306 ± 0.031 127.84 ± 14.08 10.6 ± 1.2 gaussian Das 5 0.610 ± 0.037 0.618 ± 0.044 0.622 ± 0.064 0.434 ± 0.056 0.438 ± 0.060 0.443 ± 0.069 16.2 ± 2.0 0.267 ± 0.054 15.25 ± 1.83 17.0 ± 1.9 gaussian Score 5 0.675 ± 0.050 0.592 ± 0.062 0.805 ± 0.050 0.505 ± 0.065 0.445 ± 0.069 0.601 ± 0.068 17.2 ± 3.4 0.267 ± 0.054 13.24 ± 0.40 23.6 ± 2.7 gaussian NoGAM 5 0.677 ± 0.037 0.588 ± 0.051 0.818 ± 0.048 0.494 ± 0.059 0.430 ± 0.062 0.599 ± 0.074 17.0 ± 2.3 0.305 ± 0.087 44.84 ± 0.55 23.6 ± 1.8 gaussian Avici 5 0.732 ± 0.046 1.000 0.586 ± 0.057 0.641 ± 0.045 0.877 ± 0.037 0.513 ± 0.052 8.6 ± 1.7 0.246 ± 0.047 7.52 ± 0.04 9.6 ± 0.5 gaussian NOTEARS 5 0.755 ± 0.033 0.775 ± 0.044 0.741 ± 0.035 0.568 ± 0.039 0.583 ± 0.043 0.559 ± 0.043 11.2 ± 1.2 0.318 ± 0.073 231.25 ± 41.04 16.6 ± 2.5 gaussian DAGMA 5 0.705 ± 0.045 0.765 ± 0.085 0.676 ± 0.047 0.471 ± 0.087 0.526 ± 0.125 0.443 ± 0.072 14.0 ± 3.3 0.401 ± 0.087 841.47 ± 66.42 15.6 ± 2.2 gaussian DiffAN 5 0.740 ± 0.027 0.713 ± 0.043 0.776 ± 0.027 0.346 ± 0.048 0.337 ± 0.055 0.358 ± 0.041 16.2 ± 1.9 0.529 ± 0.057 17.35 ± 0.37 18.4 ± 1.5 gaussian SciNO 5 0.755 ± 0.033 0.765 ± 0.047 0.753 ± 0.035 0.455 ± 0.044 0.461 ± 0.049 0.453 ± 0.044 13.8 ± 2.5 0.442 ± 0.051 422.57 ± 51.81 16.8 ± 1.8 gaussian Caps 5 0.816 ± 0.032 0.878 ± 0.040 0.766 ± 0.035 0.497 ± 0.054 0.540 ± 0.070 0.464 ± 0.046 11.2 ± 2.0 0.356 ± 0.041 7.29 ± 0.21 14.8 ± 1.5 gaussian PC 5 0.620 ± 0.036 0.872 ± 0.055 0.491 ± 0.042 0.392 ± 0.055 0.553 ± 0.081 0.309 ± 0.043 13.0 ± 1.3 0.492 ± 0.046 315.35 ± 83.87 9.4 ± 0.7 gaussian FCI 5 0.319 ± 0.038 1.000 0.192 ± 0.027 0.281 ± 0.049 0.860 ± 0.098 0.170 ± 0.032 14.2 ± 1.7 0.432 ± 0.043 330.16 ± 86.58 3.2 ± 0.5 multimodal Spade-Hist 5 0.779 ± 0.040 0.951 ± 0.020 0.663 ± 0.048 0.779 ± 0.040 0.951 ± 0.020 0.663 ± 0.048 6.6 ± 1.6 0.091 ± 0.029 0.54 ± 0.02 11.6 ± 0.8 multimodal Spade-Gauss 5 0.801 ± 0.051 0.913 ± 0.035 0.718 ± 0.064 0.632 ± 0.080 0.718 ± 0.080 0.569 ± 0.083 9.0 ± 2.5 0.251 ± 0.059 0.09 ± 0.01 13.0 ± 0.8 multimodal Topic 5 0.657 ± 0.011 0.599 ± 0.026 0.740 ± 0.034 0.479 ± 0.045 0.432 ± 0.035 0.546 ± 0.070 16.2 ± 1.3 0.262 ± 0.054 12.06 ± 1.10 20.6 ± 0.9 multimodal Cam 5 0.692 ± 0.025 0.566 ± 0.039 0.905 ± 0.020 0.549 ± 0.038 0.447 ± 0.035 0.722 ± 0.062 16.6 ± 1.3 0.236 ± 0.066 243.16 ± 5.87 27.2 ± 2.5 multimodal CauScale 5 0.816 ± 0.036 0.920 ± 0.049 0.739 ± 0.039 0.573 ± 0.051 0.650 ± 0.067 0.516 ± 0.043 9.4 ± 1.6 0.315 ± 0.062 5.93 ± 0.16 13.6 ± 1.2 multimodal SDCD 5 0.728 ± 0.041 0.807 ± 0.061 0.680 ± 0.055 0.558 ± 0.070 0.612 ± 0.080 0.524 ± 0.070 11.2 ± 2.1 0.304 ± 0.079 72.67 ± 2.78 15.2 ± 3.1 multimodal Globe 5 0.659 ± 0.038 0.775 ± 0.051 0.583 ± 0.051 0.602 ± 0.047 0.704 ± 0.045 0.536 ± 0.063 10.8 ± 1.1 0.279 ± 0.054 140.53 ± 7.05 12.6 ± 1.0 multimodal Das 5 0.692 ± 0.038 0.658 ± 0.019 0.739 ± 0.066 0.539 ± 0.032 0.516 ± 0.032 0.572 ± 0.044 13.4 ± 0.7 0.256 ± 0.052 14.83 ± 1.58 19.2 ± 2.6 multimodal Score 5 0.722 ± 0.024 0.620 ± 0.038 0.878 ± 0.030 0.561 ± 0.043 0.483 ± 0.049 0.679 ± 0.040 14.6 ± 1.5 0.256 ± 0.052 13.88 ± 0.40 24.4 ± 3.1 multimodal NoGAM 5 0.693 ± 0.036 0.583 ± 0.043 0.866 ± 0.038 0.651 ± 0.047 0.549 ± 0.052 0.809 ± 0.038 13.8 ± 1.9 0.129 ± 0.052 43.46 ± 0.75 25.4 ± 2.6 multimodal Avici 5 0.738 ± 0.016 0.958 ± 0.026 0.603 ± 0.022 0.456 ± 0.076 0.577 ± 0.093 0.377 ± 0.064 11.2 ± 1.5 0.409 ± 0.063 7.65 ± 0.15 10.8 ± 0.9 multimodal NOTEARS 5 0.768 ± 0.029 0.715 ± 0.043 0.837 ± 0.028 0.543 ± 0.041 0.507 ± 0.047 0.590 ± 0.035 13.0 ± 2.5 0.309 ± 0.028 187.84 ± 31.01 20.6 ± 3.6 multimodal DAGMA 5 0.775 ± 0.041 0.797 ± 0.035 0.760 ± 0.055 0.550 ± 0.053 0.574 ± 0.071 0.533 ± 0.043 11.4 ± 2.0 0.293 ± 0.060 723.17 ± 47.15 16.4 ± 2.2 multimodal DiffAN 5 0.802 ± 0.018 0.786 ± 0.017 0.821 ± 0.031 0.515 ± 0.053 0.506 ± 0.054 0.525 ± 0.053 11.8 ± 1.2 0.415 ± 0.071 17.03 ± 0.34 17.8 ± 2.0 multimodal SciNO 5 0.785 ± 0.026 0.778 ± 0.016 0.795 ± 0.039 0.463 ± 0.055 0.456 ± 0.048 0.471 ± 0.063 12.6 ± 1.2 0.470 ± 0.069 410.19 ± 12.71 17.4 ± 1.9 multimodal Caps 5 0.780 ± 0.036 0.831 ± 0.031 0.736 ± 0.042 0.621 ± 0.038 0.662 ± 0.036 0.586 ± 0.040 9.8 ± 1.9 0.256 ± 0.030 7.18 ± 0.08 15.0 ± 1.6 multimodal PC 5 0.584 ± 0.061 0.867 ± 0.057 0.464 ± 0.081 0.456 ± 0.058 0.688 ± 0.102 0.357 ± 0.056 12.2 ± 1.3 0.361 ± 0.028 342.80 ± 106.35 9.4 ± 2.1 multimodal FCI 5 0.358 ± 0.059 1.000 0.225 ± 0.045 0.358 ± 0.059 1.000 0.225 ± 0.045 13.4 ± 1.9 0.298 ± 0.033 351.93 ± 110.03 3.6 ± 0.5 unimodal_asymmetric Spade-Hist 5 0.729 ± 0.062 0.983 ± 0.017 0.603 ± 0.092 0.662 ± 0.045 0.917 ± 0.083 0.537 ± 0.049 8.2 ± 1.3 0.206 ± 0.054 0.48 ± 0.01 10.0 ± 1.0 unimodal_asymmetric Spade-Gauss 5 0.796 ± 0.032 0.919 ± 0.037 0.711 ± 0.048 0.621 ± 0.069 0.714 ± 0.076 0.556 ± 0.071 8.8 ± 1.7 0.280 ± 0.066 0.08 ± 0.00 12.8 ± 0.4 unimodal_asymmetric Topic 5 0.757 ± 0.037 0.729 ± 0.062 0.802 ± 0.026 0.539 ± 0.059 0.514 ± 0.058 0.575 ± 0.067 12.4 ± 1.4 0.303 ± 0.062 11.14 ± 1.17 18.6 ± 0.9 unimodal_asymmetric Cam 5 0.628 ± 0.034 0.482 ± 0.043 0.929 ± 0.032 0.456 ± 0.066 0.348 ± 0.051 0.677 ± 0.102 23.6 ± 4.3 0.303 ± 0.094 238.80 ± 2.24 33.2 ± 3.5 unimodal_asymmetric CauScale 5 0.816 ± 0.052 0.971 ± 0.018 0.728 ± 0.091 0.478 ± 0.093 0.551 ± 0.085 0.437 ± 0.104 10.2 ± 1.8 0.394 ± 0.075 6.09 ± 0.17 12.4 ± 1.3 unimodal_asymmetric SDCD 5 0.743 ± 0.036 0.847 ± 0.022 0.674 ± 0.062 0.578 ± 0.071 0.648 ± 0.043 0.532 ± 0.090 10.4 ± 1.9 0.342 ± 0.095 74.86 ± 1.87 13.2 ± 1.1 unimodal_asymmetric Globe 5 0.641 ± 0.053 0.873 ± 0.068 0.529 ± 0.071 0.512 ± 0.055 0.683 ± 0.019 0.428 ± 0.072 11.6 ± 1.7 0.425 ± 0.060 116.94 ± 5.96 10.0 ± 0.8 unimodal_asymmetric Das 5 0.649 ± 0.014 0.577 ± 0.027 0.752 ± 0.030 0.464 ± 0.020 0.414 ± 0.029 0.537 ± 0.022 17.2 ± 1.5 0.336 ± 0.037 14.14 ± 0.99 21.8 ± 1.1 unimodal_asymmetric Score 5 0.652 ± 0.028 0.505 ± 0.036 0.938 ± 0.029 0.433 ± 0.027 0.335 ± 0.028 0.625 ± 0.039 22.4 ± 2.8 0.336 ± 0.037 12.38 ± 0.13 31.6 ± 2.8 unimodal_asymmetric NoGAM 5 0.653 ± 0.031 0.515 ± 0.043 0.919 ± 0.039 0.472 ± 0.043 0.374 ± 0.047 0.660 ± 0.044 21.0 ± 3.0 0.301 ± 0.044 42.67 ± 0.84 30.4 ± 2.7 unimodal_asymmetric Avici 5 0.832 ± 0.026 1.000 0.716 ± 0.038 0.623 ± 0.080 0.744 ± 0.077 0.539 ± 0.080 8.2 ± 1.8 0.293 ± 0.064 8.45 ± 0.37 12.0 ± 1.0 unimodal_asymmetric NOTEARS 5 0.746 ± 0.034 0.670 ± 0.038 0.850 ± 0.042 0.506 ± 0.033 0.455 ± 0.037 0.574 ± 0.032 14.6 ± 2.0 0.379 ± 0.039 336.55 ± 97.37 21.8 ± 2.5 unimodal_asymmetric DAGMA 5 0.710 ± 0.020 0.695 ± 0.043 0.746 ± 0.053 0.374 ± 0.040 0.372 ± 0.058 0.388 ± 0.036 16.2 ± 1.9 0.524 ± 0.041 825.85 ± 84.34 18.2 ± 2.0 unimodal_asymmetric DiffAN 5 0.745 ± 0.021 0.644 ± 0.025 0.896 ± 0.047 0.413 ± 0.073 0.357 ± 0.064 0.494 ± 0.089 17.0 ± 2.0 0.464 ± 0.095 17.35 ± 0.17 23.6 ± 2.7 unimodal_asymmetric SciNO 5 0.818 ± 0.046 0.755 ± 0.069 0.908 ± 0.033 0.539 ± 0.041 0.496 ± 0.047 0.600 ± 0.043 12.8 ± 2.9 0.321 ± 0.042 394.73 ± 12.23 21.0 ± 3.0 unimodal_asymmetric Caps 5 0.818 ± 0.041 0.798 ± 0.057 0.863 ± 0.071 0.519 ± 0.039 0.506 ± 0.046 0.548 ± 0.057 12.0 ± 2.3 0.366 ± 0.034 8.19 ± 0.29 18.2 ± 1.7 unimodal_asymmetric PC 5 0.670 ± 0.082 0.885 ± 0.034 0.557 ± 0.091 0.473 ± 0.071 0.621 ± 0.046 0.394 ± 0.075 11.2 ± 1.5 0.347 ± 0.043 415.52 ± 160.34 10.6 ± 1.7 unimodal_asymmetric FCI 5 0.468 ± 0.044 1.000 0.309 ± 0.037 0.394 ± 0.056 0.850 ± 0.100 0.259 ± 0.040 12.6 ± 1.5 0.314 ± 0.069 394.81 ± 150.34 5.2 ± 0.8 unimodal_symmetric Spade-Hist 5 0.838 ± 0.031 0.988 ± 0.012 0.732 ± 0.043 0.838 ± 0.031 0.988 ± 0.012 0.732 ± 0.043 5.0 ± 1.3 0.027 ± 0.017 0.57 ± 0.03 12.4 ± 1.0 unimodal_symmetric Spade-Gauss 5 0.831 ± 0.040 0.971 ± 0.029 0.735 ± 0.058 0.748 ± 0.074 0.866 ± 0.057 0.666 ± 0.084 6.4 ± 1.7 0.149 ± 0.079 0.08 ± 0.00 12.6 ± 1.0 unimodal_symmetric Topic 5 0.756 ± 0.043 0.722 ± 0.036 0.801 ± 0.064 0.595 ± 0.055 0.569 ± 0.052 0.632 ± 0.069 11.8 ± 2.1 0.251 ± 0.052 11.33 ± 0.81 18.4 ± 0.9 unimodal_symmetric Cam 5 0.619 ± 0.019 0.468 ± 0.013 0.916 ± 0.034 0.551 ± 0.037 0.416 ± 0.027 0.816 ± 0.060 21.0 ± 2.8 0.133 ± 0.059 244.11 ± 3.15 33.2 ± 3.3 unimodal_symmetric CauScale 5 0.828 ± 0.060 0.971 ± 0.018 0.744 ± 0.089 0.488 ± 0.045 0.575 ± 0.047 0.435 ± 0.049 10.2 ± 1.6 0.402 ± 0.053 5.95 ± 0.06 12.6 ± 1.3 unimodal_symmetric SDCD 5 0.762 ± 0.043 0.790 ± 0.051 0.751 ± 0.060 0.513 ± 0.093 0.533 ± 0.091 0.502 ± 0.098 11.4 ± 1.4 0.367 ± 0.082 73.78 ± 2.57 16.2 ± 2.0 unimodal_symmetric Globe 5 0.687 ± 0.023 0.881 ± 0.044 0.573 ± 0.038 0.576 ± 0.053 0.729 ± 0.049 0.484 ± 0.058 10.2 ± 1.1 0.347 ± 0.075 120.94 ± 16.98 11.0 ± 1.1 unimodal_symmetric Das 5 0.652 ± 0.039 0.578 ± 0.043 0.752 ± 0.030 0.514 ± 0.043 0.456 ± 0.045 0.592 ± 0.040 17.0 ± 3.2 0.231 ± 0.047 14.25 ± 0.46 22.6 ± 3.1 unimodal_symmetric Score 5 0.674 ± 0.033 0.531 ± 0.033 0.929 ± 0.032 0.537 ± 0.044 0.424 ± 0.041 0.737 ± 0.045 19.0 ± 3.1 0.231 ± 0.047 14.24 ± 0.37 30.0 ± 3.1 unimodal_symmetric NoGAM 5 0.678 ± 0.043 0.541 ± 0.045 0.916 ± 0.034 0.559 ± 0.027 0.445 ± 0.028 0.757 ± 0.029 18.2 ± 3.0 0.220 ± 0.028 42.59 ± 0.44 29.4 ± 3.5 unimodal_symmetric Avici 5 0.830 ± 0.046 0.967 ± 0.033 0.743 ± 0.071 0.743 ± 0.037 0.873 ± 0.057 0.662 ± 0.052 6.4 ± 1.4 0.154 ± 0.046 7.67 ± 0.15 12.6 ± 0.5 unimodal_symmetric NOTEARS 5 0.748 ± 0.040 0.712 ± 0.087 0.835 ± 0.053 0.536 ± 0.069 0.519 ± 0.102 0.585 ± 0.048 13.4 ± 1.9 0.328 ± 0.075 238.14 ± 48.04 20.6 ± 2.6 unimodal_symmetric DAGMA 5 0.760 ± 0.049 0.785 ± 0.055 0.748 ± 0.062 0.494 ± 0.070 0.508 ± 0.068 0.489 ± 0.078 13.0 ± 3.0 0.360 ± 0.075 805.73 ± 51.06 16.4 ± 2.3 unimodal_symmetric DiffAN 5 0.786 ± 0.048 0.728 ± 0.058 0.862 ± 0.049 0.527 ± 0.090 0.491 ± 0.093 0.575 ± 0.090 12.8 ± 2.8 0.353 ± 0.072 16.89 ± 0.31 20.2 ± 2.2 unimodal_symmetric SciNO 5 0.765 ± 0.047 0.689 ± 0.051 0.869 ± 0.058 0.463 ± 0.065 0.420 ± 0.064 0.522 ± 0.071 15.0 ± 2.3 0.426 ± 0.068 410.12 ± 13.82 21.4 ± 2.0 unimodal_symmetric Caps 5 0.753 ± 0.056 0.746 ± 0.068 0.772 ± 0.053 0.563 ± 0.055 0.556 ± 0.059 0.577 ± 0.056 12.0 ± 2.3 0.296 ± 0.039 7.69 ± 0.31 17.8 ± 1.8 unimodal_symmetric PC 5 0.740 ± 0.040 0.956 ± 0.029 0.610 ± 0.051 0.620 ± 0.057 0.804 ± 0.074 0.511 ± 0.056 8.6 ± 0.9 0.283 ± 0.039 385.02 ± 117.54 10.8 ± 1.3 unimodal_symmetric FCI 5 0.458 ± 0.064 1.000 0.306 ± 0.054 0.433 ± 0.052 0.960 ± 0.040 0.288 ± 0.043 12.0 ± 1.3 0.295 ± 0.020 387.79 ± 123.78 5.4 ± 1.3 Table 19: Qualitative comparison without marginalization (n=1000n=1000, d=20d=20, degree 3, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.834 ± 0.041 0.971 ± 0.018 0.738 ± 0.057 0.808 ± 0.042 0.940 ± 0.019 0.715 ± 0.059 8.0 ± 1.9 0.056 ± 0.022 0.63 ± 0.05 19.2 ± 1.7 gaussian Spade-Gauss 5 0.802 ± 0.044 0.948 ± 0.015 0.705 ± 0.066 0.791 ± 0.053 0.932 ± 0.025 0.696 ± 0.072 8.8 ± 1.9 0.026 ± 0.026 0.11 ± 0.01 18.8 ± 1.9 gaussian Topic 5 0.737 ± 0.035 0.780 ± 0.030 0.708 ± 0.058 0.711 ± 0.024 0.755 ± 0.032 0.681 ± 0.045 13.2 ± 1.2 0.095 ± 0.034 14.52 ± 0.93 22.8 ± 1.6 gaussian Cam 5 0.735 ± 0.042 0.621 ± 0.050 0.913 ± 0.027 0.718 ± 0.057 0.608 ± 0.062 0.887 ± 0.037 17.4 ± 3.5 0.026 ± 0.026 237.08 ± 1.95 37.8 ± 2.6 gaussian CauScale 5 0.733 ± 0.037 0.863 ± 0.036 0.650 ± 0.059 0.535 ± 0.050 0.626 ± 0.044 0.478 ± 0.064 16.0 ± 1.5 0.264 ± 0.034 5.96 ± 0.15 19.0 ± 1.7 gaussian SDCD 5 0.688 ± 0.041 0.740 ± 0.015 0.655 ± 0.063 0.479 ± 0.022 0.517 ± 0.016 0.455 ± 0.038 19.8 ± 1.9 0.304 ± 0.045 76.22 ± 1.38 22.2 ± 2.0 gaussian Globe 5 0.709 ± 0.048 0.842 ± 0.034 0.626 ± 0.075 0.687 ± 0.051 0.813 ± 0.027 0.608 ± 0.079 13.0 ± 1.9 0.139 ± 0.039 176.48 ± 16.03 18.8 ± 2.1 gaussian Das 5 0.578 ± 0.031 0.541 ± 0.039 0.629 ± 0.032 0.432 ± 0.041 0.406 ± 0.046 0.466 ± 0.037 27.4 ± 2.7 0.319 ± 0.045 13.10 ± 0.19 30.0 ± 2.9 gaussian Score 5 0.684 ± 0.038 0.584 ± 0.037 0.829 ± 0.045 0.525 ± 0.045 0.451 ± 0.045 0.632 ± 0.043 24.6 ± 3.3 0.319 ± 0.045 12.19 ± 0.09 36.0 ± 2.3 gaussian NoGAM 5 0.674 ± 0.032 0.570 ± 0.030 0.829 ± 0.045 0.478 ± 0.051 0.407 ± 0.049 0.583 ± 0.054 26.8 ± 3.8 0.373 ± 0.054 39.50 ± 0.04 37.0 ± 3.2 gaussian Avici 5 0.752 ± 0.024 0.966 ± 0.015 0.620 ± 0.037 0.700 ± 0.036 0.899 ± 0.041 0.577 ± 0.037 11.2 ± 0.6 0.196 ± 0.056 7.83 ± 0.31 16.4 ± 1.7 gaussian NOTEARS 5 0.679 ± 0.010 0.632 ± 0.029 0.747 ± 0.042 0.479 ± 0.045 0.451 ± 0.056 0.521 ± 0.041 23.4 ± 2.1 0.309 ± 0.037 377.83 ± 85.44 30.4 ± 3.5 gaussian DAGMA 5 0.705 ± 0.039 0.729 ± 0.038 0.686 ± 0.048 0.491 ± 0.037 0.511 ± 0.048 0.476 ± 0.033 20.0 ± 2.7 0.321 ± 0.045 726.10 ± 50.62 24.0 ± 2.3 gaussian DiffAN 5 0.694 ± 0.022 0.651 ± 0.018 0.753 ± 0.053 0.465 ± 0.033 0.437 ± 0.036 0.503 ± 0.044 22.8 ± 1.4 0.461 ± 0.039 18.17 ± 0.52 29.2 ± 2.3 gaussian SciNO 5 0.701 ± 0.021 0.679 ± 0.057 0.745 ± 0.036 0.391 ± 0.065 0.389 ± 0.091 0.406 ± 0.048 24.8 ± 3.7 0.526 ± 0.036 391.82 ± 11.27 28.6 ± 3.5 gaussian Caps 5 0.702 ± 0.036 0.725 ± 0.039 0.686 ± 0.042 0.490 ± 0.038 0.506 ± 0.041 0.479 ± 0.041 20.2 ± 2.6 0.377 ± 0.067 6.97 ± 0.28 24.2 ± 2.5 gaussian PC 5 0.687 ± 0.068 0.935 ± 0.037 0.554 ± 0.074 0.580 ± 0.061 0.788 ± 0.028 0.468 ± 0.066 14.6 ± 2.5 0.293 ± 0.025 663.79 ± 186.92 14.6 ± 1.4 gaussian FCI 5 0.561 ± 0.060 0.987 ± 0.013 0.405 ± 0.062 0.521 ± 0.051 0.923 ± 0.032 0.375 ± 0.055 16.2 ± 1.9 0.295 ± 0.046 686.21 ± 202.25 10.4 ± 1.7 multimodal Spade-Hist 5 0.904 ± 0.018 0.991 ± 0.009 0.832 ± 0.026 0.904 ± 0.018 0.991 ± 0.009 0.832 ± 0.026 4.6 ± 1.0 0.009 ± 0.009 0.56 ± 0.01 21.2 ± 1.2 multimodal Spade-Gauss 5 0.870 ± 0.030 0.964 ± 0.017 0.798 ± 0.047 0.766 ± 0.027 0.850 ± 0.020 0.703 ± 0.041 8.6 ± 1.7 0.149 ± 0.017 0.10 ± 0.00 20.8 ± 1.2 multimodal Topic 5 0.800 ± 0.026 0.818 ± 0.022 0.789 ± 0.045 0.694 ± 0.048 0.709 ± 0.046 0.685 ± 0.059 12.6 ± 1.9 0.122 ± 0.033 15.63 ± 0.71 24.2 ± 0.7 multimodal Cam 5 0.796 ± 0.036 0.685 ± 0.040 0.953 ± 0.025 0.731 ± 0.024 0.628 ± 0.028 0.876 ± 0.018 14.6 ± 2.2 0.077 ± 0.024 236.73 ± 1.91 35.6 ± 2.8 multimodal CauScale 5 0.866 ± 0.019 0.963 ± 0.018 0.790 ± 0.030 0.710 ± 0.024 0.793 ± 0.044 0.645 ± 0.016 9.8 ± 0.9 0.214 ± 0.027 5.95 ± 0.06 20.8 ± 1.5 multimodal SDCD 5 0.783 ± 0.048 0.893 ± 0.029 0.702 ± 0.059 0.575 ± 0.070 0.654 ± 0.065 0.517 ± 0.073 14.4 ± 2.6 0.284 ± 0.080 78.54 ± 0.22 19.6 ± 1.0 multimodal Globe 5 0.750 ± 0.032 0.869 ± 0.036 0.675 ± 0.054 0.716 ± 0.035 0.829 ± 0.039 0.644 ± 0.053 12.2 ± 1.9 0.148 ± 0.044 211.19 ± 25.39 19.6 ± 1.7 multimodal Das 5 0.713 ± 0.026 0.680 ± 0.028 0.749 ± 0.025 0.599 ± 0.046 0.572 ± 0.047 0.628 ± 0.044 18.0 ± 1.2 0.179 ± 0.026 15.47 ± 0.09 28.0 ± 2.0 multimodal Score 5 0.745 ± 0.010 0.628 ± 0.011 0.923 ± 0.033 0.625 ± 0.017 0.527 ± 0.021 0.771 ± 0.020 19.6 ± 0.9 0.179 ± 0.026 14.66 ± 0.43 37.2 ± 2.3 multimodal NoGAM 5 0.734 ± 0.017 0.604 ± 0.019 0.938 ± 0.024 0.604 ± 0.028 0.498 ± 0.027 0.770 ± 0.030 21.2 ± 1.0 0.198 ± 0.035 44.62 ± 0.65 39.2 ± 1.7 multimodal Avici 5 0.611 ± 0.052 0.953 ± 0.033 0.460 ± 0.054 0.418 ± 0.066 0.675 ± 0.119 0.311 ± 0.055 18.2 ± 2.2 0.458 ± 0.083 7.62 ± 0.17 12.0 ± 1.1 multimodal NOTEARS 5 0.795 ± 0.034 0.770 ± 0.037 0.821 ± 0.033 0.544 ± 0.051 0.527 ± 0.049 0.563 ± 0.054 17.6 ± 2.9 0.331 ± 0.057 362.92 ± 57.48 27.2 ± 2.3 multimodal DAGMA 5 0.753 ± 0.029 0.758 ± 0.033 0.752 ± 0.033 0.363 ± 0.051 0.363 ± 0.047 0.364 ± 0.055 22.4 ± 2.5 0.512 ± 0.064 739.58 ± 40.44 25.0 ± 0.8 multimodal DiffAN 5 0.814 ± 0.037 0.747 ± 0.052 0.903 ± 0.018 0.593 ± 0.073 0.549 ± 0.078 0.650 ± 0.062 16.6 ± 3.2 0.314 ± 0.065 15.30 ± 0.40 31.0 ± 2.1 multimodal SciNO 5 0.821 ± 0.017 0.765 ± 0.029 0.891 ± 0.021 0.617 ± 0.048 0.577 ± 0.055 0.666 ± 0.042 15.2 ± 1.3 0.302 ± 0.045 101.98 ± 1.10 29.6 ± 1.9 multimodal Caps 5 0.824 ± 0.030 0.842 ± 0.045 0.810 ± 0.022 0.684 ± 0.054 0.702 ± 0.066 0.670 ± 0.043 12.2 ± 2.1 0.235 ± 0.044 5.68 ± 0.17 24.6 ± 1.8 multimodal PC 5 0.756 ± 0.071 0.949 ± 0.039 0.640 ± 0.083 0.576 ± 0.065 0.725 ± 0.059 0.486 ± 0.066 13.6 ± 1.8 0.386 ± 0.061 747.97 ± 190.02 17.2 ± 2.8 multimodal FCI 5 0.546 ± 0.077 1.000 0.391 ± 0.072 0.502 ± 0.062 0.936 ± 0.045 0.357 ± 0.059 16.6 ± 2.6 0.345 ± 0.060 903.23 ± 239.16 9.6 ± 1.5 unimodal_asymmetric Spade-Hist 5 0.807 ± 0.017 0.989 ± 0.011 0.685 ± 0.029 0.807 ± 0.017 0.989 ± 0.011 0.685 ± 0.029 8.4 ± 1.2 0.022 ± 0.009 0.54 ± 0.02 17.4 ± 0.6 unimodal_asymmetric Spade-Gauss 5 0.796 ± 0.025 0.959 ± 0.019 0.688 ± 0.043 0.713 ± 0.029 0.857 ± 0.008 0.617 ± 0.046 10.8 ± 1.6 0.109 ± 0.009 0.09 ± 0.00 18.0 ± 0.9 unimodal_asymmetric Topic 5 0.749 ± 0.026 0.807 ± 0.018 0.702 ± 0.039 0.625 ± 0.047 0.671 ± 0.039 0.588 ± 0.055 15.0 ± 2.3 0.200 ± 0.053 14.66 ± 1.37 21.8 ± 0.6 unimodal_asymmetric Cam 5 0.701 ± 0.022 0.594 ± 0.028 0.862 ± 0.025 0.652 ± 0.032 0.553 ± 0.035 0.800 ± 0.031 20.2 ± 2.2 0.091 ± 0.031 234.69 ± 2.53 36.8 ± 1.8 unimodal_asymmetric CauScale 5 0.729 ± 0.037 0.896 ± 0.027 0.624 ± 0.056 0.420 ± 0.011 0.520 ± 0.028 0.358 ± 0.020 18.2 ± 1.2 0.359 ± 0.032 5.89 ± 0.13 17.4 ± 1.0 unimodal_asymmetric SDCD 5 0.739 ± 0.032 0.820 ± 0.036 0.675 ± 0.035 0.524 ± 0.066 0.583 ± 0.076 0.478 ± 0.059 17.0 ± 2.2 0.308 ± 0.067 77.97 ± 0.59 20.8 ± 1.2 unimodal_asymmetric Globe 5 0.638 ± 0.032 0.789 ± 0.064 0.540 ± 0.023 0.540 ± 0.047 0.662 ± 0.058 0.460 ± 0.042 18.0 ± 2.5 0.257 ± 0.071 165.39 ± 15.61 17.6 ± 1.4 unimodal_asymmetric Das 5 0.626 ± 0.025 0.603 ± 0.032 0.662 ± 0.043 0.444 ± 0.024 0.428 ± 0.030 0.469 ± 0.031 24.6 ± 1.0 0.365 ± 0.026 15.37 ± 0.22 28.0 ± 2.5 unimodal_asymmetric Score 5 0.667 ± 0.013 0.565 ± 0.025 0.823 ± 0.023 0.461 ± 0.019 0.391 ± 0.025 0.566 ± 0.016 27.0 ± 0.8 0.365 ± 0.026 14.24 ± 0.13 36.8 ± 1.6 unimodal_asymmetric NoGAM 5 0.666 ± 0.012 0.561 ± 0.011 0.824 ± 0.035 0.466 ± 0.019 0.394 ± 0.023 0.574 ± 0.012 27.0 ± 1.2 0.366 ± 0.026 44.16 ± 0.71 37.0 ± 1.9 unimodal_asymmetric Avici 5 0.740 ± 0.033 0.953 ± 0.021 0.610 ± 0.043 0.549 ± 0.045 0.707 ± 0.050 0.453 ± 0.043 15.0 ± 2.3 0.386 ± 0.054 7.46 ± 0.11 16.0 ± 0.6 unimodal_asymmetric NOTEARS 5 0.663 ± 0.026 0.596 ± 0.014 0.754 ± 0.052 0.419 ± 0.062 0.377 ± 0.054 0.476 ± 0.077 26.8 ± 4.3 0.354 ± 0.077 290.26 ± 44.12 32.0 ± 2.9 unimodal_asymmetric DAGMA 5 0.725 ± 0.036 0.732 ± 0.014 0.723 ± 0.057 0.360 ± 0.054 0.359 ± 0.045 0.363 ± 0.063 23.2 ± 3.2 0.480 ± 0.058 699.46 ± 22.11 24.6 ± 0.7 unimodal_asymmetric DiffAN 5 0.752 ± 0.020 0.742 ± 0.042 0.771 ± 0.026 0.511 ± 0.029 0.507 ± 0.044 0.522 ± 0.019 19.2 ± 1.8 0.384 ± 0.041 15.56 ± 0.28 26.4 ± 1.4 unimodal_asymmetric SciNO 5 0.776 ± 0.022 0.784 ± 0.043 0.783 ± 0.040 0.555 ± 0.027 0.563 ± 0.048 0.557 ± 0.025 16.8 ± 0.9 0.382 ± 0.032 113.07 ± 4.62 25.4 ± 1.8 unimodal_asymmetric Caps 5 0.748 ± 0.017 0.786 ± 0.048 0.723 ± 0.025 0.542 ± 0.035 0.574 ± 0.061 0.522 ± 0.026 17.2 ± 1.0 0.361 ± 0.032 6.49 ± 0.15 23.6 ± 2.0 unimodal_asymmetric PC 5 0.671 ± 0.025 0.946 ± 0.037 0.521 ± 0.020 0.504 ± 0.057 0.712 ± 0.086 0.390 ± 0.044 16.4 ± 2.2 0.325 ± 0.049 633.54 ± 205.43 14.0 ± 1.0 unimodal_asymmetric FCI 5 0.446 ± 0.041 1.000 0.290 ± 0.033 0.372 ± 0.056 0.835 ± 0.092 0.242 ± 0.040 19.4 ± 2.1 0.346 ± 0.030 639.04 ± 192.64 7.2 ± 0.6 unimodal_symmetric Spade-Hist 5 0.870 ± 0.030 0.990 ± 0.010 0.780 ± 0.046 0.870 ± 0.030 0.990 ± 0.010 0.780 ± 0.046 6.0 ± 1.6 0.023 ± 0.010 0.54 ± 0.01 19.8 ± 1.0 unimodal_symmetric Spade-Gauss 5 0.781 ± 0.041 0.943 ± 0.027 0.674 ± 0.056 0.743 ± 0.041 0.899 ± 0.033 0.641 ± 0.052 10.4 ± 2.0 0.080 ± 0.031 0.10 ± 0.00 17.8 ± 0.7 unimodal_symmetric Topic 5 0.728 ± 0.035 0.782 ± 0.021 0.692 ± 0.060 0.642 ± 0.047 0.689 ± 0.036 0.611 ± 0.066 15.0 ± 2.1 0.155 ± 0.045 14.14 ± 1.01 22.0 ± 0.9 unimodal_symmetric Cam 5 0.738 ± 0.032 0.635 ± 0.036 0.887 ± 0.032 0.707 ± 0.034 0.607 ± 0.034 0.850 ± 0.040 17.0 ± 2.2 0.083 ± 0.022 238.56 ± 1.77 35.4 ± 1.3 unimodal_symmetric CauScale 5 0.759 ± 0.039 0.904 ± 0.031 0.659 ± 0.046 0.496 ± 0.012 0.594 ± 0.025 0.429 ± 0.014 16.4 ± 1.8 0.349 ± 0.035 5.90 ± 0.07 18.2 ± 0.4 unimodal_symmetric SDCD 5 0.702 ± 0.043 0.748 ± 0.022 0.668 ± 0.060 0.468 ± 0.049 0.500 ± 0.051 0.443 ± 0.053 19.8 ± 2.3 0.358 ± 0.059 77.24 ± 2.63 22.2 ± 1.0 unimodal_symmetric Globe 5 0.706 ± 0.062 0.878 ± 0.025 0.607 ± 0.083 0.677 ± 0.056 0.844 ± 0.036 0.580 ± 0.076 13.0 ± 2.5 0.217 ± 0.038 163.97 ± 9.35 17.0 ± 1.4 unimodal_symmetric Das 5 0.640 ± 0.028 0.600 ± 0.028 0.687 ± 0.029 0.520 ± 0.046 0.488 ± 0.043 0.558 ± 0.049 23.0 ± 2.8 0.253 ± 0.044 15.39 ± 0.15 29.0 ± 1.7 unimodal_symmetric Score 5 0.675 ± 0.043 0.585 ± 0.048 0.806 ± 0.042 0.558 ± 0.051 0.484 ± 0.052 0.665 ± 0.052 23.8 ± 3.9 0.253 ± 0.044 14.97 ± 0.36 35.2 ± 2.4 unimodal_symmetric NoGAM 5 0.717 ± 0.027 0.634 ± 0.018 0.829 ± 0.050 0.607 ± 0.020 0.538 ± 0.015 0.702 ± 0.038 19.8 ± 2.1 0.226 ± 0.015 45.10 ± 0.56 32.8 ± 1.3 unimodal_symmetric Avici 5 0.810 ± 0.048 0.980 ± 0.012 0.703 ± 0.067 0.755 ± 0.051 0.911 ± 0.020 0.655 ± 0.066 9.6 ± 2.5 0.141 ± 0.049 9.89 ± 1.71 17.8 ± 1.1 unimodal_symmetric NOTEARS 5 0.707 ± 0.034 0.668 ± 0.038 0.752 ± 0.032 0.504 ± 0.050 0.477 ± 0.052 0.535 ± 0.050 21.4 ± 2.6 0.303 ± 0.047 290.75 ± 25.05 28.6 ± 1.6 unimodal_symmetric DAGMA 5 0.730 ± 0.043 0.731 ± 0.036 0.737 ± 0.063 0.395 ± 0.051 0.394 ± 0.047 0.401 ± 0.058 22.4 ± 3.1 0.425 ± 0.064 755.10 ± 30.69 25.2 ± 1.1 unimodal_symmetric DiffAN 5 0.707 ± 0.033 0.659 ± 0.040 0.774 ± 0.046 0.478 ± 0.044 0.450 ± 0.056 0.517 ± 0.034 22.4 ± 1.9 0.367 ± 0.053 15.40 ± 0.13 29.6 ± 0.9 unimodal_symmetric SciNO 5 0.760 ± 0.036 0.761 ± 0.041 0.766 ± 0.049 0.597 ± 0.042 0.599 ± 0.049 0.600 ± 0.044 16.8 ± 3.2 0.279 ± 0.050 105.48 ± 0.92 25.6 ± 2.0 unimodal_symmetric Caps 5 0.760 ± 0.034 0.817 ± 0.052 0.716 ± 0.032 0.609 ± 0.043 0.654 ± 0.056 0.573 ± 0.041 15.6 ± 3.1 0.307 ± 0.036 6.15 ± 0.15 22.6 ± 2.4 unimodal_symmetric PC 5 0.654 ± 0.081 0.942 ± 0.026 0.518 ± 0.089 0.535 ± 0.092 0.755 ± 0.055 0.428 ± 0.095 15.4 ± 3.0 0.288 ± 0.063 626.37 ± 68.75 13.6 ± 2.0 unimodal_symmetric FCI 5 0.497 ± 0.045 1.000 0.336 ± 0.041 0.476 ± 0.056 0.947 ± 0.034 0.322 ± 0.048 17.4 ± 2.2 0.277 ± 0.055 675.14 ± 61.48 8.4 ± 0.9 Table 20: Qualitative comparison without marginalization (n=1000n=1000, d=20d=20, degree 3, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.681 ± 0.023 0.931 ± 0.023 0.541 ± 0.030 0.594 ± 0.056 0.813 ± 0.080 0.471 ± 0.049 14.4 ± 1.3 0.302 ± 0.081 0.53 ± 0.02 14.6 ± 0.7 gaussian Spade-Gauss 5 0.734 ± 0.035 0.973 ± 0.016 0.593 ± 0.039 0.702 ± 0.037 0.931 ± 0.031 0.566 ± 0.039 11.6 ± 1.9 0.146 ± 0.029 0.09 ± 0.00 15.2 ± 0.5 gaussian Topic 5 0.741 ± 0.016 0.813 ± 0.014 0.682 ± 0.019 0.673 ± 0.024 0.738 ± 0.024 0.619 ± 0.024 13.8 ± 1.7 0.148 ± 0.014 13.49 ± 0.95 21.2 ± 1.2 gaussian Cam 5 0.726 ± 0.008 0.599 ± 0.004 0.923 ± 0.019 0.675 ± 0.018 0.557 ± 0.015 0.857 ± 0.026 19.4 ± 1.9 0.097 ± 0.028 238.63 ± 2.68 39.0 ± 2.6 gaussian CauScale 5 0.745 ± 0.048 0.950 ± 0.013 0.621 ± 0.057 0.488 ± 0.058 0.620 ± 0.056 0.407 ± 0.056 16.0 ± 2.3 0.385 ± 0.044 5.83 ± 0.07 16.2 ± 0.9 gaussian SDCD 5 0.741 ± 0.029 0.897 ± 0.035 0.641 ± 0.048 0.634 ± 0.052 0.761 ± 0.042 0.551 ± 0.063 13.8 ± 2.4 0.291 ± 0.062 76.18 ± 3.53 18.0 ± 1.2 gaussian Globe 5 0.547 ± 0.045 0.794 ± 0.046 0.426 ± 0.047 0.454 ± 0.010 0.673 ± 0.059 0.351 ± 0.019 19.4 ± 1.4 0.394 ± 0.034 159.67 ± 10.50 13.4 ± 1.1 gaussian Das 5 0.657 ± 0.044 0.660 ± 0.058 0.660 ± 0.040 0.569 ± 0.055 0.574 ± 0.069 0.570 ± 0.046 19.4 ± 2.0 0.175 ± 0.022 12.90 ± 0.24 25.4 ± 0.5 gaussian Score 5 0.724 ± 0.025 0.616 ± 0.020 0.882 ± 0.045 0.624 ± 0.026 0.532 ± 0.026 0.758 ± 0.035 20.0 ± 1.6 0.175 ± 0.022 12.59 ± 0.15 36.0 ± 1.5 gaussian NoGAM 5 0.721 ± 0.032 0.616 ± 0.027 0.873 ± 0.047 0.601 ± 0.040 0.514 ± 0.037 0.726 ± 0.048 20.8 ± 2.4 0.222 ± 0.035 39.45 ± 0.06 35.8 ± 2.0 gaussian Avici 5 0.721 ± 0.044 0.976 ± 0.024 0.580 ± 0.052 0.621 ± 0.058 0.837 ± 0.047 0.502 ± 0.057 13.4 ± 2.5 0.317 ± 0.068 7.84 ± 0.28 14.8 ± 1.0 gaussian NOTEARS 5 0.710 ± 0.037 0.699 ± 0.059 0.737 ± 0.043 0.531 ± 0.051 0.518 ± 0.052 0.557 ± 0.069 20.4 ± 3.5 0.317 ± 0.064 290.88 ± 97.99 27.6 ± 3.7 gaussian DAGMA 5 0.675 ± 0.031 0.730 ± 0.053 0.632 ± 0.022 0.391 ± 0.065 0.427 ± 0.084 0.364 ± 0.053 22.6 ± 3.1 0.485 ± 0.050 781.14 ± 72.36 22.4 ± 2.3 gaussian DiffAN 5 0.795 ± 0.010 0.795 ± 0.017 0.800 ± 0.028 0.533 ± 0.022 0.532 ± 0.020 0.537 ± 0.031 17.0 ± 1.0 0.348 ± 0.034 17.08 ± 0.12 25.4 ± 1.3 gaussian SciNO 5 0.792 ± 0.018 0.772 ± 0.026 0.815 ± 0.021 0.547 ± 0.032 0.534 ± 0.035 0.563 ± 0.033 17.4 ± 1.9 0.319 ± 0.035 425.78 ± 11.90 27.0 ± 2.6 gaussian Caps 5 0.767 ± 0.021 0.824 ± 0.023 0.720 ± 0.025 0.625 ± 0.025 0.670 ± 0.025 0.586 ± 0.028 14.4 ± 1.1 0.243 ± 0.029 7.00 ± 0.11 22.2 ± 1.7 gaussian PC 5 0.747 ± 0.018 0.973 ± 0.016 0.608 ± 0.024 0.532 ± 0.034 0.692 ± 0.036 0.434 ± 0.031 14.8 ± 1.2 0.387 ± 0.027 589.98 ± 113.20 15.8 ± 1.1 gaussian FCI 5 0.517 ± 0.059 0.980 ± 0.020 0.360 ± 0.055 0.482 ± 0.042 0.930 ± 0.049 0.334 ± 0.039 17.4 ± 2.2 0.325 ± 0.018 620.40 ± 121.34 9.0 ± 0.9 multimodal Spade-Hist 5 0.629 ± 0.023 0.966 ± 0.021 0.467 ± 0.025 0.629 ± 0.023 0.966 ± 0.021 0.467 ± 0.025 14.0 ± 1.4 0.113 ± 0.018 0.53 ± 0.01 12.2 ± 0.7 multimodal Spade-Gauss 5 0.664 ± 0.049 0.933 ± 0.035 0.522 ± 0.056 0.584 ± 0.050 0.818 ± 0.038 0.459 ± 0.054 15.0 ± 2.6 0.266 ± 0.036 0.09 ± 0.01 14.0 ± 1.1 multimodal Topic 5 0.648 ± 0.057 0.710 ± 0.068 0.606 ± 0.065 0.538 ± 0.043 0.587 ± 0.048 0.504 ± 0.052 19.2 ± 2.7 0.296 ± 0.031 13.69 ± 0.71 21.6 ± 1.3 multimodal Cam 5 0.734 ± 0.021 0.636 ± 0.029 0.883 ± 0.052 0.593 ± 0.023 0.516 ± 0.036 0.708 ± 0.022 20.4 ± 1.5 0.248 ± 0.021 233.89 ± 2.92 35.2 ± 2.7 multimodal CauScale 5 0.687 ± 0.034 0.958 ± 0.018 0.538 ± 0.040 0.566 ± 0.039 0.795 ± 0.062 0.442 ± 0.030 14.6 ± 0.7 0.373 ± 0.046 5.91 ± 0.06 14.2 ± 1.2 multimodal SDCD 5 0.648 ± 0.042 0.819 ± 0.050 0.541 ± 0.046 0.501 ± 0.059 0.640 ± 0.093 0.416 ± 0.047 18.2 ± 2.8 0.412 ± 0.046 76.37 ± 1.18 16.8 ± 1.7 multimodal Globe 5 0.590 ± 0.053 0.818 ± 0.052 0.464 ± 0.051 0.562 ± 0.055 0.781 ± 0.058 0.443 ± 0.052 17.0 ± 2.9 0.371 ± 0.034 145.18 ± 10.14 14.2 ± 1.1 multimodal Das 5 0.673 ± 0.028 0.694 ± 0.043 0.670 ± 0.049 0.575 ± 0.016 0.593 ± 0.034 0.572 ± 0.037 19.2 ± 2.7 0.161 ± 0.017 15.86 ± 0.30 25.0 ± 3.2 multimodal Score 5 0.709 ± 0.021 0.643 ± 0.022 0.802 ± 0.048 0.623 ± 0.012 0.565 ± 0.019 0.704 ± 0.036 19.2 ± 2.1 0.161 ± 0.017 14.58 ± 0.04 31.8 ± 3.2 multimodal NoGAM 5 0.718 ± 0.024 0.645 ± 0.029 0.818 ± 0.036 0.615 ± 0.040 0.554 ± 0.044 0.697 ± 0.038 19.6 ± 2.9 0.173 ± 0.036 45.41 ± 0.25 32.4 ± 3.0 multimodal Avici 5 0.656 ± 0.033 0.933 ± 0.028 0.514 ± 0.046 0.483 ± 0.019 0.691 ± 0.040 0.377 ± 0.025 17.0 ± 1.7 0.363 ± 0.042 7.56 ± 0.13 13.8 ± 1.0 multimodal NOTEARS 5 0.686 ± 0.037 0.664 ± 0.042 0.715 ± 0.042 0.429 ± 0.060 0.417 ± 0.064 0.444 ± 0.056 23.8 ± 3.5 0.403 ± 0.082 413.10 ± 140.62 27.4 ± 1.8 multimodal DAGMA 5 0.682 ± 0.030 0.774 ± 0.031 0.620 ± 0.050 0.443 ± 0.029 0.502 ± 0.028 0.404 ± 0.040 20.0 ± 1.7 0.413 ± 0.089 846.60 ± 56.90 20.2 ± 1.6 multimodal DiffAN 5 0.755 ± 0.019 0.817 ± 0.016 0.705 ± 0.031 0.539 ± 0.025 0.584 ± 0.028 0.503 ± 0.028 16.6 ± 1.0 0.277 ± 0.022 14.58 ± 0.21 21.8 ± 1.4 multimodal SciNO 5 0.735 ± 0.037 0.804 ± 0.021 0.692 ± 0.066 0.499 ± 0.041 0.547 ± 0.046 0.467 ± 0.048 18.0 ± 1.9 0.317 ± 0.042 105.23 ± 1.26 21.8 ± 2.3 multimodal Caps 5 0.742 ± 0.020 0.851 ± 0.023 0.659 ± 0.024 0.594 ± 0.032 0.683 ± 0.039 0.527 ± 0.029 15.0 ± 1.5 0.202 ± 0.027 6.18 ± 0.26 19.6 ± 1.4 multimodal PC 5 0.801 ± 0.052 0.965 ± 0.016 0.694 ± 0.069 0.637 ± 0.058 0.771 ± 0.064 0.550 ± 0.062 12.2 ± 2.2 0.306 ± 0.035 653.08 ± 150.85 18.4 ± 2.5 multimodal FCI 5 0.483 ± 0.061 1.000 0.327 ± 0.056 0.422 ± 0.078 0.848 ± 0.056 0.289 ± 0.066 18.2 ± 2.3 0.341 ± 0.050 719.63 ± 182.53 8.2 ± 1.3 unimodal_asymmetric Spade-Hist 5 0.686 ± 0.059 0.962 ± 0.025 0.541 ± 0.060 0.616 ± 0.096 0.847 ± 0.101 0.489 ± 0.083 13.2 ± 2.0 0.259 ± 0.082 0.51 ± 0.01 14.2 ± 1.7 unimodal_asymmetric Spade-Gauss 5 0.759 ± 0.024 0.978 ± 0.022 0.626 ± 0.038 0.645 ± 0.047 0.835 ± 0.070 0.530 ± 0.039 12.2 ± 0.9 0.277 ± 0.056 0.09 ± 0.00 16.2 ± 1.2 unimodal_asymmetric Topic 5 0.676 ± 0.026 0.727 ± 0.050 0.639 ± 0.028 0.486 ± 0.049 0.529 ± 0.072 0.456 ± 0.036 19.8 ± 1.3 0.349 ± 0.065 13.18 ± 0.87 22.4 ± 1.4 unimodal_asymmetric Cam 5 0.686 ± 0.026 0.549 ± 0.031 0.921 ± 0.017 0.516 ± 0.063 0.415 ± 0.057 0.687 ± 0.068 27.0 ± 2.9 0.293 ± 0.079 242.57 ± 5.64 42.8 ± 2.5 unimodal_asymmetric CauScale 5 0.720 ± 0.053 0.965 ± 0.023 0.585 ± 0.067 0.476 ± 0.033 0.644 ± 0.044 0.385 ± 0.041 16.4 ± 2.2 0.395 ± 0.033 5.78 ± 0.08 15.2 ± 1.5 unimodal_asymmetric SDCD 5 0.659 ± 0.047 0.772 ± 0.031 0.590 ± 0.064 0.460 ± 0.053 0.535 ± 0.040 0.415 ± 0.067 19.4 ± 1.6 0.412 ± 0.045 77.03 ± 0.53 20.0 ± 3.8 unimodal_asymmetric Globe 5 0.524 ± 0.060 0.870 ± 0.047 0.390 ± 0.062 0.391 ± 0.049 0.669 ± 0.091 0.290 ± 0.051 19.8 ± 2.1 0.512 ± 0.066 116.21 ± 11.12 11.6 ± 2.1 unimodal_asymmetric Das 5 0.651 ± 0.034 0.660 ± 0.046 0.647 ± 0.032 0.453 ± 0.071 0.459 ± 0.072 0.450 ± 0.073 22.6 ± 2.9 0.343 ± 0.071 15.31 ± 0.21 25.6 ± 3.2 unimodal_asymmetric Score 5 0.716 ± 0.031 0.605 ± 0.034 0.880 ± 0.025 0.493 ± 0.053 0.417 ± 0.049 0.603 ± 0.059 24.6 ± 2.7 0.343 ± 0.071 14.69 ± 0.23 37.2 ± 3.1 unimodal_asymmetric NoGAM 5 0.710 ± 0.035 0.597 ± 0.035 0.880 ± 0.037 0.476 ± 0.049 0.401 ± 0.045 0.588 ± 0.054 25.2 ± 2.0 0.367 ± 0.066 44.54 ± 0.87 37.4 ± 1.9 unimodal_asymmetric Avici 5 0.724 ± 0.025 0.963 ± 0.016 0.582 ± 0.030 0.482 ± 0.026 0.642 ± 0.035 0.387 ± 0.023 16.0 ± 0.7 0.400 ± 0.069 7.54 ± 0.10 15.4 ± 1.4 unimodal_asymmetric NOTEARS 5 0.722 ± 0.029 0.665 ± 0.033 0.804 ± 0.054 0.504 ± 0.020 0.465 ± 0.030 0.560 ± 0.033 22.0 ± 2.5 0.290 ± 0.059 250.99 ± 47.49 31.4 ± 4.5 unimodal_asymmetric DAGMA 5 0.692 ± 0.033 0.698 ± 0.007 0.701 ± 0.067 0.361 ± 0.053 0.362 ± 0.044 0.368 ± 0.066 23.6 ± 1.3 0.468 ± 0.058 730.65 ± 24.32 25.6 ± 3.4 unimodal_asymmetric DiffAN 5 0.739 ± 0.044 0.681 ± 0.047 0.816 ± 0.054 0.447 ± 0.055 0.411 ± 0.054 0.494 ± 0.062 23.0 ± 3.5 0.430 ± 0.046 14.55 ± 0.37 30.8 ± 3.4 unimodal_asymmetric SciNO 5 0.731 ± 0.036 0.692 ± 0.035 0.782 ± 0.053 0.478 ± 0.060 0.457 ± 0.062 0.504 ± 0.056 21.2 ± 2.2 0.374 ± 0.065 106.22 ± 1.20 28.6 ± 2.2 unimodal_asymmetric Caps 5 0.721 ± 0.022 0.716 ± 0.043 0.739 ± 0.037 0.478 ± 0.068 0.482 ± 0.081 0.482 ± 0.059 20.6 ± 2.1 0.362 ± 0.075 6.99 ± 0.28 26.6 ± 2.7 unimodal_asymmetric PC 5 0.693 ± 0.039 0.983 ± 0.017 0.540 ± 0.044 0.527 ± 0.060 0.740 ± 0.060 0.413 ± 0.054 15.2 ± 1.9 0.365 ± 0.049 690.29 ± 201.14 14.0 ± 1.8 unimodal_asymmetric FCI 5 0.515 ± 0.051 1.000 0.354 ± 0.049 0.477 ± 0.059 0.919 ± 0.034 0.329 ± 0.054 17.2 ± 2.1 0.326 ± 0.065 728.96 ± 227.48 8.8 ± 1.0 unimodal_symmetric Spade-Hist 5 0.684 ± 0.032 0.960 ± 0.016 0.538 ± 0.041 0.654 ± 0.041 0.914 ± 0.012 0.516 ± 0.048 13.2 ± 2.0 0.183 ± 0.028 0.52 ± 0.01 14.0 ± 0.7 unimodal_symmetric Spade-Gauss 5 0.731 ± 0.034 0.908 ± 0.010 0.619 ± 0.050 0.650 ± 0.031 0.811 ± 0.035 0.549 ± 0.042 13.2 ± 1.6 0.200 ± 0.051 0.09 ± 0.00 17.0 ± 0.8 unimodal_symmetric Topic 5 0.727 ± 0.022 0.781 ± 0.038 0.692 ± 0.046 0.585 ± 0.028 0.624 ± 0.008 0.561 ± 0.054 16.4 ± 1.2 0.227 ± 0.037 14.23 ± 0.67 22.4 ± 1.5 unimodal_symmetric Cam 5 0.650 ± 0.028 0.512 ± 0.026 0.894 ± 0.036 0.509 ± 0.037 0.401 ± 0.032 0.699 ± 0.047 29.8 ± 4.1 0.251 ± 0.041 232.43 ± 1.94 44.6 ± 3.9 unimodal_symmetric CauScale 5 0.713 ± 0.052 0.907 ± 0.033 0.593 ± 0.061 0.503 ± 0.033 0.647 ± 0.052 0.416 ± 0.032 16.6 ± 2.3 0.345 ± 0.066 5.96 ± 0.11 16.4 ± 1.4 unimodal_symmetric SDCD 5 0.708 ± 0.048 0.779 ± 0.026 0.656 ± 0.069 0.573 ± 0.054 0.629 ± 0.041 0.532 ± 0.068 16.6 ± 2.3 0.272 ± 0.059 77.08 ± 1.83 21.0 ± 1.4 unimodal_symmetric Globe 5 0.582 ± 0.044 0.909 ± 0.041 0.438 ± 0.049 0.487 ± 0.035 0.768 ± 0.057 0.364 ± 0.033 17.6 ± 2.0 0.411 ± 0.032 134.17 ± 4.30 12.0 ± 1.0 unimodal_symmetric Das 5 0.582 ± 0.031 0.550 ± 0.040 0.628 ± 0.040 0.445 ± 0.042 0.425 ± 0.054 0.477 ± 0.037 27.4 ± 4.7 0.245 ± 0.028 15.11 ± 0.07 29.8 ± 4.2 unimodal_symmetric Score 5 0.637 ± 0.018 0.510 ± 0.019 0.856 ± 0.044 0.473 ± 0.029 0.380 ± 0.029 0.633 ± 0.037 30.6 ± 3.7 0.245 ± 0.028 14.34 ± 0.25 42.8 ± 4.3 unimodal_symmetric NoGAM 5 0.683 ± 0.029 0.568 ± 0.036 0.869 ± 0.039 0.523 ± 0.044 0.437 ± 0.047 0.661 ± 0.044 26.8 ± 5.6 0.230 ± 0.044 43.61 ± 0.56 40.0 ± 5.8 unimodal_symmetric Avici 5 0.742 ± 0.030 0.946 ± 0.028 0.612 ± 0.033 0.670 ± 0.043 0.852 ± 0.040 0.554 ± 0.043 12.4 ± 1.9 0.214 ± 0.052 7.60 ± 0.12 16.4 ± 1.2 unimodal_symmetric NOTEARS 5 0.710 ± 0.022 0.644 ± 0.036 0.813 ± 0.058 0.589 ± 0.038 0.531 ± 0.035 0.679 ± 0.074 20.4 ± 2.5 0.214 ± 0.052 289.91 ± 59.99 32.8 ± 4.5 unimodal_symmetric DAGMA 5 0.734 ± 0.044 0.772 ± 0.054 0.707 ± 0.047 0.487 ± 0.057 0.513 ± 0.066 0.467 ± 0.052 19.6 ± 3.7 0.360 ± 0.055 732.83 ± 32.52 24.0 ± 3.8 unimodal_symmetric DiffAN 5 0.715 ± 0.018 0.646 ± 0.037 0.821 ± 0.050 0.392 ± 0.022 0.357 ± 0.034 0.445 ± 0.019 26.6 ± 4.1 0.471 ± 0.034 15.04 ± 0.43 33.2 ± 4.9 unimodal_symmetric SciNO 5 0.742 ± 0.035 0.740 ± 0.066 0.774 ± 0.057 0.500 ± 0.066 0.503 ± 0.087 0.516 ± 0.064 21.0 ± 5.0 0.350 ± 0.072 112.91 ± 6.25 28.2 ± 5.5 unimodal_symmetric Caps 5 0.731 ± 0.026 0.761 ± 0.067 0.719 ± 0.025 0.515 ± 0.038 0.538 ± 0.065 0.507 ± 0.036 19.4 ± 3.2 0.308 ± 0.041 5.96 ± 0.17 25.0 ± 3.6 unimodal_symmetric PC 5 0.745 ± 0.045 0.951 ± 0.021 0.623 ± 0.062 0.581 ± 0.044 0.744 ± 0.050 0.484 ± 0.052 13.8 ± 1.2 0.309 ± 0.050 614.37 ± 122.47 16.6 ± 1.9 unimodal_symmetric FCI 5 0.476 ± 0.063 1.000 0.320 ± 0.050 0.476 ± 0.063 1.000 0.320 ± 0.050 17.2 ± 1.6 0.309 ± 0.074 666.61 ± 138.07 8.2 ± 1.4 Table 21: Qualitative comparison without marginalization (n=1000n=1000, d=20d=20, degree 4, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.780 ± 0.034 0.967 ± 0.016 0.659 ± 0.047 0.759 ± 0.038 0.941 ± 0.033 0.641 ± 0.048 13.8 ± 2.5 0.052 ± 0.021 0.59 ± 0.01 23.6 ± 0.8 gaussian Spade-Gauss 5 0.785 ± 0.035 0.983 ± 0.011 0.660 ± 0.051 0.785 ± 0.035 0.983 ± 0.011 0.660 ± 0.051 12.8 ± 2.4 0.027 ± 0.018 0.11 ± 0.01 23.2 ± 0.7 gaussian Topic 5 0.726 ± 0.024 0.859 ± 0.045 0.640 ± 0.046 0.699 ± 0.031 0.826 ± 0.045 0.618 ± 0.052 17.6 ± 1.7 0.092 ± 0.030 17.30 ± 1.32 26.0 ± 1.2 gaussian Cam 5 0.815 ± 0.014 0.734 ± 0.023 0.922 ± 0.033 0.815 ± 0.014 0.734 ± 0.023 0.922 ± 0.033 14.6 ± 1.2 0.006 ± 0.006 242.91 ± 1.35 43.8 ± 1.1 gaussian CauScale 5 0.693 ± 0.043 0.915 ± 0.028 0.564 ± 0.052 0.501 ± 0.036 0.660 ± 0.019 0.408 ± 0.042 23.0 ± 2.9 0.377 ± 0.025 6.19 ± 0.26 21.2 ± 0.4 gaussian SDCD 5 0.654 ± 0.037 0.746 ± 0.029 0.584 ± 0.043 0.403 ± 0.065 0.456 ± 0.067 0.362 ± 0.062 29.8 ± 3.9 0.476 ± 0.068 77.22 ± 2.55 27.2 ± 1.4 gaussian Globe 5 0.664 ± 0.023 0.879 ± 0.025 0.535 ± 0.025 0.641 ± 0.014 0.850 ± 0.017 0.517 ± 0.020 19.8 ± 2.0 0.208 ± 0.035 197.57 ± 20.56 21.6 ± 2.2 gaussian Das 5 0.652 ± 0.025 0.665 ± 0.041 0.655 ± 0.042 0.482 ± 0.044 0.498 ± 0.067 0.477 ± 0.031 30.2 ± 2.6 0.271 ± 0.051 20.82 ± 3.98 35.0 ± 3.4 gaussian Score 5 0.757 ± 0.014 0.664 ± 0.021 0.885 ± 0.024 0.580 ± 0.034 0.511 ± 0.041 0.674 ± 0.023 27.2 ± 2.4 0.271 ± 0.051 13.33 ± 0.55 46.8 ± 2.8 gaussian NoGAM 5 0.768 ± 0.011 0.685 ± 0.020 0.879 ± 0.025 0.656 ± 0.027 0.587 ± 0.037 0.748 ± 0.014 23.0 ± 1.7 0.203 ± 0.040 41.47 ± 0.84 45.0 ± 2.6 gaussian Avici 5 0.651 ± 0.040 0.956 ± 0.011 0.501 ± 0.049 0.578 ± 0.061 0.834 ± 0.038 0.448 ± 0.064 21.0 ± 3.3 0.294 ± 0.069 8.80 ± 0.34 18.2 ± 0.7 gaussian NOTEARS 5 0.724 ± 0.044 0.710 ± 0.051 0.744 ± 0.046 0.467 ± 0.067 0.461 ± 0.073 0.476 ± 0.063 30.4 ± 6.6 0.369 ± 0.066 307.33 ± 40.73 37.4 ± 4.1 gaussian DAGMA 5 0.646 ± 0.038 0.659 ± 0.028 0.639 ± 0.055 0.301 ± 0.084 0.307 ± 0.087 0.299 ± 0.083 36.8 ± 5.4 0.503 ± 0.087 738.55 ± 36.77 33.6 ± 1.8 gaussian DiffAN 5 0.778 ± 0.037 0.773 ± 0.058 0.793 ± 0.032 0.541 ± 0.066 0.544 ± 0.081 0.545 ± 0.052 24.2 ± 3.5 0.359 ± 0.057 17.50 ± 0.41 36.2 ± 1.7 gaussian SciNO 5 0.738 ± 0.019 0.736 ± 0.036 0.745 ± 0.018 0.458 ± 0.060 0.461 ± 0.070 0.458 ± 0.052 28.4 ± 3.0 0.441 ± 0.066 387.21 ± 16.87 35.6 ± 1.9 gaussian Caps 5 0.776 ± 0.020 0.836 ± 0.014 0.729 ± 0.039 0.595 ± 0.042 0.640 ± 0.044 0.560 ± 0.048 20.6 ± 1.7 0.297 ± 0.035 6.85 ± 0.15 30.4 ± 1.5 gaussian PC 5 0.782 ± 0.032 0.960 ± 0.018 0.666 ± 0.048 0.534 ± 0.051 0.649 ± 0.033 0.458 ± 0.060 20.6 ± 3.2 0.377 ± 0.034 1502.41 ± 313.47 24.0 ± 0.8 gaussian FCI 5 0.460 ± 0.068 0.967 ± 0.033 0.310 ± 0.057 0.383 ± 0.077 0.787 ± 0.067 0.260 ± 0.062 26.6 ± 3.4 0.369 ± 0.076 1912.55 ± 427.16 10.8 ± 1.4 multimodal Spade-Hist 5 0.856 ± 0.038 0.993 ± 0.007 0.759 ± 0.059 0.856 ± 0.038 0.993 ± 0.007 0.759 ± 0.059 9.0 ± 2.5 0.041 ± 0.022 0.60 ± 0.03 26.6 ± 1.9 multimodal Spade-Gauss 5 0.775 ± 0.042 0.917 ± 0.029 0.676 ± 0.052 0.697 ± 0.064 0.821 ± 0.054 0.609 ± 0.069 16.4 ± 3.5 0.108 ± 0.034 0.12 ± 0.01 25.6 ± 1.4 multimodal Topic 5 0.752 ± 0.032 0.836 ± 0.024 0.687 ± 0.042 0.634 ± 0.051 0.704 ± 0.046 0.580 ± 0.056 20.0 ± 3.2 0.178 ± 0.042 18.51 ± 1.28 28.6 ± 1.3 multimodal Cam 5 0.767 ± 0.012 0.701 ± 0.023 0.854 ± 0.030 0.728 ± 0.028 0.664 ± 0.028 0.812 ± 0.047 19.8 ± 2.2 0.070 ± 0.026 238.72 ± 2.07 42.6 ± 1.6 multimodal CauScale 5 0.684 ± 0.062 0.874 ± 0.029 0.572 ± 0.074 0.625 ± 0.047 0.803 ± 0.029 0.521 ± 0.059 20.0 ± 3.5 0.169 ± 0.026 6.05 ± 0.09 22.6 ± 2.2 multimodal SDCD 5 0.688 ± 0.058 0.770 ± 0.043 0.627 ± 0.068 0.557 ± 0.070 0.620 ± 0.060 0.510 ± 0.076 24.0 ± 4.4 0.245 ± 0.030 79.01 ± 1.10 28.2 ± 2.1 multimodal Globe 5 0.623 ± 0.033 0.829 ± 0.014 0.505 ± 0.043 0.558 ± 0.028 0.744 ± 0.017 0.453 ± 0.038 23.0 ± 1.8 0.321 ± 0.027 218.61 ± 22.51 21.2 ± 1.7 multimodal Das 5 0.676 ± 0.020 0.666 ± 0.030 0.697 ± 0.042 0.563 ± 0.036 0.555 ± 0.044 0.579 ± 0.045 27.2 ± 1.8 0.200 ± 0.019 13.37 ± 0.60 36.4 ± 1.0 multimodal Score 5 0.699 ± 0.011 0.615 ± 0.032 0.826 ± 0.040 0.576 ± 0.014 0.506 ± 0.023 0.683 ± 0.044 29.8 ± 0.9 0.200 ± 0.019 12.29 ± 0.31 47.2 ± 2.6 multimodal NoGAM 5 0.706 ± 0.012 0.615 ± 0.013 0.837 ± 0.040 0.614 ± 0.022 0.534 ± 0.011 0.730 ± 0.050 28.2 ± 2.3 0.172 ± 0.027 43.70 ± 1.28 47.6 ± 2.8 multimodal Avici 5 0.726 ± 0.037 0.892 ± 0.017 0.616 ± 0.047 0.605 ± 0.055 0.739 ± 0.044 0.515 ± 0.059 20.0 ± 3.1 0.243 ± 0.064 8.26 ± 0.36 24.0 ± 1.5 multimodal NOTEARS 5 0.680 ± 0.059 0.643 ± 0.064 0.731 ± 0.063 0.414 ± 0.083 0.392 ± 0.081 0.443 ± 0.088 36.0 ± 8.5 0.406 ± 0.080 440.40 ± 63.23 41.2 ± 5.5 multimodal DAGMA 5 0.676 ± 0.060 0.694 ± 0.062 0.672 ± 0.077 0.326 ± 0.059 0.335 ± 0.061 0.324 ± 0.065 35.6 ± 6.8 0.509 ± 0.033 729.03 ± 32.99 34.2 ± 3.5 multimodal DiffAN 5 0.778 ± 0.031 0.778 ± 0.023 0.783 ± 0.050 0.571 ± 0.063 0.570 ± 0.059 0.575 ± 0.071 23.4 ± 3.9 0.309 ± 0.070 17.69 ± 0.76 35.2 ± 2.3 multimodal SciNO 5 0.720 ± 0.015 0.702 ± 0.051 0.764 ± 0.044 0.490 ± 0.058 0.486 ± 0.083 0.513 ± 0.048 29.2 ± 2.9 0.372 ± 0.072 367.16 ± 7.42 38.6 ± 3.1 multimodal Caps 5 0.733 ± 0.034 0.805 ± 0.047 0.683 ± 0.048 0.625 ± 0.041 0.685 ± 0.045 0.585 ± 0.055 20.6 ± 1.7 0.226 ± 0.036 6.72 ± 0.20 29.6 ± 1.4 multimodal PC 5 0.642 ± 0.085 0.981 ± 0.019 0.502 ± 0.094 0.398 ± 0.034 0.644 ± 0.084 0.305 ± 0.043 25.0 ± 2.7 0.396 ± 0.042 1464.45 ± 362.81 17.4 ± 2.8 multimodal FCI 5 0.479 ± 0.063 1.000 0.325 ± 0.057 0.420 ± 0.038 0.897 ± 0.044 0.282 ± 0.038 25.4 ± 2.7 0.300 ± 0.026 1718.79 ± 458.14 11.2 ± 1.8 unimodal_asymmetric Spade-Hist 5 0.794 ± 0.036 0.960 ± 0.018 0.681 ± 0.045 0.776 ± 0.049 0.936 ± 0.030 0.667 ± 0.056 13.2 ± 3.1 0.103 ± 0.051 0.66 ± 0.05 24.6 ± 1.0 unimodal_asymmetric Spade-Gauss 5 0.740 ± 0.032 0.924 ± 0.044 0.621 ± 0.032 0.680 ± 0.039 0.850 ± 0.057 0.569 ± 0.035 17.4 ± 2.8 0.143 ± 0.058 0.12 ± 0.01 23.6 ± 1.4 unimodal_asymmetric Topic 5 0.721 ± 0.036 0.844 ± 0.033 0.636 ± 0.048 0.606 ± 0.049 0.717 ± 0.071 0.532 ± 0.047 21.0 ± 3.1 0.208 ± 0.063 17.50 ± 0.86 26.2 ± 1.5 unimodal_asymmetric Cam 5 0.783 ± 0.018 0.711 ± 0.029 0.878 ± 0.019 0.728 ± 0.023 0.662 ± 0.034 0.815 ± 0.013 19.4 ± 2.2 0.095 ± 0.023 241.51 ± 2.31 43.6 ± 3.1 unimodal_asymmetric CauScale 5 0.676 ± 0.022 0.887 ± 0.023 0.550 ± 0.030 0.394 ± 0.030 0.512 ± 0.022 0.322 ± 0.033 26.6 ± 2.6 0.417 ± 0.037 6.07 ± 0.03 21.8 ± 1.7 unimodal_asymmetric SDCD 5 0.707 ± 0.036 0.800 ± 0.031 0.638 ± 0.048 0.535 ± 0.027 0.607 ± 0.032 0.482 ± 0.033 23.6 ± 1.4 0.309 ± 0.012 77.82 ± 1.04 27.8 ± 1.7 unimodal_asymmetric Globe 5 0.608 ± 0.033 0.868 ± 0.031 0.472 ± 0.036 0.480 ± 0.056 0.677 ± 0.054 0.375 ± 0.053 24.8 ± 2.9 0.402 ± 0.079 172.64 ± 20.42 19.0 ± 1.6 unimodal_asymmetric Das 5 0.641 ± 0.028 0.613 ± 0.037 0.679 ± 0.036 0.495 ± 0.008 0.472 ± 0.012 0.526 ± 0.028 32.4 ± 2.9 0.294 ± 0.029 13.06 ± 0.22 39.2 ± 3.1 unimodal_asymmetric Score 5 0.707 ± 0.020 0.618 ± 0.032 0.835 ± 0.026 0.524 ± 0.009 0.457 ± 0.014 0.621 ± 0.029 32.0 ± 2.4 0.294 ± 0.029 13.51 ± 0.57 47.8 ± 3.4 unimodal_asymmetric NoGAM 5 0.675 ± 0.027 0.570 ± 0.030 0.831 ± 0.035 0.493 ± 0.021 0.415 ± 0.014 0.611 ± 0.042 36.6 ± 4.3 0.298 ± 0.037 40.06 ± 0.12 51.8 ± 4.5 unimodal_asymmetric Avici 5 0.657 ± 0.037 0.958 ± 0.019 0.507 ± 0.045 0.504 ± 0.014 0.742 ± 0.039 0.387 ± 0.022 22.6 ± 2.1 0.395 ± 0.035 8.51 ± 0.64 18.2 ± 0.7 unimodal_asymmetric NOTEARS 5 0.691 ± 0.037 0.624 ± 0.049 0.783 ± 0.028 0.442 ± 0.056 0.402 ± 0.062 0.495 ± 0.046 34.6 ± 4.1 0.386 ± 0.051 419.90 ± 124.20 44.6 ± 2.9 unimodal_asymmetric DAGMA 5 0.690 ± 0.030 0.698 ± 0.050 0.694 ± 0.041 0.384 ± 0.036 0.389 ± 0.044 0.386 ± 0.039 32.4 ± 2.4 0.457 ± 0.045 731.27 ± 51.54 35.2 ± 2.7 unimodal_asymmetric DiffAN 5 0.722 ± 0.029 0.669 ± 0.037 0.789 ± 0.031 0.500 ± 0.052 0.465 ± 0.054 0.544 ± 0.050 30.2 ± 4.2 0.341 ± 0.047 17.06 ± 0.08 41.6 ± 2.8 unimodal_asymmetric SciNO 5 0.735 ± 0.021 0.717 ± 0.033 0.769 ± 0.050 0.553 ± 0.040 0.545 ± 0.059 0.573 ± 0.032 26.0 ± 2.5 0.290 ± 0.059 374.33 ± 13.16 37.8 ± 3.3 unimodal_asymmetric Caps 5 0.693 ± 0.031 0.698 ± 0.033 0.691 ± 0.037 0.529 ± 0.030 0.531 ± 0.023 0.529 ± 0.039 27.4 ± 3.2 0.296 ± 0.026 7.96 ± 0.49 35.0 ± 2.9 unimodal_asymmetric PC 5 0.738 ± 0.033 0.979 ± 0.013 0.596 ± 0.041 0.581 ± 0.023 0.778 ± 0.051 0.467 ± 0.021 19.0 ± 0.8 0.311 ± 0.046 1855.16 ± 395.71 21.4 ± 2.0 unimodal_asymmetric FCI 5 0.500 ± 0.041 1.000 0.338 ± 0.038 0.451 ± 0.047 0.895 ± 0.032 0.305 ± 0.040 24.8 ± 2.9 0.323 ± 0.033 2320.90 ± 458.11 11.6 ± 0.7 unimodal_symmetric Spade-Hist 5 0.809 ± 0.024 0.993 ± 0.007 0.687 ± 0.035 0.777 ± 0.045 0.950 ± 0.027 0.661 ± 0.051 12.4 ± 2.3 0.095 ± 0.038 0.63 ± 0.02 24.2 ± 1.6 unimodal_symmetric Spade-Gauss 5 0.776 ± 0.027 0.984 ± 0.010 0.644 ± 0.036 0.729 ± 0.027 0.924 ± 0.017 0.605 ± 0.034 14.6 ± 2.2 0.123 ± 0.034 0.12 ± 0.01 22.8 ± 1.2 unimodal_symmetric Topic 5 0.785 ± 0.022 0.945 ± 0.008 0.676 ± 0.036 0.718 ± 0.013 0.865 ± 0.016 0.617 ± 0.027 15.0 ± 1.5 0.198 ± 0.034 15.72 ± 1.47 25.0 ± 1.6 unimodal_symmetric Cam 5 0.735 ± 0.030 0.650 ± 0.039 0.853 ± 0.031 0.688 ± 0.026 0.609 ± 0.038 0.797 ± 0.015 23.6 ± 2.8 0.067 ± 0.023 247.61 ± 2.57 46.4 ± 3.3 unimodal_symmetric CauScale 5 0.693 ± 0.043 0.863 ± 0.046 0.581 ± 0.043 0.464 ± 0.053 0.576 ± 0.061 0.390 ± 0.048 25.0 ± 3.7 0.363 ± 0.039 5.80 ± 0.07 23.4 ± 1.0 unimodal_symmetric SDCD 5 0.651 ± 0.011 0.737 ± 0.020 0.589 ± 0.025 0.438 ± 0.033 0.501 ± 0.054 0.393 ± 0.022 29.2 ± 3.2 0.389 ± 0.045 80.46 ± 1.69 28.6 ± 3.5 unimodal_symmetric Globe 5 0.557 ± 0.043 0.795 ± 0.029 0.441 ± 0.055 0.527 ± 0.041 0.754 ± 0.033 0.418 ± 0.054 25.0 ± 2.7 0.251 ± 0.054 180.33 ± 9.99 19.0 ± 1.5 unimodal_symmetric Das 5 0.628 ± 0.028 0.617 ± 0.032 0.641 ± 0.027 0.485 ± 0.034 0.476 ± 0.034 0.495 ± 0.035 31.8 ± 2.9 0.305 ± 0.027 16.45 ± 2.29 36.6 ± 2.4 unimodal_symmetric Score 5 0.669 ± 0.022 0.559 ± 0.026 0.836 ± 0.020 0.476 ± 0.029 0.398 ± 0.028 0.594 ± 0.029 37.4 ± 2.5 0.305 ± 0.027 13.12 ± 0.45 52.6 ± 2.9 unimodal_symmetric NoGAM 5 0.665 ± 0.021 0.561 ± 0.031 0.827 ± 0.024 0.490 ± 0.027 0.413 ± 0.029 0.608 ± 0.032 36.6 ± 1.5 0.286 ± 0.029 64.07 ± 13.26 51.8 ± 2.0 unimodal_symmetric Avici 5 0.699 ± 0.022 0.909 ± 0.013 0.570 ± 0.031 0.625 ± 0.029 0.819 ± 0.058 0.508 ± 0.020 19.4 ± 1.7 0.246 ± 0.062 8.39 ± 0.62 21.8 ± 0.7 unimodal_symmetric NOTEARS 5 0.691 ± 0.014 0.663 ± 0.028 0.725 ± 0.011 0.502 ± 0.048 0.482 ± 0.051 0.526 ± 0.047 30.2 ± 4.0 0.334 ± 0.072 394.05 ± 47.27 39.2 ± 4.4 unimodal_symmetric DAGMA 5 0.629 ± 0.025 0.678 ± 0.014 0.589 ± 0.035 0.325 ± 0.015 0.352 ± 0.019 0.304 ± 0.017 34.4 ± 2.6 0.524 ± 0.040 677.96 ± 11.86 30.2 ± 1.3 unimodal_symmetric DiffAN 5 0.745 ± 0.016 0.737 ± 0.024 0.758 ± 0.026 0.560 ± 0.022 0.552 ± 0.013 0.571 ± 0.036 25.0 ± 2.5 0.326 ± 0.039 17.91 ± 0.27 36.4 ± 3.2 unimodal_symmetric SciNO 5 0.714 ± 0.022 0.716 ± 0.035 0.714 ± 0.013 0.515 ± 0.015 0.516 ± 0.020 0.516 ± 0.016 27.4 ± 2.6 0.356 ± 0.037 395.76 ± 14.62 35.4 ± 2.8 unimodal_symmetric Caps 5 0.723 ± 0.023 0.781 ± 0.017 0.677 ± 0.037 0.551 ± 0.036 0.595 ± 0.035 0.516 ± 0.041 23.6 ± 1.8 0.304 ± 0.037 6.85 ± 0.30 30.4 ± 2.3 unimodal_symmetric PC 5 0.700 ± 0.048 0.954 ± 0.028 0.558 ± 0.051 0.577 ± 0.043 0.787 ± 0.040 0.460 ± 0.045 20.2 ± 2.9 0.304 ± 0.017 1519.51 ± 445.15 20.2 ± 1.2 unimodal_symmetric FCI 5 0.550 ± 0.038 1.000 0.384 ± 0.039 0.487 ± 0.036 0.885 ± 0.022 0.340 ± 0.035 23.4 ± 2.5 0.270 ± 0.035 1886.64 ± 597.47 13.4 ± 1.4 Table 22: Qualitative comparison without marginalization (n=1000n=1000, d=20d=20, degree 4, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.593 ± 0.029 0.960 ± 0.027 0.432 ± 0.029 0.549 ± 0.033 0.895 ± 0.062 0.399 ± 0.027 21.8 ± 1.9 0.349 ± 0.021 0.53 ± 0.01 15.6 ± 0.4 gaussian Spade-Gauss 5 0.684 ± 0.044 0.951 ± 0.023 0.539 ± 0.048 0.663 ± 0.042 0.921 ± 0.025 0.522 ± 0.047 18.2 ± 3.1 0.075 ± 0.022 0.10 ± 0.00 19.6 ± 1.2 gaussian Topic 5 0.708 ± 0.027 0.855 ± 0.037 0.610 ± 0.035 0.654 ± 0.020 0.791 ± 0.034 0.563 ± 0.029 19.2 ± 1.7 0.136 ± 0.028 15.26 ± 0.54 24.8 ± 0.4 gaussian Cam 5 0.690 ± 0.021 0.570 ± 0.022 0.881 ± 0.039 0.637 ± 0.019 0.526 ± 0.019 0.814 ± 0.042 30.6 ± 3.6 0.084 ± 0.016 249.86 ± 1.41 54.6 ± 4.5 gaussian CauScale 5 0.538 ± 0.016 0.879 ± 0.035 0.388 ± 0.012 0.332 ± 0.021 0.541 ± 0.032 0.240 ± 0.017 28.8 ± 2.5 0.456 ± 0.058 7.12 ± 1.09 15.6 ± 1.2 gaussian SDCD 5 0.657 ± 0.010 0.839 ± 0.030 0.542 ± 0.014 0.561 ± 0.014 0.718 ± 0.033 0.463 ± 0.013 22.8 ± 2.0 0.261 ± 0.020 75.47 ± 3.23 22.8 ± 1.7 gaussian Globe 5 0.419 ± 0.034 0.836 ± 0.042 0.281 ± 0.026 0.403 ± 0.033 0.808 ± 0.052 0.270 ± 0.025 27.8 ± 2.7 0.433 ± 0.022 127.39 ± 6.86 11.8 ± 1.2 gaussian Das 5 0.596 ± 0.032 0.590 ± 0.049 0.608 ± 0.014 0.456 ± 0.041 0.452 ± 0.055 0.463 ± 0.029 34.4 ± 4.1 0.257 ± 0.046 13.07 ± 0.13 37.0 ± 3.4 gaussian Score 5 0.688 ± 0.026 0.568 ± 0.028 0.877 ± 0.032 0.528 ± 0.045 0.436 ± 0.041 0.673 ± 0.056 35.6 ± 5.0 0.257 ± 0.046 12.99 ± 0.51 55.0 ± 5.3 gaussian NoGAM 5 0.722 ± 0.020 0.606 ± 0.029 0.899 ± 0.024 0.534 ± 0.032 0.449 ± 0.035 0.663 ± 0.031 33.4 ± 4.5 0.272 ± 0.049 43.72 ± 0.43 53.2 ± 5.7 gaussian Avici 5 0.663 ± 0.028 0.907 ± 0.060 0.526 ± 0.020 0.522 ± 0.068 0.724 ± 0.112 0.411 ± 0.049 23.8 ± 5.0 0.342 ± 0.070 7.56 ± 0.08 21.0 ± 2.8 gaussian NOTEARS 5 0.684 ± 0.040 0.635 ± 0.048 0.743 ± 0.029 0.497 ± 0.064 0.462 ± 0.067 0.539 ± 0.060 31.6 ± 4.4 0.287 ± 0.059 355.19 ± 96.83 41.4 ± 2.2 gaussian DAGMA 5 0.694 ± 0.019 0.773 ± 0.027 0.634 ± 0.029 0.463 ± 0.035 0.520 ± 0.055 0.420 ± 0.025 27.4 ± 3.4 0.404 ± 0.043 714.99 ± 34.67 29.2 ± 3.1 gaussian DiffAN 5 0.755 ± 0.026 0.747 ± 0.033 0.765 ± 0.026 0.495 ± 0.052 0.491 ± 0.056 0.501 ± 0.049 26.8 ± 3.3 0.398 ± 0.062 17.22 ± 0.35 36.0 ± 2.0 gaussian SciNO 5 0.756 ± 0.034 0.750 ± 0.046 0.766 ± 0.032 0.482 ± 0.038 0.481 ± 0.050 0.486 ± 0.028 27.8 ± 4.2 0.398 ± 0.023 379.47 ± 10.27 36.4 ± 3.3 gaussian Caps 5 0.712 ± 0.032 0.760 ± 0.039 0.670 ± 0.029 0.492 ± 0.037 0.526 ± 0.043 0.462 ± 0.033 26.4 ± 2.8 0.371 ± 0.052 7.07 ± 0.14 31.2 ± 2.5 gaussian PC 5 0.721 ± 0.032 0.972 ± 0.018 0.577 ± 0.038 0.504 ± 0.038 0.692 ± 0.083 0.400 ± 0.022 21.8 ± 1.8 0.373 ± 0.074 1394.96 ± 258.75 21.2 ± 2.6 gaussian FCI 5 0.449 ± 0.034 1.000 0.292 ± 0.028 0.409 ± 0.044 0.906 ± 0.047 0.266 ± 0.034 26.0 ± 2.6 0.384 ± 0.049 1854.78 ± 457.89 10.2 ± 1.1 multimodal Spade-Hist 5 0.623 ± 0.026 0.975 ± 0.015 0.461 ± 0.029 0.616 ± 0.032 0.963 ± 0.015 0.456 ± 0.033 19.8 ± 2.3 0.199 ± 0.071 0.61 ± 0.07 16.4 ± 0.7 multimodal Spade-Gauss 5 0.645 ± 0.020 0.916 ± 0.021 0.499 ± 0.021 0.567 ± 0.031 0.804 ± 0.036 0.439 ± 0.029 21.6 ± 2.4 0.177 ± 0.055 0.10 ± 0.00 19.0 ± 0.7 multimodal Topic 5 0.625 ± 0.015 0.765 ± 0.041 0.533 ± 0.018 0.520 ± 0.024 0.633 ± 0.024 0.446 ± 0.034 25.6 ± 1.8 0.193 ± 0.039 15.73 ± 1.08 24.4 ± 0.5 multimodal Cam 5 0.733 ± 0.028 0.617 ± 0.037 0.909 ± 0.020 0.599 ± 0.050 0.506 ± 0.051 0.742 ± 0.055 29.8 ± 5.1 0.208 ± 0.048 246.74 ± 1.91 52.2 ± 4.0 multimodal CauScale 5 0.696 ± 0.029 0.952 ± 0.022 0.552 ± 0.035 0.558 ± 0.028 0.765 ± 0.034 0.442 ± 0.028 20.8 ± 2.1 0.217 ± 0.043 5.81 ± 0.07 20.2 ± 1.2 multimodal SDCD 5 0.602 ± 0.021 0.693 ± 0.031 0.538 ± 0.029 0.425 ± 0.025 0.495 ± 0.049 0.377 ± 0.017 30.2 ± 1.2 0.349 ± 0.039 77.39 ± 0.81 27.4 ± 2.2 multimodal Globe 5 0.540 ± 0.026 0.855 ± 0.054 0.409 ± 0.044 0.501 ± 0.037 0.790 ± 0.057 0.381 ± 0.050 25.0 ± 1.9 0.400 ± 0.046 155.89 ± 11.87 16.8 ± 1.7 multimodal Das 5 0.638 ± 0.019 0.619 ± 0.039 0.666 ± 0.017 0.516 ± 0.028 0.498 ± 0.030 0.540 ± 0.037 31.2 ± 3.0 0.231 ± 0.042 14.16 ± 0.52 38.4 ± 3.8 multimodal Score 5 0.695 ± 0.015 0.589 ± 0.023 0.852 ± 0.018 0.548 ± 0.019 0.463 ± 0.012 0.674 ± 0.038 32.6 ± 2.1 0.231 ± 0.042 13.89 ± 0.40 51.2 ± 4.2 multimodal NoGAM 5 0.708 ± 0.030 0.602 ± 0.035 0.862 ± 0.025 0.569 ± 0.032 0.484 ± 0.035 0.692 ± 0.028 31.6 ± 4.5 0.217 ± 0.014 47.87 ± 0.79 50.8 ± 4.3 multimodal Avici 5 0.589 ± 0.053 0.898 ± 0.036 0.451 ± 0.063 0.457 ± 0.046 0.696 ± 0.043 0.350 ± 0.051 25.0 ± 2.7 0.331 ± 0.048 7.59 ± 0.05 17.2 ± 1.5 multimodal NOTEARS 5 0.682 ± 0.015 0.642 ± 0.031 0.736 ± 0.024 0.472 ± 0.032 0.443 ± 0.030 0.512 ± 0.045 32.0 ± 2.6 0.327 ± 0.056 505.76 ± 100.51 41.2 ± 5.2 multimodal DAGMA 5 0.691 ± 0.036 0.715 ± 0.047 0.672 ± 0.032 0.459 ± 0.035 0.475 ± 0.040 0.446 ± 0.033 28.8 ± 2.2 0.384 ± 0.044 713.09 ± 38.39 33.4 ± 3.2 multimodal DiffAN 5 0.728 ± 0.042 0.740 ± 0.037 0.720 ± 0.052 0.555 ± 0.054 0.564 ± 0.053 0.547 ± 0.058 25.6 ± 5.1 0.303 ± 0.043 17.46 ± 0.44 34.2 ± 2.9 multimodal SciNO 5 0.708 ± 0.028 0.697 ± 0.028 0.721 ± 0.031 0.482 ± 0.026 0.476 ± 0.031 0.490 ± 0.021 29.2 ± 3.3 0.354 ± 0.058 398.22 ± 16.32 36.4 ± 2.6 multimodal Caps 5 0.673 ± 0.033 0.717 ± 0.050 0.644 ± 0.041 0.516 ± 0.037 0.546 ± 0.037 0.496 ± 0.048 27.8 ± 4.3 0.279 ± 0.042 6.85 ± 0.13 32.2 ± 3.7 multimodal PC 5 0.754 ± 0.062 0.980 ± 0.013 0.624 ± 0.069 0.542 ± 0.042 0.709 ± 0.034 0.447 ± 0.047 20.2 ± 3.1 0.325 ± 0.044 1307.07 ± 270.67 21.8 ± 1.7 multimodal FCI 5 0.450 ± 0.020 1.000 0.291 ± 0.016 0.364 ± 0.010 0.816 ± 0.047 0.235 ± 0.007 27.0 ± 2.1 0.404 ± 0.030 1500.69 ± 473.67 10.2 ± 0.8 unimodal_asymmetric Spade-Hist 5 0.693 ± 0.033 1.000 0.534 ± 0.041 0.666 ± 0.037 0.963 ± 0.037 0.513 ± 0.041 17.2 ± 2.1 0.178 ± 0.051 0.56 ± 0.01 18.6 ± 1.3 unimodal_asymmetric Spade-Gauss 5 0.694 ± 0.020 0.859 ± 0.015 0.586 ± 0.031 0.556 ± 0.031 0.686 ± 0.022 0.470 ± 0.039 22.2 ± 2.1 0.259 ± 0.040 0.11 ± 0.00 23.8 ± 1.3 unimodal_asymmetric Topic 5 0.678 ± 0.023 0.770 ± 0.015 0.608 ± 0.033 0.523 ± 0.032 0.593 ± 0.025 0.471 ± 0.038 25.2 ± 2.5 0.317 ± 0.050 19.00 ± 2.14 27.6 ± 1.7 unimodal_asymmetric Cam 5 0.658 ± 0.016 0.523 ± 0.017 0.888 ± 0.018 0.522 ± 0.045 0.416 ± 0.039 0.703 ± 0.054 38.8 ± 3.3 0.232 ± 0.057 242.80 ± 3.55 59.8 ± 4.1 unimodal_asymmetric CauScale 5 0.648 ± 0.015 0.896 ± 0.050 0.519 ± 0.035 0.397 ± 0.051 0.545 ± 0.073 0.317 ± 0.042 26.4 ± 1.4 0.419 ± 0.050 6.07 ± 0.11 20.4 ± 1.6 unimodal_asymmetric SDCD 5 0.669 ± 0.036 0.719 ± 0.029 0.629 ± 0.045 0.509 ± 0.028 0.547 ± 0.024 0.478 ± 0.034 27.4 ± 3.5 0.294 ± 0.045 78.77 ± 0.55 30.6 ± 2.2 unimodal_asymmetric Globe 5 0.481 ± 0.038 0.868 ± 0.023 0.336 ± 0.035 0.368 ± 0.048 0.658 ± 0.052 0.258 ± 0.040 28.2 ± 3.2 0.458 ± 0.039 142.47 ± 12.81 13.4 ± 1.0 unimodal_asymmetric Das 5 0.601 ± 0.020 0.561 ± 0.023 0.649 ± 0.023 0.458 ± 0.045 0.429 ± 0.045 0.494 ± 0.047 35.4 ± 1.9 0.258 ± 0.052 17.54 ± 2.30 41.0 ± 3.6 unimodal_asymmetric Score 5 0.662 ± 0.023 0.533 ± 0.025 0.881 ± 0.032 0.502 ± 0.041 0.405 ± 0.037 0.665 ± 0.045 38.4 ± 1.6 0.258 ± 0.052 13.09 ± 0.65 58.0 ± 3.0 unimodal_asymmetric NoGAM 5 0.670 ± 0.012 0.543 ± 0.015 0.881 ± 0.028 0.493 ± 0.037 0.400 ± 0.034 0.645 ± 0.042 38.2 ± 1.2 0.272 ± 0.055 41.25 ± 0.52 57.0 ± 3.6 unimodal_asymmetric Avici 5 0.632 ± 0.036 0.974 ± 0.016 0.470 ± 0.038 0.471 ± 0.020 0.728 ± 0.018 0.350 ± 0.022 23.4 ± 2.1 0.394 ± 0.047 9.56 ± 1.17 16.8 ± 1.2 unimodal_asymmetric NOTEARS 5 0.685 ± 0.018 0.590 ± 0.028 0.824 ± 0.028 0.472 ± 0.025 0.406 ± 0.024 0.570 ± 0.039 35.6 ± 2.3 0.348 ± 0.046 356.73 ± 67.88 49.4 ± 3.7 unimodal_asymmetric DAGMA 5 0.664 ± 0.038 0.613 ± 0.050 0.733 ± 0.028 0.379 ± 0.050 0.354 ± 0.055 0.413 ± 0.042 38.6 ± 6.7 0.471 ± 0.056 786.91 ± 50.44 43.4 ± 5.1 unimodal_asymmetric DiffAN 5 0.728 ± 0.010 0.661 ± 0.017 0.814 ± 0.020 0.514 ± 0.030 0.468 ± 0.035 0.573 ± 0.025 29.6 ± 1.8 0.332 ± 0.017 17.66 ± 0.38 43.2 ± 2.2 unimodal_asymmetric SciNO 5 0.712 ± 0.017 0.633 ± 0.012 0.819 ± 0.041 0.475 ± 0.043 0.424 ± 0.041 0.544 ± 0.050 32.4 ± 1.4 0.338 ± 0.054 382.11 ± 15.46 45.6 ± 4.2 unimodal_asymmetric Caps 5 0.717 ± 0.013 0.714 ± 0.016 0.721 ± 0.017 0.557 ± 0.046 0.557 ± 0.050 0.559 ± 0.042 25.2 ± 1.0 0.275 ± 0.058 7.51 ± 0.21 35.4 ± 2.0 unimodal_asymmetric PC 5 0.706 ± 0.039 0.958 ± 0.031 0.565 ± 0.045 0.512 ± 0.020 0.698 ± 0.026 0.409 ± 0.026 21.8 ± 2.2 0.365 ± 0.037 1675.96 ± 467.56 20.4 ± 0.9 unimodal_asymmetric FCI 5 0.535 ± 0.030 1.000 0.368 ± 0.029 0.480 ± 0.023 0.905 ± 0.056 0.328 ± 0.017 23.6 ± 1.5 0.345 ± 0.029 1868.42 ± 540.69 12.8 ± 0.9 unimodal_symmetric Spade-Hist 5 0.638 ± 0.053 0.961 ± 0.028 0.489 ± 0.063 0.580 ± 0.073 0.861 ± 0.040 0.448 ± 0.078 20.8 ± 3.6 0.264 ± 0.076 0.55 ± 0.01 17.6 ± 1.7 unimodal_symmetric Spade-Gauss 5 0.689 ± 0.039 0.877 ± 0.028 0.570 ± 0.046 0.601 ± 0.034 0.767 ± 0.037 0.497 ± 0.038 20.8 ± 2.7 0.191 ± 0.021 0.10 ± 0.01 22.6 ± 1.3 unimodal_symmetric Topic 5 0.674 ± 0.038 0.806 ± 0.022 0.584 ± 0.052 0.583 ± 0.040 0.696 ± 0.023 0.506 ± 0.053 22.6 ± 2.9 0.217 ± 0.040 16.19 ± 1.49 25.0 ± 1.0 unimodal_symmetric Cam 5 0.682 ± 0.015 0.538 ± 0.013 0.935 ± 0.017 0.549 ± 0.025 0.433 ± 0.021 0.752 ± 0.031 37.6 ± 4.6 0.193 ± 0.028 240.69 ± 1.27 61.4 ± 4.8 unimodal_symmetric CauScale 5 0.609 ± 0.066 0.895 ± 0.023 0.474 ± 0.076 0.384 ± 0.074 0.547 ± 0.066 0.303 ± 0.072 27.0 ± 4.3 0.496 ± 0.049 6.14 ± 0.11 17.8 ± 1.2 unimodal_symmetric SDCD 5 0.701 ± 0.027 0.758 ± 0.030 0.654 ± 0.034 0.510 ± 0.021 0.552 ± 0.023 0.476 ± 0.024 26.2 ± 3.1 0.346 ± 0.046 79.33 ± 2.54 30.4 ± 2.4 unimodal_symmetric Globe 5 0.452 ± 0.033 0.823 ± 0.032 0.319 ± 0.037 0.397 ± 0.047 0.715 ± 0.054 0.282 ± 0.045 28.2 ± 2.7 0.441 ± 0.048 135.08 ± 7.34 13.6 ± 1.5 unimodal_symmetric Das 5 0.619 ± 0.022 0.597 ± 0.031 0.646 ± 0.022 0.430 ± 0.029 0.412 ± 0.026 0.450 ± 0.035 34.8 ± 2.1 0.318 ± 0.069 13.25 ± 0.26 38.4 ± 3.5 unimodal_symmetric Score 5 0.674 ± 0.009 0.538 ± 0.014 0.907 ± 0.020 0.474 ± 0.049 0.378 ± 0.041 0.638 ± 0.065 40.0 ± 2.7 0.318 ± 0.069 14.60 ± 0.62 59.2 ± 3.5 unimodal_symmetric NoGAM 5 0.683 ± 0.012 0.551 ± 0.012 0.902 ± 0.022 0.498 ± 0.039 0.401 ± 0.031 0.659 ± 0.055 38.0 ± 3.4 0.297 ± 0.058 44.46 ± 1.32 57.6 ± 4.0 unimodal_symmetric Avici 5 0.705 ± 0.036 0.933 ± 0.034 0.580 ± 0.054 0.607 ± 0.031 0.814 ± 0.070 0.495 ± 0.031 19.6 ± 1.8 0.284 ± 0.032 7.65 ± 0.07 22.0 ± 2.6 unimodal_symmetric NOTEARS 5 0.659 ± 0.034 0.546 ± 0.039 0.836 ± 0.032 0.549 ± 0.043 0.456 ± 0.043 0.697 ± 0.048 36.4 ± 6.2 0.222 ± 0.047 369.38 ± 33.61 54.6 ± 5.2 unimodal_symmetric DAGMA 5 0.716 ± 0.021 0.696 ± 0.038 0.758 ± 0.057 0.431 ± 0.053 0.414 ± 0.050 0.462 ± 0.071 31.6 ± 3.9 0.445 ± 0.063 744.57 ± 40.90 38.2 ± 2.8 unimodal_symmetric DiffAN 5 0.722 ± 0.026 0.673 ± 0.023 0.781 ± 0.033 0.482 ± 0.020 0.449 ± 0.014 0.522 ± 0.028 30.4 ± 2.9 0.351 ± 0.045 18.00 ± 0.38 40.8 ± 2.9 unimodal_symmetric SciNO 5 0.703 ± 0.012 0.655 ± 0.018 0.762 ± 0.028 0.423 ± 0.048 0.396 ± 0.050 0.457 ± 0.049 33.0 ± 1.9 0.429 ± 0.072 445.04 ± 48.75 40.6 ± 1.4 unimodal_symmetric Caps 5 0.706 ± 0.023 0.693 ± 0.019 0.723 ± 0.036 0.501 ± 0.044 0.494 ± 0.049 0.511 ± 0.042 28.4 ± 2.6 0.348 ± 0.070 6.75 ± 0.07 36.4 ± 1.5 unimodal_symmetric PC 5 0.723 ± 0.040 0.939 ± 0.024 0.592 ± 0.048 0.549 ± 0.037 0.711 ± 0.026 0.450 ± 0.041 21.0 ± 2.9 0.283 ± 0.037 1728.94 ± 515.67 22.0 ± 1.7 unimodal_symmetric FCI 5 0.478 ± 0.055 0.980 ± 0.020 0.323 ± 0.047 0.403 ± 0.057 0.836 ± 0.082 0.272 ± 0.048 26.0 ± 2.8 0.342 ± 0.082 2126.96 ± 755.07 11.6 ± 1.8 Table 23: Qualitative comparison without marginalization (n=1000n=1000, d=50d=50, degree 2, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.885 ± 0.016 0.990 ± 0.006 0.802 ± 0.022 0.876 ± 0.013 0.979 ± 0.005 0.793 ± 0.019 10.2 ± 1.7 0.036 ± 0.014 1.50 ± 0.05 37.4 ± 2.2 gaussian Spade-Gauss 5 0.884 ± 0.020 0.995 ± 0.005 0.797 ± 0.028 0.884 ± 0.020 0.995 ± 0.005 0.797 ± 0.028 10.0 ± 2.4 0.029 ± 0.008 0.40 ± 0.03 36.8 ± 1.5 gaussian Topic 5 0.763 ± 0.019 0.713 ± 0.020 0.822 ± 0.026 0.730 ± 0.027 0.683 ± 0.028 0.786 ± 0.031 25.6 ± 3.5 0.076 ± 0.012 82.75 ± 4.35 53.2 ± 3.1 gaussian Cam 5 0.153 ± 0.009 0.084 ± 0.006 0.948 ± 0.015 0.150 ± 0.009 0.082 ± 0.005 0.930 ± 0.013 490.0 ± 33.0 0.035 ± 0.011 30345.19 ± 310.65 530.4 ± 34.3 gaussian CauScale 5 0.879 ± 0.021 0.919 ± 0.012 0.846 ± 0.035 0.604 ± 0.028 0.631 ± 0.023 0.581 ± 0.035 23.2 ± 3.1 0.325 ± 0.024 12.59 ± 1.50 42.2 ± 1.5 gaussian SDCD 5 0.743 ± 0.025 0.737 ± 0.016 0.750 ± 0.035 0.550 ± 0.012 0.546 ± 0.013 0.555 ± 0.016 33.2 ± 3.2 0.259 ± 0.033 76.03 ± 1.83 46.8 ± 2.3 gaussian Globe 5 0.667 ± 0.022 0.830 ± 0.036 0.562 ± 0.028 0.644 ± 0.029 0.800 ± 0.037 0.544 ± 0.035 27.0 ± 3.3 0.217 ± 0.014 747.05 ± 50.03 31.6 ± 2.8 gaussian Das 5 0.590 ± 0.028 0.528 ± 0.028 0.671 ± 0.033 0.420 ± 0.044 0.377 ± 0.041 0.477 ± 0.047 52.6 ± 5.8 0.345 ± 0.040 48.11 ± 0.98 58.8 ± 3.4 gaussian Score 5 0.319 ± 0.032 0.195 ± 0.024 0.896 ± 0.033 0.222 ± 0.033 0.137 ± 0.023 0.620 ± 0.050 200.4 ± 27.9 0.345 ± 0.040 615.93 ± 7.65 223.6 ± 27.3 gaussian NoGAM 5 0.324 ± 0.033 0.199 ± 0.024 0.907 ± 0.026 0.233 ± 0.034 0.143 ± 0.024 0.646 ± 0.046 198.8 ± 29.9 0.316 ± 0.035 788.73 ± 9.76 223.8 ± 29.7 gaussian Avici 5 0.708 ± 0.040 0.972 ± 0.013 0.563 ± 0.049 0.663 ± 0.048 0.909 ± 0.034 0.528 ± 0.053 23.2 ± 3.8 0.247 ± 0.031 15.48 ± 0.66 26.4 ± 1.5 gaussian NOTEARS 5 0.637 ± 0.047 0.535 ± 0.056 0.806 ± 0.019 0.486 ± 0.044 0.408 ± 0.048 0.613 ± 0.032 54.8 ± 11.3 0.242 ± 0.039 2285.84 ± 385.93 73.8 ± 10.7 gaussian DAGMA 5 0.753 ± 0.034 0.765 ± 0.058 0.751 ± 0.036 0.481 ± 0.065 0.493 ± 0.085 0.476 ± 0.055 36.6 ± 7.0 0.343 ± 0.061 2289.39 ± 157.69 46.2 ± 4.5 gaussian DiffAN 5 0.377 ± 0.048 0.244 ± 0.038 0.872 ± 0.028 0.248 ± 0.032 0.160 ± 0.025 0.576 ± 0.040 162.2 ± 33.1 0.376 ± 0.038 483.38 ± 7.42 182.6 ± 33.0 gaussian SciNO 5 0.238 ± 0.023 0.140 ± 0.017 0.873 ± 0.032 0.136 ± 0.019 0.080 ± 0.013 0.496 ± 0.051 286.8 ± 34.1 0.444 ± 0.048 887.66 ± 16.86 303.2 ± 34.5 gaussian Caps 5 0.720 ± 0.060 0.661 ± 0.075 0.817 ± 0.043 0.483 ± 0.075 0.447 ± 0.077 0.538 ± 0.067 45.0 ± 11.4 0.384 ± 0.038 311.65 ± 17.41 60.4 ± 8.5 gaussian PC 5 0.650 ± 0.029 0.811 ± 0.041 0.546 ± 0.033 0.511 ± 0.030 0.637 ± 0.032 0.430 ± 0.032 32.6 ± 2.7 0.339 ± 0.040 1997.04 ± 314.27 31.0 ± 1.7 gaussian FCI 5 0.420 ± 0.033 0.983 ± 0.017 0.269 ± 0.026 0.383 ± 0.033 0.899 ± 0.036 0.246 ± 0.025 35.4 ± 3.4 0.355 ± 0.045 2085.31 ± 300.07 12.6 ± 1.2 multimodal Spade-Hist 5 0.885 ± 0.017 0.982 ± 0.012 0.806 ± 0.022 0.885 ± 0.017 0.982 ± 0.012 0.806 ± 0.022 9.8 ± 1.9 0.043 ± 0.019 1.59 ± 0.06 38.2 ± 3.2 multimodal Spade-Gauss 5 0.817 ± 0.017 0.902 ± 0.009 0.749 ± 0.029 0.689 ± 0.037 0.760 ± 0.035 0.633 ± 0.042 21.0 ± 2.8 0.155 ± 0.025 0.42 ± 0.01 38.4 ± 2.6 multimodal Topic 5 0.683 ± 0.018 0.620 ± 0.016 0.764 ± 0.033 0.561 ± 0.038 0.510 ± 0.035 0.627 ± 0.047 38.8 ± 3.2 0.182 ± 0.036 81.59 ± 2.75 56.6 ± 2.2 multimodal Cam 5 0.156 ± 0.009 0.085 ± 0.005 0.963 ± 0.018 0.139 ± 0.008 0.076 ± 0.005 0.863 ± 0.043 487.8 ± 16.6 0.115 ± 0.041 29874.83 ± 313.04 526.4 ± 20.1 multimodal CauScale 5 0.882 ± 0.011 0.960 ± 0.009 0.816 ± 0.019 0.764 ± 0.034 0.830 ± 0.028 0.708 ± 0.038 15.4 ± 2.5 0.160 ± 0.035 11.64 ± 0.35 39.4 ± 2.9 multimodal SDCD 5 0.704 ± 0.009 0.682 ± 0.022 0.733 ± 0.025 0.486 ± 0.032 0.468 ± 0.021 0.510 ± 0.051 38.8 ± 2.6 0.366 ± 0.047 75.97 ± 1.97 50.0 ± 4.1 multimodal Globe 5 0.671 ± 0.048 0.865 ± 0.012 0.565 ± 0.070 0.638 ± 0.052 0.819 ± 0.012 0.538 ± 0.071 26.0 ± 3.8 0.200 ± 0.051 820.89 ± 59.56 30.0 ± 3.7 multimodal Das 5 0.615 ± 0.021 0.556 ± 0.028 0.695 ± 0.029 0.456 ± 0.026 0.411 ± 0.025 0.517 ± 0.038 48.6 ± 4.0 0.318 ± 0.046 52.53 ± 0.95 58.2 ± 4.3 multimodal Score 5 0.332 ± 0.026 0.208 ± 0.022 0.858 ± 0.031 0.243 ± 0.014 0.152 ± 0.011 0.638 ± 0.053 179.2 ± 27.8 0.318 ± 0.046 592.77 ± 24.44 201.8 ± 29.7 multimodal NoGAM 5 0.310 ± 0.025 0.191 ± 0.020 0.857 ± 0.029 0.227 ± 0.008 0.139 ± 0.008 0.640 ± 0.050 196.2 ± 28.2 0.317 ± 0.042 790.85 ± 1.41 219.2 ± 31.0 multimodal Avici 5 0.534 ± 0.025 0.876 ± 0.025 0.388 ± 0.028 0.387 ± 0.033 0.631 ± 0.040 0.282 ± 0.030 36.0 ± 2.7 0.420 ± 0.038 13.41 ± 1.51 20.4 ± 1.6 multimodal NOTEARS 5 0.597 ± 0.025 0.483 ± 0.036 0.799 ± 0.038 0.358 ± 0.022 0.290 ± 0.025 0.478 ± 0.027 64.4 ± 3.7 0.404 ± 0.018 2401.07 ± 469.28 76.8 ± 4.0 multimodal DAGMA 5 0.712 ± 0.035 0.737 ± 0.033 0.704 ± 0.059 0.398 ± 0.051 0.416 ± 0.055 0.390 ± 0.052 40.4 ± 4.7 0.433 ± 0.059 2510.75 ± 148.30 43.8 ± 3.3 multimodal DiffAN 5 0.407 ± 0.042 0.276 ± 0.040 0.841 ± 0.020 0.264 ± 0.025 0.178 ± 0.024 0.549 ± 0.030 136.4 ± 23.3 0.405 ± 0.033 483.71 ± 6.55 154.6 ± 25.3 multimodal SciNO 5 0.207 ± 0.003 0.118 ± 0.002 0.865 ± 0.026 0.105 ± 0.014 0.060 ± 0.008 0.440 ± 0.059 326.2 ± 22.2 0.504 ± 0.067 879.08 ± 19.75 340.4 ± 23.6 multimodal Caps 5 0.622 ± 0.051 0.553 ± 0.086 0.757 ± 0.024 0.409 ± 0.024 0.358 ± 0.040 0.505 ± 0.038 58.0 ± 10.4 0.355 ± 0.040 342.71 ± 22.19 70.0 ± 11.8 multimodal PC 5 0.578 ± 0.032 0.833 ± 0.066 0.446 ± 0.025 0.474 ± 0.071 0.693 ± 0.114 0.363 ± 0.051 33.4 ± 2.8 0.316 ± 0.043 1911.69 ± 303.54 25.0 ± 1.6 multimodal FCI 5 0.445 ± 0.041 0.988 ± 0.012 0.291 ± 0.034 0.404 ± 0.028 0.906 ± 0.031 0.264 ± 0.025 34.6 ± 3.5 0.336 ± 0.047 1782.41 ± 241.96 13.4 ± 1.3 unimodal_asymmetric Spade-Hist 5 0.907 ± 0.016 0.995 ± 0.005 0.835 ± 0.027 0.896 ± 0.016 0.983 ± 0.012 0.825 ± 0.026 8.4 ± 1.5 0.031 ± 0.015 1.58 ± 0.08 38.8 ± 2.5 unimodal_asymmetric Spade-Gauss 5 0.829 ± 0.011 0.877 ± 0.024 0.791 ± 0.026 0.691 ± 0.029 0.733 ± 0.042 0.658 ± 0.028 21.4 ± 2.8 0.176 ± 0.037 0.43 ± 0.01 41.6 ± 2.3 unimodal_asymmetric Topic 5 0.709 ± 0.023 0.635 ± 0.030 0.812 ± 0.036 0.560 ± 0.019 0.501 ± 0.022 0.640 ± 0.030 38.4 ± 2.3 0.211 ± 0.009 87.63 ± 3.14 58.8 ± 1.6 unimodal_asymmetric Cam 5 0.147 ± 0.008 0.079 ± 0.004 0.970 ± 0.008 0.124 ± 0.007 0.067 ± 0.004 0.818 ± 0.031 528.4 ± 16.9 0.160 ± 0.037 31125.65 ± 1043.44 565.0 ± 19.3 unimodal_asymmetric CauScale 5 0.877 ± 0.021 0.917 ± 0.026 0.843 ± 0.024 0.629 ± 0.021 0.657 ± 0.021 0.605 ± 0.026 22.0 ± 1.8 0.289 ± 0.043 9.81 ± 1.68 42.6 ± 2.8 unimodal_asymmetric SDCD 5 0.730 ± 0.003 0.662 ± 0.019 0.820 ± 0.022 0.524 ± 0.012 0.474 ± 0.011 0.590 ± 0.027 39.0 ± 3.5 0.284 ± 0.021 72.03 ± 1.26 58.2 ± 5.8 unimodal_asymmetric Globe 5 0.536 ± 0.068 0.827 ± 0.026 0.424 ± 0.082 0.421 ± 0.047 0.657 ± 0.035 0.331 ± 0.057 35.8 ± 3.2 0.423 ± 0.020 613.75 ± 82.17 23.6 ± 4.4 unimodal_asymmetric Das 5 0.632 ± 0.017 0.565 ± 0.038 0.727 ± 0.014 0.441 ± 0.044 0.396 ± 0.049 0.505 ± 0.045 48.8 ± 1.9 0.336 ± 0.039 46.96 ± 2.18 59.6 ± 2.0 unimodal_asymmetric Score 5 0.193 ± 0.012 0.108 ± 0.008 0.946 ± 0.016 0.130 ± 0.012 0.072 ± 0.007 0.634 ± 0.050 381.8 ± 19.3 0.336 ± 0.039 585.44 ± 2.98 408.8 ± 20.9 unimodal_asymmetric NoGAM 5 0.166 ± 0.006 0.091 ± 0.004 0.951 ± 0.014 0.112 ± 0.009 0.061 ± 0.005 0.639 ± 0.043 455.4 ± 13.4 0.330 ± 0.034 790.29 ± 7.55 482.8 ± 16.0 unimodal_asymmetric Avici 5 0.636 ± 0.065 0.971 ± 0.013 0.484 ± 0.063 0.471 ± 0.048 0.722 ± 0.022 0.358 ± 0.047 31.0 ± 4.8 0.359 ± 0.028 11.39 ± 0.26 22.2 ± 1.9 unimodal_asymmetric NOTEARS 5 0.481 ± 0.041 0.338 ± 0.038 0.879 ± 0.022 0.304 ± 0.025 0.213 ± 0.022 0.560 ± 0.032 109.4 ± 20.4 0.358 ± 0.018 1711.93 ± 187.49 129.2 ± 21.5 unimodal_asymmetric DAGMA 5 0.695 ± 0.019 0.615 ± 0.030 0.809 ± 0.031 0.346 ± 0.027 0.307 ± 0.029 0.401 ± 0.030 52.4 ± 6.0 0.510 ± 0.033 2218.57 ± 139.21 62.0 ± 6.5 unimodal_asymmetric DiffAN 5 0.166 ± 0.013 0.091 ± 0.008 0.909 ± 0.007 0.115 ± 0.010 0.063 ± 0.006 0.631 ± 0.036 442.6 ± 30.7 0.337 ± 0.032 490.58 ± 6.22 468.0 ± 32.5 unimodal_asymmetric SciNO 5 0.175 ± 0.013 0.097 ± 0.008 0.930 ± 0.021 0.090 ± 0.005 0.050 ± 0.003 0.484 ± 0.032 429.4 ± 26.7 0.492 ± 0.030 911.16 ± 45.91 448.2 ± 27.7 unimodal_asymmetric Caps 5 0.316 ± 0.019 0.195 ± 0.015 0.855 ± 0.017 0.212 ± 0.020 0.131 ± 0.014 0.572 ± 0.035 184.6 ± 9.2 0.349 ± 0.040 350.22 ± 13.89 204.6 ± 10.6 unimodal_asymmetric PC 5 0.667 ± 0.031 0.864 ± 0.024 0.550 ± 0.042 0.511 ± 0.032 0.676 ± 0.076 0.415 ± 0.016 31.4 ± 3.0 0.334 ± 0.053 2456.83 ± 284.80 30.0 ± 4.1 unimodal_asymmetric FCI 5 0.439 ± 0.040 1.000 0.284 ± 0.033 0.389 ± 0.036 0.889 ± 0.038 0.252 ± 0.029 34.8 ± 3.3 0.340 ± 0.053 2371.96 ± 277.56 13.2 ± 1.7 unimodal_symmetric Spade-Hist 5 0.891 ± 0.011 0.965 ± 0.005 0.828 ± 0.019 0.875 ± 0.008 0.949 ± 0.010 0.813 ± 0.014 10.2 ± 1.2 0.057 ± 0.018 1.57 ± 0.06 39.6 ± 2.4 unimodal_symmetric Spade-Gauss 5 0.874 ± 0.016 0.948 ± 0.018 0.812 ± 0.025 0.804 ± 0.019 0.874 ± 0.030 0.747 ± 0.020 14.2 ± 2.4 0.094 ± 0.012 0.44 ± 0.02 39.6 ± 2.7 unimodal_symmetric Topic 5 0.716 ± 0.006 0.651 ± 0.019 0.802 ± 0.031 0.651 ± 0.021 0.593 ± 0.029 0.728 ± 0.031 32.8 ± 2.7 0.131 ± 0.034 84.72 ± 5.51 56.6 ± 2.0 unimodal_symmetric Cam 5 0.146 ± 0.008 0.079 ± 0.005 0.975 ± 0.012 0.137 ± 0.007 0.074 ± 0.004 0.914 ± 0.017 528.4 ± 18.6 0.060 ± 0.007 29806.36 ± 249.77 569.6 ± 20.7 unimodal_symmetric CauScale 5 0.854 ± 0.016 0.883 ± 0.010 0.830 ± 0.030 0.615 ± 0.015 0.637 ± 0.020 0.597 ± 0.018 23.8 ± 1.7 0.304 ± 0.015 12.55 ± 0.53 43.4 ± 2.7 unimodal_symmetric SDCD 5 0.680 ± 0.023 0.651 ± 0.011 0.714 ± 0.039 0.459 ± 0.039 0.439 ± 0.032 0.484 ± 0.049 41.6 ± 3.8 0.365 ± 0.048 72.83 ± 1.62 51.0 ± 4.6 unimodal_symmetric Globe 5 0.631 ± 0.028 0.879 ± 0.031 0.499 ± 0.039 0.603 ± 0.030 0.840 ± 0.032 0.477 ± 0.039 28.0 ± 3.4 0.285 ± 0.047 646.36 ± 40.49 26.2 ± 2.3 unimodal_symmetric Das 5 0.560 ± 0.026 0.501 ± 0.030 0.636 ± 0.019 0.390 ± 0.014 0.348 ± 0.013 0.445 ± 0.018 55.6 ± 5.0 0.358 ± 0.019 47.85 ± 1.89 59.2 ± 4.5 unimodal_symmetric Score 5 0.200 ± 0.011 0.113 ± 0.007 0.883 ± 0.015 0.139 ± 0.005 0.078 ± 0.003 0.615 ± 0.023 342.4 ± 29.5 0.358 ± 0.019 551.48 ± 19.40 365.6 ± 31.7 unimodal_symmetric NoGAM 5 0.188 ± 0.008 0.105 ± 0.005 0.890 ± 0.023 0.144 ± 0.004 0.080 ± 0.003 0.684 ± 0.028 368.8 ± 31.2 0.293 ± 0.029 788.58 ± 11.56 395.2 ± 32.9 unimodal_symmetric Avici 5 0.683 ± 0.023 0.971 ± 0.019 0.530 ± 0.030 0.601 ± 0.027 0.854 ± 0.027 0.467 ± 0.029 25.4 ± 1.6 0.288 ± 0.020 13.50 ± 1.44 25.6 ± 2.9 unimodal_symmetric NOTEARS 5 0.534 ± 0.035 0.405 ± 0.034 0.800 ± 0.027 0.374 ± 0.036 0.283 ± 0.031 0.560 ± 0.043 80.4 ± 17.2 0.330 ± 0.035 1965.87 ± 284.18 97.2 ± 17.8 unimodal_symmetric DAGMA 5 0.685 ± 0.044 0.679 ± 0.059 0.700 ± 0.044 0.403 ± 0.055 0.404 ± 0.068 0.406 ± 0.043 44.6 ± 8.5 0.408 ± 0.053 2613.99 ± 188.22 49.6 ± 7.3 unimodal_symmetric DiffAN 5 0.200 ± 0.011 0.113 ± 0.007 0.859 ± 0.029 0.120 ± 0.009 0.068 ± 0.005 0.524 ± 0.051 336.2 ± 21.4 0.446 ± 0.053 480.25 ± 9.22 353.2 ± 21.3 unimodal_symmetric SciNO 5 0.202 ± 0.010 0.114 ± 0.006 0.893 ± 0.026 0.103 ± 0.016 0.058 ± 0.009 0.447 ± 0.046 346.8 ± 18.3 0.532 ± 0.049 857.97 ± 6.53 363.0 ± 19.0 unimodal_symmetric Caps 5 0.294 ± 0.007 0.182 ± 0.005 0.776 ± 0.032 0.209 ± 0.017 0.129 ± 0.011 0.549 ± 0.038 184.4 ± 17.5 0.394 ± 0.016 365.11 ± 14.72 199.0 ± 17.7 unimodal_symmetric PC 5 0.683 ± 0.035 0.880 ± 0.026 0.560 ± 0.038 0.535 ± 0.039 0.690 ± 0.042 0.439 ± 0.037 29.4 ± 2.9 0.327 ± 0.025 2452.38 ± 308.83 29.4 ± 2.5 unimodal_symmetric FCI 5 0.502 ± 0.032 0.985 ± 0.015 0.339 ± 0.029 0.453 ± 0.044 0.884 ± 0.050 0.306 ± 0.036 32.6 ± 3.7 0.312 ± 0.055 2338.07 ± 305.32 15.8 ± 1.3 Table 24: Qualitative comparison without marginalization (n=1000n=1000, d=50d=50, degree 2, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.731 ± 0.042 0.966 ± 0.019 0.600 ± 0.058 0.666 ± 0.034 0.881 ± 0.027 0.545 ± 0.050 22.6 ± 3.4 0.259 ± 0.024 1.37 ± 0.05 28.2 ± 1.8 gaussian Spade-Gauss 5 0.745 ± 0.044 0.951 ± 0.021 0.621 ± 0.057 0.735 ± 0.047 0.937 ± 0.028 0.612 ± 0.060 20.0 ± 4.0 0.083 ± 0.017 0.37 ± 0.01 29.6 ± 1.2 gaussian Topic 5 0.651 ± 0.028 0.629 ± 0.021 0.677 ± 0.041 0.591 ± 0.028 0.571 ± 0.018 0.616 ± 0.042 36.6 ± 4.0 0.177 ± 0.026 73.34 ± 3.67 49.4 ± 2.2 gaussian Cam 5 0.172 ± 0.012 0.095 ± 0.007 0.951 ± 0.023 0.160 ± 0.010 0.088 ± 0.006 0.888 ± 0.036 431.8 ± 34.6 0.070 ± 0.019 29545.85 ± 174.38 470.8 ± 37.1 gaussian CauScale 5 0.811 ± 0.025 0.895 ± 0.019 0.743 ± 0.035 0.559 ± 0.074 0.613 ± 0.070 0.516 ± 0.077 26.6 ± 4.3 0.319 ± 0.051 5.79 ± 0.07 38.4 ± 2.7 gaussian SDCD 5 0.654 ± 0.038 0.694 ± 0.026 0.631 ± 0.061 0.464 ± 0.038 0.489 ± 0.019 0.450 ± 0.055 38.4 ± 2.8 0.364 ± 0.031 75.13 ± 1.78 42.0 ± 4.5 gaussian Globe 5 0.542 ± 0.042 0.886 ± 0.038 0.394 ± 0.040 0.537 ± 0.045 0.877 ± 0.043 0.390 ± 0.042 31.2 ± 4.5 0.385 ± 0.025 569.36 ± 34.14 20.2 ± 1.2 gaussian Das 5 0.518 ± 0.021 0.481 ± 0.024 0.571 ± 0.040 0.343 ± 0.050 0.314 ± 0.038 0.386 ± 0.070 57.2 ± 3.3 0.374 ± 0.063 48.75 ± 2.53 54.4 ± 2.3 gaussian Score 5 0.320 ± 0.013 0.194 ± 0.010 0.910 ± 0.029 0.210 ± 0.023 0.127 ± 0.015 0.600 ± 0.070 195.2 ± 15.4 0.374 ± 0.063 570.58 ± 8.03 218.8 ± 17.5 gaussian NoGAM 5 0.302 ± 0.016 0.183 ± 0.012 0.890 ± 0.028 0.202 ± 0.021 0.122 ± 0.013 0.605 ± 0.074 209.2 ± 24.1 0.367 ± 0.073 793.73 ± 5.62 232.4 ± 28.2 gaussian Avici 5 0.688 ± 0.063 0.969 ± 0.031 0.542 ± 0.068 0.645 ± 0.055 0.913 ± 0.042 0.508 ± 0.060 24.2 ± 5.1 0.281 ± 0.032 12.05 ± 0.70 25.4 ± 2.4 gaussian NOTEARS 5 0.546 ± 0.041 0.432 ± 0.048 0.762 ± 0.020 0.416 ± 0.031 0.328 ± 0.034 0.583 ± 0.034 68.0 ± 7.5 0.321 ± 0.033 2019.99 ± 360.07 83.8 ± 6.6 gaussian DAGMA 5 0.666 ± 0.041 0.651 ± 0.070 0.708 ± 0.046 0.436 ± 0.050 0.433 ± 0.075 0.456 ± 0.034 44.6 ± 6.0 0.411 ± 0.036 2497.64 ± 200.64 51.8 ± 5.1 gaussian DiffAN 5 0.357 ± 0.023 0.228 ± 0.019 0.839 ± 0.023 0.247 ± 0.031 0.158 ± 0.021 0.586 ± 0.069 156.2 ± 18.5 0.335 ± 0.071 489.24 ± 7.88 175.6 ± 19.7 gaussian SciNO 5 0.269 ± 0.049 0.167 ± 0.039 0.823 ± 0.010 0.156 ± 0.028 0.097 ± 0.022 0.481 ± 0.033 252.6 ± 43.0 0.438 ± 0.047 955.66 ± 35.32 267.0 ± 44.2 gaussian Caps 5 0.574 ± 0.035 0.456 ± 0.042 0.797 ± 0.023 0.370 ± 0.044 0.292 ± 0.038 0.523 ± 0.073 69.6 ± 8.5 0.416 ± 0.075 381.20 ± 11.54 84.4 ± 10.7 gaussian PC 5 0.748 ± 0.011 0.866 ± 0.026 0.662 ± 0.023 0.528 ± 0.033 0.616 ± 0.056 0.464 ± 0.020 29.8 ± 2.7 0.377 ± 0.053 2012.07 ± 260.94 35.6 ± 3.0 gaussian FCI 5 0.418 ± 0.061 0.987 ± 0.013 0.272 ± 0.047 0.343 ± 0.055 0.810 ± 0.051 0.224 ± 0.041 36.4 ± 3.9 0.427 ± 0.037 1986.92 ± 295.35 12.6 ± 1.9 multimodal Spade-Hist 5 0.735 ± 0.046 0.960 ± 0.018 0.604 ± 0.058 0.725 ± 0.052 0.945 ± 0.024 0.596 ± 0.062 20.6 ± 4.3 0.079 ± 0.013 1.39 ± 0.04 28.6 ± 2.0 multimodal Spade-Gauss 5 0.732 ± 0.045 0.928 ± 0.018 0.612 ± 0.060 0.609 ± 0.055 0.770 ± 0.050 0.510 ± 0.060 25.2 ± 3.8 0.232 ± 0.039 0.37 ± 0.02 30.2 ± 2.6 multimodal Topic 5 0.617 ± 0.033 0.590 ± 0.026 0.654 ± 0.051 0.476 ± 0.043 0.454 ± 0.037 0.505 ± 0.056 43.8 ± 3.8 0.315 ± 0.011 71.78 ± 2.39 50.6 ± 1.7 multimodal Cam 5 0.166 ± 0.010 0.092 ± 0.006 0.913 ± 0.031 0.129 ± 0.012 0.071 ± 0.007 0.711 ± 0.056 435.0 ± 22.4 0.239 ± 0.055 30182.63 ± 244.38 464.2 ± 25.7 multimodal CauScale 5 0.809 ± 0.038 0.950 ± 0.015 0.710 ± 0.052 0.677 ± 0.053 0.793 ± 0.040 0.596 ± 0.060 20.8 ± 3.7 0.202 ± 0.039 6.04 ± 0.13 34.4 ± 2.6 multimodal SDCD 5 0.678 ± 0.015 0.710 ± 0.030 0.655 ± 0.031 0.433 ± 0.018 0.453 ± 0.019 0.419 ± 0.026 39.8 ± 3.4 0.429 ± 0.043 73.17 ± 1.47 43.2 ± 4.5 multimodal Globe 5 0.530 ± 0.044 0.834 ± 0.042 0.391 ± 0.039 0.504 ± 0.051 0.788 ± 0.047 0.374 ± 0.044 32.6 ± 3.3 0.406 ± 0.042 673.63 ± 39.75 21.8 ± 2.6 multimodal Das 5 0.522 ± 0.018 0.501 ± 0.023 0.551 ± 0.033 0.395 ± 0.009 0.379 ± 0.014 0.417 ± 0.023 52.4 ± 1.8 0.283 ± 0.007 52.37 ± 5.20 50.6 ± 2.2 multimodal Score 5 0.306 ± 0.012 0.188 ± 0.009 0.843 ± 0.031 0.235 ± 0.012 0.144 ± 0.009 0.645 ± 0.022 189.6 ± 22.1 0.283 ± 0.007 559.70 ± 20.50 212.4 ± 24.8 multimodal NoGAM 5 0.313 ± 0.019 0.192 ± 0.015 0.864 ± 0.021 0.240 ± 0.014 0.147 ± 0.011 0.664 ± 0.015 191.2 ± 24.6 0.282 ± 0.016 798.99 ± 4.19 215.8 ± 27.0 multimodal Avici 5 0.500 ± 0.048 0.882 ± 0.031 0.355 ± 0.043 0.386 ± 0.045 0.677 ± 0.040 0.274 ± 0.040 36.0 ± 3.6 0.439 ± 0.030 13.15 ± 0.72 18.4 ± 2.1 multimodal NOTEARS 5 0.557 ± 0.023 0.471 ± 0.051 0.733 ± 0.049 0.387 ± 0.030 0.327 ± 0.041 0.510 ± 0.051 66.6 ± 10.8 0.389 ± 0.042 1630.16 ± 225.81 77.6 ± 13.6 multimodal DAGMA 5 0.689 ± 0.020 0.720 ± 0.046 0.675 ± 0.042 0.424 ± 0.037 0.450 ± 0.064 0.410 ± 0.022 40.6 ± 4.8 0.480 ± 0.025 2125.12 ± 101.16 44.0 ± 4.4 multimodal DiffAN 5 0.368 ± 0.041 0.255 ± 0.049 0.769 ± 0.045 0.224 ± 0.006 0.151 ± 0.011 0.491 ± 0.058 146.4 ± 23.6 0.409 ± 0.064 503.66 ± 8.16 158.4 ± 27.3 multimodal SciNO 5 0.229 ± 0.014 0.136 ± 0.010 0.751 ± 0.042 0.128 ± 0.017 0.076 ± 0.010 0.416 ± 0.042 253.6 ± 28.5 0.503 ± 0.046 863.86 ± 13.73 261.8 ± 30.7 multimodal Caps 5 0.510 ± 0.031 0.398 ± 0.038 0.734 ± 0.038 0.343 ± 0.023 0.267 ± 0.023 0.495 ± 0.035 80.2 ± 14.2 0.341 ± 0.019 388.01 ± 8.96 90.6 ± 15.1 multimodal PC 5 0.691 ± 0.025 0.835 ± 0.024 0.593 ± 0.031 0.495 ± 0.027 0.601 ± 0.043 0.423 ± 0.022 32.0 ± 1.9 0.389 ± 0.051 1662.51 ± 144.45 33.2 ± 3.6 multimodal FCI 5 0.404 ± 0.044 0.943 ± 0.027 0.260 ± 0.032 0.353 ± 0.037 0.830 ± 0.034 0.227 ± 0.028 36.6 ± 3.1 0.427 ± 0.026 1725.15 ± 161.95 12.6 ± 1.5 unimodal_asymmetric Spade-Hist 5 0.759 ± 0.025 0.982 ± 0.018 0.623 ± 0.035 0.730 ± 0.032 0.942 ± 0.022 0.599 ± 0.040 19.4 ± 2.8 0.140 ± 0.014 1.40 ± 0.05 29.2 ± 1.8 unimodal_asymmetric Spade-Gauss 5 0.735 ± 0.037 0.877 ± 0.020 0.639 ± 0.048 0.568 ± 0.062 0.671 ± 0.052 0.497 ± 0.065 28.4 ± 5.3 0.318 ± 0.056 0.38 ± 0.01 33.2 ± 1.6 unimodal_asymmetric Topic 5 0.624 ± 0.009 0.587 ± 0.028 0.675 ± 0.025 0.418 ± 0.035 0.389 ± 0.024 0.458 ± 0.050 47.8 ± 3.4 0.402 ± 0.060 78.07 ± 3.33 53.0 ± 1.6 unimodal_asymmetric Cam 5 0.149 ± 0.013 0.081 ± 0.008 0.962 ± 0.012 0.105 ± 0.006 0.057 ± 0.003 0.689 ± 0.045 528.0 ± 22.5 0.286 ± 0.046 29620.12 ± 154.55 557.6 ± 22.2 unimodal_asymmetric CauScale 5 0.820 ± 0.029 0.885 ± 0.021 0.766 ± 0.040 0.550 ± 0.030 0.597 ± 0.040 0.513 ± 0.028 27.2 ± 2.4 0.357 ± 0.021 5.91 ± 0.11 39.6 ± 1.2 unimodal_asymmetric SDCD 5 0.673 ± 0.016 0.653 ± 0.019 0.702 ± 0.038 0.447 ± 0.028 0.435 ± 0.032 0.464 ± 0.032 42.4 ± 3.5 0.381 ± 0.035 76.02 ± 1.22 49.6 ± 3.3 unimodal_asymmetric Globe 5 0.574 ± 0.032 0.868 ± 0.034 0.436 ± 0.036 0.440 ± 0.050 0.659 ± 0.052 0.336 ± 0.045 34.6 ± 4.0 0.458 ± 0.044 561.07 ± 20.90 23.0 ± 1.6 unimodal_asymmetric Das 5 0.536 ± 0.005 0.474 ± 0.018 0.624 ± 0.027 0.374 ± 0.020 0.329 ± 0.009 0.440 ± 0.040 58.4 ± 3.2 0.340 ± 0.039 47.50 ± 1.06 61.0 ± 4.2 unimodal_asymmetric Score 5 0.207 ± 0.013 0.117 ± 0.008 0.919 ± 0.022 0.138 ± 0.006 0.078 ± 0.004 0.615 ± 0.032 342.0 ± 22.3 0.340 ± 0.039 548.07 ± 16.76 366.6 ± 23.7 unimodal_asymmetric NoGAM 5 0.184 ± 0.013 0.103 ± 0.008 0.914 ± 0.012 0.118 ± 0.006 0.066 ± 0.004 0.591 ± 0.025 396.4 ± 33.0 0.358 ± 0.031 800.35 ± 0.98 419.6 ± 34.2 unimodal_asymmetric Avici 5 0.593 ± 0.028 0.929 ± 0.032 0.442 ± 0.037 0.417 ± 0.039 0.654 ± 0.060 0.311 ± 0.033 34.2 ± 3.8 0.466 ± 0.031 12.11 ± 0.52 22.0 ± 2.2 unimodal_asymmetric NOTEARS 5 0.389 ± 0.027 0.258 ± 0.025 0.842 ± 0.039 0.245 ± 0.017 0.162 ± 0.016 0.529 ± 0.019 141.0 ± 19.3 0.394 ± 0.023 1538.22 ± 182.24 157.6 ± 21.0 unimodal_asymmetric DAGMA 5 0.589 ± 0.040 0.512 ± 0.069 0.749 ± 0.036 0.316 ± 0.041 0.279 ± 0.054 0.391 ± 0.021 67.6 ± 12.4 0.543 ± 0.034 2368.78 ± 139.65 73.6 ± 12.9 unimodal_asymmetric DiffAN 5 0.196 ± 0.026 0.111 ± 0.016 0.893 ± 0.019 0.131 ± 0.018 0.074 ± 0.011 0.607 ± 0.053 373.2 ± 42.5 0.318 ± 0.042 500.59 ± 5.33 395.8 ± 41.5 unimodal_asymmetric SciNO 5 0.186 ± 0.008 0.104 ± 0.005 0.882 ± 0.028 0.087 ± 0.003 0.049 ± 0.002 0.417 ± 0.027 379.2 ± 18.6 0.532 ± 0.022 885.27 ± 18.06 392.6 ± 20.2 unimodal_asymmetric Caps 5 0.336 ± 0.034 0.216 ± 0.027 0.787 ± 0.030 0.210 ± 0.016 0.134 ± 0.012 0.500 ± 0.038 164.0 ± 20.5 0.373 ± 0.045 391.25 ± 19.04 176.6 ± 19.5 unimodal_asymmetric PC 5 0.658 ± 0.035 0.809 ± 0.022 0.561 ± 0.047 0.513 ± 0.005 0.635 ± 0.026 0.436 ± 0.020 32.2 ± 1.7 0.341 ± 0.040 2045.53 ± 213.57 32.2 ± 3.5 unimodal_asymmetric FCI 5 0.412 ± 0.043 0.972 ± 0.018 0.266 ± 0.037 0.393 ± 0.043 0.923 ± 0.021 0.254 ± 0.037 35.0 ± 3.0 0.364 ± 0.047 1911.36 ± 184.72 12.8 ± 2.1 unimodal_symmetric Spade-Hist 5 0.738 ± 0.029 0.975 ± 0.015 0.597 ± 0.036 0.729 ± 0.034 0.963 ± 0.024 0.590 ± 0.040 20.2 ± 3.4 0.185 ± 0.026 1.38 ± 0.04 28.2 ± 1.9 unimodal_symmetric Spade-Gauss 5 0.730 ± 0.021 0.906 ± 0.018 0.615 ± 0.032 0.629 ± 0.033 0.780 ± 0.036 0.530 ± 0.036 25.2 ± 3.5 0.162 ± 0.036 0.38 ± 0.02 31.2 ± 1.8 unimodal_symmetric Topic 5 0.622 ± 0.021 0.588 ± 0.027 0.665 ± 0.026 0.514 ± 0.036 0.485 ± 0.035 0.550 ± 0.042 42.6 ± 3.6 0.248 ± 0.015 75.89 ± 3.20 52.0 ± 0.8 unimodal_symmetric Cam 5 0.173 ± 0.010 0.095 ± 0.006 0.960 ± 0.014 0.153 ± 0.005 0.084 ± 0.003 0.859 ± 0.028 434.2 ± 29.0 0.106 ± 0.032 29954.83 ± 217.74 472.0 ± 30.7 unimodal_symmetric CauScale 5 0.775 ± 0.024 0.872 ± 0.030 0.701 ± 0.033 0.473 ± 0.024 0.530 ± 0.014 0.429 ± 0.030 31.2 ± 2.0 0.392 ± 0.034 5.69 ± 0.03 37.4 ± 3.2 unimodal_symmetric SDCD 5 0.641 ± 0.018 0.712 ± 0.018 0.587 ± 0.029 0.484 ± 0.029 0.541 ± 0.042 0.441 ± 0.026 36.6 ± 1.4 0.365 ± 0.043 69.99 ± 2.33 38.4 ± 3.6 unimodal_symmetric Globe 5 0.514 ± 0.018 0.857 ± 0.028 0.368 ± 0.014 0.460 ± 0.028 0.767 ± 0.049 0.328 ± 0.019 34.0 ± 3.0 0.466 ± 0.024 565.45 ± 31.60 20.0 ± 1.9 unimodal_symmetric Das 5 0.515 ± 0.024 0.485 ± 0.023 0.553 ± 0.034 0.400 ± 0.017 0.377 ± 0.016 0.430 ± 0.026 53.8 ± 4.1 0.269 ± 0.016 46.41 ± 0.96 53.4 ± 5.9 unimodal_symmetric Score 5 0.216 ± 0.011 0.123 ± 0.007 0.896 ± 0.012 0.165 ± 0.010 0.094 ± 0.006 0.682 ± 0.020 313.8 ± 25.0 0.269 ± 0.016 543.83 ± 12.13 340.8 ± 27.2 unimodal_symmetric NoGAM 5 0.203 ± 0.007 0.115 ± 0.005 0.893 ± 0.015 0.157 ± 0.007 0.089 ± 0.004 0.692 ± 0.019 333.4 ± 18.8 0.264 ± 0.012 772.59 ± 23.59 360.6 ± 21.3 unimodal_symmetric Avici 5 0.667 ± 0.044 0.939 ± 0.035 0.524 ± 0.048 0.598 ± 0.052 0.845 ± 0.061 0.469 ± 0.049 26.6 ± 4.2 0.258 ± 0.023 11.94 ± 0.55 25.4 ± 1.5 unimodal_symmetric NOTEARS 5 0.505 ± 0.041 0.388 ± 0.051 0.780 ± 0.028 0.385 ± 0.039 0.297 ± 0.046 0.589 ± 0.007 90.4 ± 25.6 0.311 ± 0.023 1873.55 ± 227.20 107.8 ± 28.3 unimodal_symmetric DAGMA 5 0.647 ± 0.015 0.682 ± 0.023 0.623 ± 0.035 0.433 ± 0.017 0.458 ± 0.028 0.416 ± 0.025 41.0 ± 3.5 0.463 ± 0.034 2313.28 ± 295.43 43.2 ± 5.8 unimodal_symmetric DiffAN 5 0.227 ± 0.015 0.131 ± 0.010 0.839 ± 0.033 0.164 ± 0.014 0.095 ± 0.008 0.609 ± 0.044 276.2 ± 14.4 0.335 ± 0.034 479.93 ± 7.45 297.0 ± 15.2 unimodal_symmetric SciNO 5 0.207 ± 0.009 0.119 ± 0.006 0.818 ± 0.023 0.109 ± 0.007 0.063 ± 0.004 0.431 ± 0.013 311.6 ± 27.4 0.489 ± 0.032 890.61 ± 15.64 323.0 ± 28.2 unimodal_symmetric Caps 5 0.358 ± 0.026 0.233 ± 0.023 0.800 ± 0.028 0.253 ± 0.017 0.165 ± 0.015 0.568 ± 0.019 148.2 ± 17.9 0.328 ± 0.019 389.77 ± 14.02 165.4 ± 20.1 unimodal_symmetric PC 5 0.690 ± 0.056 0.859 ± 0.021 0.586 ± 0.066 0.548 ± 0.039 0.687 ± 0.029 0.463 ± 0.048 28.6 ± 1.4 0.320 ± 0.038 2374.94 ± 330.20 31.8 ± 4.7 unimodal_symmetric FCI 5 0.455 ± 0.038 0.988 ± 0.012 0.299 ± 0.033 0.378 ± 0.012 0.841 ± 0.066 0.247 ± 0.012 35.2 ± 2.8 0.405 ± 0.032 2383.62 ± 471.85 13.8 ± 1.2 Table 25: Qualitative comparison without marginalization (n=1000n=1000, d=50d=50, degree 3, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.759 ± 0.033 0.953 ± 0.007 0.635 ± 0.045 0.743 ± 0.033 0.933 ± 0.010 0.621 ± 0.044 29.8 ± 4.6 0.111 ± 0.024 1.64 ± 0.08 47.4 ± 2.8 gaussian Spade-Gauss 5 0.773 ± 0.023 0.966 ± 0.011 0.647 ± 0.033 0.766 ± 0.022 0.958 ± 0.013 0.641 ± 0.031 27.6 ± 3.1 0.046 ± 0.010 0.51 ± 0.04 47.8 ± 2.8 gaussian Topic 5 0.700 ± 0.019 0.748 ± 0.018 0.662 ± 0.033 0.665 ± 0.032 0.711 ± 0.036 0.628 ± 0.040 42.6 ± 3.0 0.118 ± 0.032 102.20 ± 6.57 63.0 ± 2.3 gaussian Cam 5 0.191 ± 0.010 0.107 ± 0.007 0.941 ± 0.023 0.189 ± 0.011 0.105 ± 0.007 0.926 ± 0.022 573.4 ± 30.4 0.039 ± 0.011 27844.70 ± 114.38 635.0 ± 31.3 gaussian CauScale 5 0.791 ± 0.015 0.922 ± 0.025 0.695 ± 0.021 0.627 ± 0.014 0.731 ± 0.018 0.552 ± 0.020 36.8 ± 3.3 0.237 ± 0.016 5.91 ± 0.12 54.0 ± 2.7 gaussian SDCD 5 0.662 ± 0.017 0.674 ± 0.025 0.658 ± 0.032 0.502 ± 0.028 0.510 ± 0.028 0.499 ± 0.038 59.4 ± 4.6 0.276 ± 0.027 78.46 ± 1.66 69.8 ± 3.6 gaussian Globe 5 0.585 ± 0.046 0.871 ± 0.033 0.448 ± 0.051 0.570 ± 0.045 0.849 ± 0.038 0.436 ± 0.049 45.4 ± 4.6 0.241 ± 0.027 839.45 ± 38.32 36.4 ± 3.1 gaussian Das 5 0.576 ± 0.016 0.571 ± 0.014 0.585 ± 0.031 0.441 ± 0.033 0.436 ± 0.023 0.450 ± 0.046 71.2 ± 5.2 0.347 ± 0.033 43.18 ± 0.60 73.2 ± 4.2 gaussian Score 5 0.271 ± 0.037 0.163 ± 0.028 0.869 ± 0.009 0.192 ± 0.020 0.116 ± 0.016 0.629 ± 0.031 386.8 ± 60.1 0.347 ± 0.033 503.64 ± 1.88 422.4 ± 62.8 gaussian NoGAM 5 0.256 ± 0.027 0.152 ± 0.020 0.855 ± 0.016 0.191 ± 0.015 0.113 ± 0.012 0.646 ± 0.027 396.8 ± 55.8 0.328 ± 0.029 683.96 ± 1.42 432.8 ± 58.8 gaussian Avici 5 0.617 ± 0.032 0.962 ± 0.013 0.457 ± 0.033 0.589 ± 0.037 0.918 ± 0.030 0.436 ± 0.036 41.8 ± 3.9 0.249 ± 0.011 13.08 ± 1.24 33.8 ± 2.2 gaussian NOTEARS 5 0.582 ± 0.013 0.478 ± 0.015 0.751 ± 0.029 0.419 ± 0.023 0.343 ± 0.016 0.543 ± 0.042 92.4 ± 7.1 0.287 ± 0.023 2154.72 ± 227.81 112.6 ± 6.1 gaussian DAGMA 5 0.648 ± 0.021 0.665 ± 0.057 0.651 ± 0.026 0.342 ± 0.026 0.354 ± 0.042 0.340 ± 0.012 73.4 ± 7.3 0.465 ± 0.029 2367.59 ± 83.86 72.4 ± 8.2 gaussian DiffAN 5 0.286 ± 0.031 0.177 ± 0.024 0.809 ± 0.020 0.199 ± 0.018 0.123 ± 0.015 0.570 ± 0.028 329.0 ± 50.9 0.360 ± 0.029 478.67 ± 3.10 356.0 ± 53.8 gaussian SciNO 5 0.228 ± 0.018 0.134 ± 0.014 0.816 ± 0.039 0.107 ± 0.007 0.063 ± 0.005 0.385 ± 0.026 444.0 ± 53.8 0.547 ± 0.017 926.35 ± 14.32 458.2 ± 57.1 gaussian Caps 5 0.497 ± 0.067 0.413 ± 0.100 0.708 ± 0.030 0.364 ± 0.038 0.299 ± 0.061 0.526 ± 0.036 129.2 ± 24.0 0.386 ± 0.040 274.00 ± 28.76 145.2 ± 25.1 gaussian PC 5 0.666 ± 0.039 0.910 ± 0.019 0.531 ± 0.047 0.536 ± 0.037 0.731 ± 0.020 0.428 ± 0.043 45.2 ± 4.9 0.294 ± 0.015 4754.26 ± 865.63 41.2 ± 2.2 gaussian FCI 5 0.461 ± 0.058 0.977 ± 0.014 0.312 ± 0.054 0.391 ± 0.051 0.827 ± 0.033 0.264 ± 0.047 54.0 ± 6.0 0.342 ± 0.026 4783.08 ± 924.05 22.4 ± 3.4 multimodal Spade-Hist 5 0.816 ± 0.032 0.972 ± 0.014 0.706 ± 0.041 0.816 ± 0.032 0.972 ± 0.014 0.706 ± 0.041 22.8 ± 4.6 0.044 ± 0.027 1.70 ± 0.07 51.8 ± 2.9 multimodal Spade-Gauss 5 0.762 ± 0.019 0.950 ± 0.007 0.640 ± 0.031 0.674 ± 0.019 0.841 ± 0.022 0.565 ± 0.025 34.0 ± 3.6 0.122 ± 0.023 0.50 ± 0.04 48.0 ± 2.4 multimodal Topic 5 0.699 ± 0.028 0.743 ± 0.027 0.662 ± 0.034 0.587 ± 0.034 0.624 ± 0.034 0.556 ± 0.037 48.4 ± 5.1 0.164 ± 0.019 101.84 ± 7.59 63.4 ± 1.7 multimodal Cam 5 0.201 ± 0.010 0.112 ± 0.006 0.956 ± 0.010 0.186 ± 0.008 0.104 ± 0.005 0.887 ± 0.016 552.2 ± 23.7 0.078 ± 0.012 34612.73 ± 939.23 612.6 ± 25.6 multimodal CauScale 5 0.826 ± 0.009 0.943 ± 0.007 0.735 ± 0.016 0.712 ± 0.040 0.812 ± 0.039 0.635 ± 0.042 29.4 ± 3.1 0.152 ± 0.041 6.02 ± 0.10 55.8 ± 3.0 multimodal SDCD 5 0.704 ± 0.013 0.761 ± 0.023 0.656 ± 0.017 0.529 ± 0.021 0.571 ± 0.014 0.495 ± 0.027 51.0 ± 2.2 0.258 ± 0.024 77.09 ± 0.63 62.0 ± 4.0 multimodal Globe 5 0.558 ± 0.043 0.875 ± 0.033 0.416 ± 0.046 0.530 ± 0.052 0.826 ± 0.050 0.396 ± 0.051 48.6 ± 6.9 0.321 ± 0.033 901.76 ± 76.06 33.6 ± 2.8 multimodal Das 5 0.610 ± 0.033 0.611 ± 0.032 0.610 ± 0.039 0.480 ± 0.047 0.481 ± 0.045 0.480 ± 0.051 64.6 ± 4.5 0.264 ± 0.049 59.16 ± 0.46 71.2 ± 2.9 multimodal Score 5 0.252 ± 0.012 0.149 ± 0.009 0.850 ± 0.036 0.199 ± 0.013 0.117 ± 0.008 0.679 ± 0.057 378.8 ± 33.3 0.264 ± 0.049 739.96 ± 9.50 417.2 ± 39.2 multimodal NoGAM 5 0.256 ± 0.015 0.151 ± 0.011 0.857 ± 0.027 0.209 ± 0.008 0.124 ± 0.006 0.708 ± 0.041 376.2 ± 36.8 0.241 ± 0.040 921.71 ± 16.15 416.8 ± 41.4 multimodal Avici 5 0.392 ± 0.026 0.947 ± 0.026 0.250 ± 0.022 0.327 ± 0.029 0.783 ± 0.019 0.208 ± 0.023 58.0 ± 3.8 0.393 ± 0.018 11.31 ± 0.30 19.2 ± 2.4 multimodal NOTEARS 5 0.605 ± 0.028 0.540 ± 0.056 0.715 ± 0.028 0.338 ± 0.037 0.306 ± 0.052 0.392 ± 0.013 91.4 ± 10.6 0.457 ± 0.038 2236.49 ± 261.08 99.0 ± 11.1 multimodal DAGMA 5 0.651 ± 0.015 0.696 ± 0.030 0.620 ± 0.033 0.284 ± 0.023 0.306 ± 0.036 0.268 ± 0.017 73.0 ± 6.3 0.556 ± 0.031 2388.58 ± 117.40 64.4 ± 5.5 multimodal DiffAN 5 0.290 ± 0.012 0.178 ± 0.011 0.801 ± 0.029 0.214 ± 0.006 0.131 ± 0.006 0.594 ± 0.029 302.6 ± 31.4 0.332 ± 0.029 490.19 ± 6.66 331.2 ± 35.9 multimodal SciNO 5 0.270 ± 0.022 0.163 ± 0.017 0.811 ± 0.023 0.179 ± 0.021 0.109 ± 0.015 0.542 ± 0.051 340.2 ± 28.2 0.367 ± 0.043 597.15 ± 5.35 365.8 ± 31.4 multimodal Caps 5 0.482 ± 0.025 0.378 ± 0.030 0.689 ± 0.046 0.357 ± 0.029 0.278 ± 0.024 0.517 ± 0.061 118.8 ± 8.9 0.286 ± 0.047 276.11 ± 26.07 133.6 ± 13.6 multimodal PC 5 0.693 ± 0.037 0.941 ± 0.009 0.556 ± 0.047 0.475 ± 0.040 0.643 ± 0.029 0.382 ± 0.043 46.8 ± 3.1 0.390 ± 0.027 4287.54 ± 672.48 42.8 ± 5.2 multimodal FCI 5 0.391 ± 0.032 0.981 ± 0.012 0.247 ± 0.024 0.335 ± 0.031 0.838 ± 0.042 0.211 ± 0.022 57.0 ± 3.6 0.393 ± 0.024 4587.85 ± 773.94 18.2 ± 2.2 unimodal_asymmetric Spade-Hist 5 0.815 ± 0.034 0.977 ± 0.007 0.705 ± 0.049 0.812 ± 0.032 0.973 ± 0.009 0.702 ± 0.048 22.8 ± 4.0 0.028 ± 0.002 1.81 ± 0.08 51.6 ± 3.9 unimodal_asymmetric Spade-Gauss 5 0.776 ± 0.027 0.912 ± 0.025 0.676 ± 0.030 0.680 ± 0.036 0.799 ± 0.037 0.593 ± 0.036 34.2 ± 5.0 0.147 ± 0.017 0.52 ± 0.02 53.0 ± 2.8 unimodal_asymmetric Topic 5 0.723 ± 0.029 0.738 ± 0.025 0.711 ± 0.038 0.619 ± 0.042 0.631 ± 0.038 0.609 ± 0.049 46.6 ± 6.3 0.186 ± 0.026 109.79 ± 7.73 68.8 ± 3.4 unimodal_asymmetric Cam 5 0.184 ± 0.007 0.102 ± 0.004 0.964 ± 0.014 0.163 ± 0.006 0.090 ± 0.004 0.855 ± 0.030 618.0 ± 18.8 0.117 ± 0.025 33688.27 ± 862.31 676.4 ± 20.8 unimodal_asymmetric CauScale 5 0.813 ± 0.019 0.923 ± 0.014 0.731 ± 0.036 0.631 ± 0.035 0.716 ± 0.031 0.569 ± 0.044 35.4 ± 3.0 0.256 ± 0.038 6.07 ± 0.06 56.6 ± 3.6 unimodal_asymmetric SDCD 5 0.693 ± 0.024 0.675 ± 0.046 0.725 ± 0.030 0.524 ± 0.027 0.511 ± 0.040 0.547 ± 0.027 59.6 ± 6.9 0.256 ± 0.015 76.54 ± 1.14 78.4 ± 7.4 unimodal_asymmetric Globe 5 0.520 ± 0.022 0.898 ± 0.015 0.369 ± 0.024 0.463 ± 0.018 0.803 ± 0.041 0.327 ± 0.018 51.4 ± 3.3 0.394 ± 0.014 640.28 ± 29.86 29.2 ± 1.6 unimodal_asymmetric Das 5 0.583 ± 0.014 0.540 ± 0.013 0.635 ± 0.018 0.438 ± 0.016 0.406 ± 0.018 0.476 ± 0.014 76.6 ± 5.9 0.329 ± 0.049 60.59 ± 0.85 84.4 ± 5.4 unimodal_asymmetric Score 5 0.210 ± 0.009 0.119 ± 0.006 0.908 ± 0.020 0.148 ± 0.008 0.084 ± 0.005 0.645 ± 0.052 510.0 ± 31.0 0.329 ± 0.049 732.64 ± 10.83 549.8 ± 35.7 unimodal_asymmetric NoGAM 5 0.195 ± 0.008 0.109 ± 0.005 0.922 ± 0.015 0.140 ± 0.007 0.078 ± 0.004 0.665 ± 0.038 565.8 ± 25.6 0.303 ± 0.037 929.09 ± 8.79 607.8 ± 28.9 unimodal_asymmetric Avici 5 0.572 ± 0.022 0.925 ± 0.025 0.419 ± 0.029 0.418 ± 0.032 0.677 ± 0.052 0.306 ± 0.029 52.6 ± 3.8 0.390 ± 0.029 11.16 ± 0.20 32.4 ± 2.5 unimodal_asymmetric NOTEARS 5 0.432 ± 0.029 0.297 ± 0.026 0.813 ± 0.030 0.252 ± 0.032 0.174 ± 0.026 0.466 ± 0.033 184.0 ± 26.9 0.416 ± 0.036 2173.98 ± 276.19 203.6 ± 25.1 unimodal_asymmetric DAGMA 5 0.618 ± 0.030 0.562 ± 0.036 0.693 ± 0.037 0.287 ± 0.023 0.262 ± 0.027 0.320 ± 0.022 89.4 ± 10.4 0.514 ± 0.015 2180.45 ± 117.11 89.8 ± 8.3 unimodal_asymmetric DiffAN 5 0.205 ± 0.009 0.116 ± 0.006 0.885 ± 0.022 0.139 ± 0.004 0.079 ± 0.003 0.604 ± 0.029 513.8 ± 18.5 0.336 ± 0.018 473.72 ± 5.09 548.2 ± 19.2 unimodal_asymmetric SciNO 5 0.248 ± 0.013 0.145 ± 0.009 0.875 ± 0.038 0.170 ± 0.004 0.100 ± 0.003 0.606 ± 0.033 405.2 ± 34.9 0.338 ± 0.022 598.47 ± 6.94 439.4 ± 37.9 unimodal_asymmetric Caps 5 0.283 ± 0.016 0.176 ± 0.012 0.736 ± 0.022 0.209 ± 0.015 0.130 ± 0.010 0.546 ± 0.041 282.0 ± 17.0 0.349 ± 0.036 327.16 ± 16.66 302.4 ± 20.0 unimodal_asymmetric PC 5 0.708 ± 0.031 0.967 ± 0.017 0.563 ± 0.038 0.545 ± 0.055 0.738 ± 0.048 0.436 ± 0.055 42.2 ± 4.8 0.325 ± 0.040 5588.23 ± 956.54 42.0 ± 4.2 unimodal_asymmetric FCI 5 0.490 ± 0.013 1.000 0.324 ± 0.011 0.438 ± 0.014 0.897 ± 0.032 0.290 ± 0.010 51.0 ± 3.5 0.307 ± 0.014 6244.24 ± 1244.72 23.2 ± 1.2 unimodal_symmetric Spade-Hist 5 0.799 ± 0.014 0.948 ± 0.017 0.693 ± 0.025 0.777 ± 0.024 0.922 ± 0.028 0.674 ± 0.028 26.6 ± 3.4 0.069 ± 0.021 1.78 ± 0.08 52.2 ± 2.2 unimodal_symmetric Spade-Gauss 5 0.761 ± 0.026 0.921 ± 0.019 0.653 ± 0.037 0.725 ± 0.026 0.877 ± 0.021 0.622 ± 0.036 31.4 ± 3.5 0.081 ± 0.009 0.55 ± 0.05 50.6 ± 2.7 unimodal_symmetric Topic 5 0.696 ± 0.016 0.736 ± 0.017 0.662 ± 0.023 0.629 ± 0.010 0.666 ± 0.015 0.599 ± 0.017 46.0 ± 3.1 0.129 ± 0.021 106.17 ± 8.42 64.2 ± 2.2 unimodal_symmetric Cam 5 0.195 ± 0.009 0.108 ± 0.005 0.956 ± 0.009 0.182 ± 0.008 0.101 ± 0.005 0.892 ± 0.012 571.0 ± 14.4 0.075 ± 0.008 33780.84 ± 1099.64 631.8 ± 15.7 unimodal_symmetric CauScale 5 0.814 ± 0.022 0.910 ± 0.004 0.740 ± 0.037 0.605 ± 0.039 0.674 ± 0.028 0.551 ± 0.047 37.6 ± 4.0 0.271 ± 0.025 5.84 ± 0.13 58.0 ± 3.1 unimodal_symmetric SDCD 5 0.670 ± 0.012 0.681 ± 0.026 0.667 ± 0.027 0.526 ± 0.023 0.536 ± 0.037 0.522 ± 0.021 57.0 ± 3.3 0.253 ± 0.035 77.29 ± 0.44 70.4 ± 4.5 unimodal_symmetric Globe 5 0.510 ± 0.024 0.854 ± 0.032 0.365 ± 0.021 0.482 ± 0.028 0.808 ± 0.046 0.344 ± 0.022 52.0 ± 5.2 0.362 ± 0.024 699.86 ± 67.87 31.0 ± 3.5 unimodal_symmetric Das 5 0.588 ± 0.015 0.547 ± 0.015 0.639 ± 0.027 0.452 ± 0.047 0.420 ± 0.041 0.491 ± 0.056 74.4 ± 5.4 0.333 ± 0.052 59.51 ± 0.55 83.2 ± 2.5 unimodal_symmetric Score 5 0.226 ± 0.012 0.130 ± 0.009 0.878 ± 0.029 0.156 ± 0.004 0.090 ± 0.002 0.619 ± 0.057 456.6 ± 37.7 0.333 ± 0.052 722.93 ± 16.27 491.8 ± 42.8 unimodal_symmetric NoGAM 5 0.216 ± 0.008 0.124 ± 0.006 0.874 ± 0.031 0.157 ± 0.004 0.090 ± 0.002 0.646 ± 0.055 474.8 ± 34.2 0.311 ± 0.048 911.61 ± 24.38 511.6 ± 39.1 unimodal_symmetric Avici 5 0.679 ± 0.030 0.952 ± 0.023 0.530 ± 0.034 0.632 ± 0.041 0.887 ± 0.051 0.494 ± 0.039 38.2 ± 3.7 0.207 ± 0.027 11.39 ± 0.45 39.6 ± 1.9 unimodal_symmetric NOTEARS 5 0.517 ± 0.029 0.404 ± 0.041 0.755 ± 0.036 0.372 ± 0.047 0.294 ± 0.049 0.529 ± 0.025 120.4 ± 17.6 0.314 ± 0.038 2485.35 ± 339.40 140.4 ± 18.4 unimodal_symmetric DAGMA 5 0.615 ± 0.026 0.605 ± 0.051 0.639 ± 0.025 0.317 ± 0.063 0.320 ± 0.076 0.320 ± 0.050 81.6 ± 11.4 0.463 ± 0.066 2491.60 ± 127.99 78.0 ± 8.5 unimodal_symmetric DiffAN 5 0.240 ± 0.013 0.140 ± 0.009 0.841 ± 0.019 0.169 ± 0.007 0.099 ± 0.005 0.597 ± 0.030 405.8 ± 33.5 0.365 ± 0.026 494.78 ± 6.50 437.2 ± 36.7 unimodal_symmetric SciNO 5 0.259 ± 0.012 0.155 ± 0.009 0.812 ± 0.031 0.189 ± 0.008 0.113 ± 0.006 0.595 ± 0.028 352.8 ± 29.6 0.315 ± 0.019 587.53 ± 3.85 381.4 ± 31.0 unimodal_symmetric Caps 5 0.384 ± 0.026 0.263 ± 0.021 0.725 ± 0.040 0.273 ± 0.024 0.186 ± 0.015 0.524 ± 0.060 184.6 ± 18.8 0.349 ± 0.053 274.71 ± 18.94 201.6 ± 18.4 unimodal_symmetric PC 5 0.690 ± 0.030 0.940 ± 0.016 0.548 ± 0.036 0.502 ± 0.037 0.685 ± 0.045 0.399 ± 0.034 45.2 ± 1.8 0.358 ± 0.043 6028.76 ± 1278.80 41.8 ± 3.4 unimodal_symmetric FCI 5 0.474 ± 0.048 0.993 ± 0.007 0.317 ± 0.040 0.420 ± 0.039 0.885 ± 0.018 0.281 ± 0.033 51.6 ± 3.0 0.314 ± 0.039 6282.44 ± 1241.51 23.2 ± 3.5 Table 26: Qualitative comparison without marginalization (n=1000n=1000, d=50d=50, degree 3, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.628 ± 0.043 0.960 ± 0.007 0.472 ± 0.048 0.575 ± 0.040 0.880 ± 0.015 0.433 ± 0.044 42.8 ± 5.8 0.290 ± 0.044 1.42 ± 0.04 34.6 ± 2.0 gaussian Spade-Gauss 5 0.664 ± 0.028 0.958 ± 0.010 0.511 ± 0.033 0.639 ± 0.029 0.922 ± 0.018 0.492 ± 0.033 38.6 ± 4.7 0.110 ± 0.023 0.43 ± 0.02 37.8 ± 1.0 gaussian Topic 5 0.621 ± 0.029 0.702 ± 0.018 0.561 ± 0.040 0.565 ± 0.024 0.639 ± 0.015 0.509 ± 0.033 52.6 ± 5.2 0.157 ± 0.023 84.96 ± 4.81 56.4 ± 0.7 gaussian Cam 5 0.210 ± 0.006 0.118 ± 0.004 0.935 ± 0.012 0.191 ± 0.007 0.108 ± 0.004 0.850 ± 0.026 511.2 ± 28.6 0.107 ± 0.023 26618.99 ± 346.44 567.2 ± 30.6 gaussian CauScale 5 0.734 ± 0.034 0.891 ± 0.020 0.627 ± 0.042 0.512 ± 0.046 0.620 ± 0.044 0.438 ± 0.047 46.2 ± 5.8 0.317 ± 0.033 5.73 ± 0.04 49.8 ± 1.1 gaussian SDCD 5 0.631 ± 0.040 0.716 ± 0.053 0.565 ± 0.035 0.491 ± 0.051 0.559 ± 0.066 0.439 ± 0.043 57.4 ± 8.1 0.318 ± 0.040 72.92 ± 1.21 57.0 ± 4.3 gaussian Globe 5 0.393 ± 0.034 0.853 ± 0.038 0.259 ± 0.029 0.376 ± 0.040 0.813 ± 0.049 0.248 ± 0.032 57.8 ± 5.7 0.442 ± 0.034 605.40 ± 49.58 21.6 ± 2.1 gaussian Das 5 0.549 ± 0.018 0.532 ± 0.011 0.572 ± 0.033 0.383 ± 0.035 0.372 ± 0.037 0.397 ± 0.038 79.6 ± 6.6 0.351 ± 0.047 43.56 ± 0.71 76.8 ± 4.7 gaussian Score 5 0.268 ± 0.021 0.160 ± 0.015 0.857 ± 0.019 0.181 ± 0.007 0.108 ± 0.005 0.588 ± 0.039 365.4 ± 35.7 0.351 ± 0.047 507.57 ± 6.30 397.0 ± 38.1 gaussian NoGAM 5 0.272 ± 0.023 0.162 ± 0.016 0.876 ± 0.013 0.179 ± 0.008 0.106 ± 0.006 0.584 ± 0.029 368.8 ± 36.9 0.364 ± 0.036 685.00 ± 3.61 401.6 ± 39.2 gaussian Avici 5 0.541 ± 0.033 0.931 ± 0.020 0.387 ± 0.036 0.459 ± 0.025 0.792 ± 0.030 0.328 ± 0.028 50.8 ± 4.5 0.406 ± 0.040 14.31 ± 1.34 29.6 ± 2.6 gaussian NOTEARS 5 0.566 ± 0.042 0.462 ± 0.048 0.741 ± 0.037 0.419 ± 0.047 0.343 ± 0.048 0.545 ± 0.045 98.2 ± 13.1 0.321 ± 0.045 1801.42 ± 180.17 118.4 ± 11.5 gaussian DAGMA 5 0.588 ± 0.028 0.617 ± 0.039 0.566 ± 0.030 0.352 ± 0.018 0.369 ± 0.020 0.340 ± 0.022 73.2 ± 5.3 0.481 ± 0.018 2434.54 ± 158.05 66.6 ± 6.3 gaussian DiffAN 5 0.296 ± 0.035 0.186 ± 0.027 0.759 ± 0.025 0.207 ± 0.016 0.130 ± 0.014 0.543 ± 0.032 291.8 ± 38.0 0.354 ± 0.040 490.84 ± 4.02 312.8 ± 39.1 gaussian SciNO 5 0.284 ± 0.023 0.176 ± 0.017 0.763 ± 0.039 0.156 ± 0.009 0.096 ± 0.007 0.420 ± 0.021 308.4 ± 33.1 0.501 ± 0.030 872.28 ± 10.82 321.2 ± 34.7 gaussian Caps 5 0.450 ± 0.045 0.336 ± 0.046 0.701 ± 0.034 0.303 ± 0.021 0.224 ± 0.020 0.479 ± 0.037 145.6 ± 20.0 0.386 ± 0.048 336.53 ± 15.62 157.6 ± 18.2 gaussian PC 5 0.739 ± 0.021 0.901 ± 0.017 0.631 ± 0.031 0.501 ± 0.058 0.606 ± 0.056 0.430 ± 0.058 46.4 ± 5.7 0.346 ± 0.051 3718.27 ± 458.39 50.0 ± 2.8 gaussian FCI 5 0.428 ± 0.045 0.953 ± 0.026 0.280 ± 0.036 0.362 ± 0.037 0.808 ± 0.038 0.236 ± 0.029 55.8 ± 4.7 0.391 ± 0.022 3926.71 ± 500.57 20.8 ± 2.4 multimodal Spade-Hist 5 0.644 ± 0.033 0.969 ± 0.011 0.487 ± 0.038 0.639 ± 0.033 0.962 ± 0.008 0.483 ± 0.039 38.8 ± 4.9 0.169 ± 0.037 1.74 ± 0.15 35.6 ± 2.2 multimodal Spade-Gauss 5 0.683 ± 0.029 0.942 ± 0.021 0.540 ± 0.037 0.570 ± 0.031 0.788 ± 0.036 0.451 ± 0.034 42.2 ± 4.3 0.227 ± 0.029 0.52 ± 0.05 40.6 ± 1.5 multimodal Topic 5 0.645 ± 0.013 0.711 ± 0.019 0.593 ± 0.022 0.498 ± 0.019 0.548 ± 0.016 0.458 ± 0.026 56.4 ± 3.6 0.291 ± 0.035 91.53 ± 6.52 59.4 ± 1.4 multimodal Cam 5 0.220 ± 0.019 0.126 ± 0.012 0.899 ± 0.007 0.187 ± 0.016 0.107 ± 0.010 0.765 ± 0.013 479.2 ± 41.4 0.150 ± 0.017 33779.65 ± 1065.01 527.0 ± 41.4 multimodal CauScale 5 0.756 ± 0.021 0.931 ± 0.012 0.639 ± 0.029 0.662 ± 0.017 0.815 ± 0.010 0.559 ± 0.024 35.4 ± 3.6 0.185 ± 0.018 5.88 ± 0.05 48.8 ± 1.3 multimodal SDCD 5 0.649 ± 0.021 0.721 ± 0.026 0.594 ± 0.029 0.468 ± 0.032 0.520 ± 0.034 0.429 ± 0.035 57.4 ± 3.4 0.350 ± 0.030 78.70 ± 1.59 58.8 ± 2.6 multimodal Globe 5 0.513 ± 0.049 0.864 ± 0.026 0.373 ± 0.047 0.470 ± 0.048 0.786 ± 0.020 0.342 ± 0.046 52.2 ± 6.1 0.401 ± 0.028 810.08 ± 41.90 30.2 ± 2.7 multimodal Das 5 0.525 ± 0.025 0.539 ± 0.031 0.514 ± 0.022 0.441 ± 0.017 0.452 ± 0.021 0.432 ± 0.014 72.8 ± 5.6 0.226 ± 0.023 59.13 ± 0.39 68.6 ± 3.7 multimodal Score 5 0.287 ± 0.027 0.174 ± 0.019 0.842 ± 0.006 0.230 ± 0.019 0.139 ± 0.014 0.680 ± 0.025 327.8 ± 42.9 0.226 ± 0.023 713.66 ± 20.66 365.4 ± 44.8 multimodal NoGAM 5 0.290 ± 0.026 0.176 ± 0.019 0.849 ± 0.012 0.224 ± 0.018 0.136 ± 0.013 0.665 ± 0.038 326.6 ± 42.8 0.256 ± 0.041 922.46 ± 22.26 363.2 ± 44.9 multimodal Avici 5 0.451 ± 0.069 0.957 ± 0.016 0.305 ± 0.059 0.373 ± 0.068 0.777 ± 0.046 0.254 ± 0.056 54.6 ± 5.7 0.398 ± 0.044 11.32 ± 0.27 22.4 ± 3.7 multimodal NOTEARS 5 0.631 ± 0.030 0.553 ± 0.037 0.741 ± 0.030 0.463 ± 0.044 0.409 ± 0.049 0.540 ± 0.036 76.6 ± 7.3 0.329 ± 0.031 2176.30 ± 197.75 96.4 ± 4.1 multimodal DAGMA 5 0.654 ± 0.035 0.732 ± 0.024 0.598 ± 0.048 0.348 ± 0.041 0.388 ± 0.039 0.319 ± 0.045 65.0 ± 6.3 0.510 ± 0.040 2177.82 ± 116.74 57.8 ± 2.3 multimodal DiffAN 5 0.326 ± 0.024 0.212 ± 0.022 0.745 ± 0.035 0.228 ± 0.018 0.149 ± 0.016 0.523 ± 0.037 247.6 ± 35.7 0.343 ± 0.042 481.76 ± 5.62 266.4 ± 39.2 multimodal SciNO 5 0.309 ± 0.014 0.198 ± 0.012 0.719 ± 0.017 0.220 ± 0.014 0.141 ± 0.009 0.516 ± 0.040 247.4 ± 20.2 0.315 ± 0.047 582.57 ± 3.57 264.4 ± 22.7 multimodal Caps 5 0.493 ± 0.053 0.396 ± 0.063 0.700 ± 0.018 0.364 ± 0.031 0.290 ± 0.039 0.523 ± 0.026 126.0 ± 22.9 0.284 ± 0.026 334.55 ± 23.45 141.8 ± 24.4 multimodal PC 5 0.742 ± 0.033 0.928 ± 0.015 0.625 ± 0.049 0.493 ± 0.014 0.619 ± 0.023 0.414 ± 0.023 45.8 ± 3.6 0.376 ± 0.016 4056.00 ± 486.76 48.0 ± 3.3 multimodal FCI 5 0.375 ± 0.039 1.000 0.233 ± 0.030 0.339 ± 0.039 0.905 ± 0.031 0.211 ± 0.029 56.8 ± 4.8 0.462 ± 0.035 4232.47 ± 553.04 16.6 ± 2.0 unimodal_asymmetric Spade-Hist 5 0.596 ± 0.031 0.971 ± 0.013 0.433 ± 0.032 0.569 ± 0.031 0.928 ± 0.019 0.413 ± 0.031 43.2 ± 3.8 0.275 ± 0.025 1.56 ± 0.10 32.0 ± 2.8 unimodal_asymmetric Spade-Gauss 5 0.639 ± 0.021 0.850 ± 0.018 0.515 ± 0.026 0.511 ± 0.027 0.678 ± 0.022 0.412 ± 0.029 49.2 ± 4.7 0.314 ± 0.013 0.44 ± 0.01 43.2 ± 2.2 unimodal_asymmetric Topic 5 0.628 ± 0.022 0.702 ± 0.020 0.571 ± 0.031 0.467 ± 0.031 0.522 ± 0.034 0.424 ± 0.033 58.8 ± 4.5 0.333 ± 0.028 92.40 ± 4.71 57.8 ± 1.9 unimodal_asymmetric Cam 5 0.206 ± 0.014 0.116 ± 0.009 0.938 ± 0.011 0.153 ± 0.009 0.086 ± 0.006 0.698 ± 0.031 541.0 ± 35.7 0.259 ± 0.037 34808.25 ± 382.28 586.6 ± 37.3 unimodal_asymmetric CauScale 5 0.770 ± 0.025 0.930 ± 0.008 0.662 ± 0.037 0.555 ± 0.045 0.666 ± 0.037 0.478 ± 0.046 40.8 ± 3.0 0.288 ± 0.035 5.88 ± 0.08 51.2 ± 4.3 unimodal_asymmetric SDCD 5 0.614 ± 0.019 0.671 ± 0.030 0.572 ± 0.029 0.465 ± 0.031 0.506 ± 0.033 0.434 ± 0.035 61.0 ± 3.4 0.345 ± 0.043 75.33 ± 2.10 61.4 ± 4.3 unimodal_asymmetric Globe 5 0.391 ± 0.013 0.820 ± 0.037 0.258 ± 0.011 0.291 ± 0.030 0.606 ± 0.057 0.192 ± 0.021 62.2 ± 3.9 0.567 ± 0.034 596.88 ± 43.53 22.8 ± 1.9 unimodal_asymmetric Das 5 0.514 ± 0.015 0.467 ± 0.017 0.578 ± 0.038 0.381 ± 0.029 0.345 ± 0.022 0.433 ± 0.050 88.2 ± 3.8 0.334 ± 0.054 61.01 ± 1.61 88.6 ± 6.1 unimodal_asymmetric Score 5 0.220 ± 0.010 0.125 ± 0.006 0.923 ± 0.019 0.148 ± 0.011 0.084 ± 0.006 0.625 ± 0.054 492.8 ± 23.7 0.334 ± 0.054 736.91 ± 10.20 532.4 ± 28.4 unimodal_asymmetric NoGAM 5 0.195 ± 0.009 0.110 ± 0.006 0.896 ± 0.014 0.129 ± 0.011 0.073 ± 0.006 0.595 ± 0.046 552.0 ± 22.6 0.340 ± 0.043 942.53 ± 12.21 587.6 ± 26.3 unimodal_asymmetric Avici 5 0.506 ± 0.049 0.961 ± 0.025 0.348 ± 0.042 0.360 ± 0.066 0.669 ± 0.097 0.249 ± 0.049 54.6 ± 4.6 0.476 ± 0.030 11.31 ± 0.24 25.6 ± 2.5 unimodal_asymmetric NOTEARS 5 0.441 ± 0.020 0.314 ± 0.023 0.759 ± 0.022 0.297 ± 0.017 0.211 ± 0.016 0.512 ± 0.032 158.8 ± 18.5 0.374 ± 0.046 1657.09 ± 264.55 177.8 ± 19.4 unimodal_asymmetric DAGMA 5 0.578 ± 0.035 0.546 ± 0.049 0.627 ± 0.027 0.325 ± 0.039 0.309 ± 0.044 0.348 ± 0.031 87.0 ± 10.8 0.488 ± 0.033 1971.15 ± 37.66 85.0 ± 9.3 unimodal_asymmetric DiffAN 5 0.214 ± 0.014 0.123 ± 0.010 0.844 ± 0.022 0.139 ± 0.009 0.080 ± 0.006 0.550 ± 0.031 468.0 ± 26.2 0.363 ± 0.019 483.59 ± 7.48 496.0 ± 28.3 unimodal_asymmetric SciNO 5 0.276 ± 0.008 0.167 ± 0.006 0.820 ± 0.019 0.189 ± 0.009 0.114 ± 0.006 0.561 ± 0.030 327.4 ± 21.6 0.331 ± 0.042 604.51 ± 8.61 354.4 ± 23.4 unimodal_asymmetric Caps 5 0.317 ± 0.010 0.201 ± 0.008 0.756 ± 0.029 0.224 ± 0.017 0.142 ± 0.010 0.539 ± 0.052 248.6 ± 14.0 0.330 ± 0.054 352.57 ± 9.13 270.0 ± 18.0 unimodal_asymmetric PC 5 0.712 ± 0.026 0.916 ± 0.030 0.585 ± 0.032 0.507 ± 0.019 0.656 ± 0.043 0.415 ± 0.013 46.0 ± 3.3 0.371 ± 0.034 4045.10 ± 669.43 46.2 ± 4.7 unimodal_asymmetric FCI 5 0.396 ± 0.048 1.000 0.251 ± 0.038 0.355 ± 0.052 0.885 ± 0.026 0.226 ± 0.039 55.8 ± 5.3 0.379 ± 0.025 4387.45 ± 755.06 17.8 ± 2.4 unimodal_symmetric Spade-Hist 5 0.639 ± 0.037 0.980 ± 0.014 0.478 ± 0.040 0.611 ± 0.047 0.936 ± 0.035 0.458 ± 0.046 40.0 ± 5.2 0.263 ± 0.018 1.71 ± 0.10 35.0 ± 3.4 unimodal_symmetric Spade-Gauss 5 0.703 ± 0.036 0.965 ± 0.014 0.558 ± 0.044 0.646 ± 0.030 0.888 ± 0.012 0.513 ± 0.037 36.6 ± 3.9 0.186 ± 0.030 0.47 ± 0.02 41.6 ± 4.0 unimodal_symmetric Topic 5 0.626 ± 0.028 0.685 ± 0.033 0.578 ± 0.029 0.537 ± 0.034 0.589 ± 0.042 0.496 ± 0.032 55.4 ± 5.0 0.266 ± 0.024 88.75 ± 5.31 60.2 ± 2.0 unimodal_symmetric Cam 5 0.208 ± 0.012 0.118 ± 0.008 0.900 ± 0.021 0.180 ± 0.013 0.102 ± 0.008 0.778 ± 0.023 504.6 ± 34.2 0.155 ± 0.018 33817.52 ± 334.91 553.0 ± 34.7 unimodal_symmetric CauScale 5 0.759 ± 0.014 0.935 ± 0.016 0.641 ± 0.022 0.540 ± 0.040 0.665 ± 0.050 0.456 ± 0.036 42.0 ± 2.2 0.336 ± 0.037 5.89 ± 0.13 49.0 ± 2.3 unimodal_symmetric SDCD 5 0.587 ± 0.021 0.667 ± 0.048 0.534 ± 0.026 0.456 ± 0.030 0.517 ± 0.047 0.415 ± 0.032 62.0 ± 3.6 0.366 ± 0.032 75.59 ± 1.86 58.4 ± 5.3 unimodal_symmetric Globe 5 0.422 ± 0.040 0.857 ± 0.026 0.285 ± 0.037 0.378 ± 0.048 0.760 ± 0.041 0.257 ± 0.041 57.4 ± 5.7 0.497 ± 0.028 634.71 ± 39.22 23.8 ± 2.8 unimodal_symmetric Das 5 0.529 ± 0.013 0.534 ± 0.019 0.526 ± 0.018 0.410 ± 0.027 0.413 ± 0.026 0.409 ± 0.031 75.8 ± 5.6 0.310 ± 0.034 60.33 ± 0.87 71.2 ± 5.9 unimodal_symmetric Score 5 0.226 ± 0.012 0.131 ± 0.008 0.849 ± 0.030 0.165 ± 0.009 0.095 ± 0.006 0.621 ± 0.034 439.0 ± 39.3 0.310 ± 0.034 728.11 ± 9.41 472.6 ± 43.0 unimodal_symmetric NoGAM 5 0.229 ± 0.015 0.133 ± 0.011 0.862 ± 0.022 0.170 ± 0.012 0.098 ± 0.008 0.640 ± 0.030 439.8 ± 42.4 0.309 ± 0.033 949.28 ± 11.91 475.6 ± 45.1 unimodal_symmetric Avici 5 0.616 ± 0.033 0.944 ± 0.013 0.461 ± 0.037 0.569 ± 0.039 0.873 ± 0.039 0.426 ± 0.039 43.4 ± 4.2 0.288 ± 0.034 11.07 ± 0.14 35.0 ± 3.0 unimodal_symmetric NOTEARS 5 0.491 ± 0.028 0.380 ± 0.037 0.724 ± 0.031 0.368 ± 0.036 0.286 ± 0.039 0.537 ± 0.028 122.8 ± 13.8 0.316 ± 0.026 1615.11 ± 272.42 141.4 ± 15.4 unimodal_symmetric DAGMA 5 0.591 ± 0.022 0.616 ± 0.052 0.582 ± 0.026 0.350 ± 0.034 0.369 ± 0.050 0.341 ± 0.022 75.4 ± 7.1 0.489 ± 0.041 2240.34 ± 101.75 70.0 ± 8.1 unimodal_symmetric DiffAN 5 0.242 ± 0.012 0.144 ± 0.009 0.792 ± 0.042 0.165 ± 0.005 0.098 ± 0.004 0.543 ± 0.033 378.2 ± 36.1 0.352 ± 0.028 481.56 ± 5.75 401.6 ± 39.6 unimodal_symmetric SciNO 5 0.301 ± 0.013 0.185 ± 0.011 0.821 ± 0.031 0.221 ± 0.008 0.136 ± 0.006 0.606 ± 0.040 292.8 ± 27.0 0.273 ± 0.018 606.87 ± 3.28 322.8 ± 29.7 unimodal_symmetric Caps 5 0.349 ± 0.021 0.233 ± 0.019 0.727 ± 0.028 0.255 ± 0.010 0.170 ± 0.011 0.536 ± 0.031 211.2 ± 19.6 0.327 ± 0.045 367.04 ± 16.66 229.4 ± 22.5 unimodal_symmetric PC 5 0.767 ± 0.014 0.938 ± 0.022 0.651 ± 0.024 0.460 ± 0.032 0.563 ± 0.038 0.391 ± 0.031 47.4 ± 4.8 0.479 ± 0.046 3983.03 ± 491.63 49.6 ± 2.4 unimodal_symmetric FCI 5 0.425 ± 0.058 0.967 ± 0.016 0.280 ± 0.052 0.353 ± 0.049 0.805 ± 0.019 0.233 ± 0.044 56.0 ± 5.3 0.416 ± 0.032 4247.76 ± 485.95 20.4 ± 3.2 Table 27: Qualitative comparison without marginalization (n=1000n=1000, d=50d=50, degree 4, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.718 ± 0.030 0.946 ± 0.011 0.582 ± 0.040 0.702 ± 0.029 0.926 ± 0.016 0.569 ± 0.038 45.4 ± 6.1 0.097 ± 0.024 1.99 ± 0.11 58.4 ± 2.1 gaussian Spade-Gauss 5 0.735 ± 0.025 0.963 ± 0.011 0.596 ± 0.030 0.735 ± 0.025 0.963 ± 0.011 0.596 ± 0.030 41.6 ± 5.3 0.047 ± 0.012 0.64 ± 0.04 59.0 ± 2.4 gaussian Topic 5 0.714 ± 0.023 0.833 ± 0.021 0.627 ± 0.026 0.687 ± 0.027 0.802 ± 0.031 0.603 ± 0.028 50.6 ± 5.3 0.105 ± 0.014 114.17 ± 4.14 71.8 ± 2.1 gaussian Cam 5 0.233 ± 0.016 0.133 ± 0.010 0.939 ± 0.014 0.226 ± 0.018 0.129 ± 0.012 0.910 ± 0.022 598.4 ± 13.1 0.043 ± 0.016 30816.27 ± 99.34 680.8 ± 10.0 gaussian CauScale 5 0.783 ± 0.020 0.923 ± 0.008 0.682 ± 0.030 0.634 ± 0.018 0.747 ± 0.017 0.552 ± 0.023 49.0 ± 4.9 0.264 ± 0.021 5.83 ± 0.05 70.6 ± 3.3 gaussian SDCD 5 0.635 ± 0.022 0.687 ± 0.028 0.593 ± 0.026 0.490 ± 0.022 0.530 ± 0.027 0.458 ± 0.022 78.6 ± 5.8 0.292 ± 0.033 77.32 ± 0.96 82.8 ± 3.8 gaussian Globe 5 0.478 ± 0.048 0.883 ± 0.029 0.338 ± 0.047 0.462 ± 0.043 0.858 ± 0.034 0.327 ± 0.043 70.4 ± 7.2 0.327 ± 0.025 852.55 ± 81.60 36.0 ± 4.0 gaussian Das 5 0.557 ± 0.017 0.556 ± 0.020 0.558 ± 0.017 0.426 ± 0.010 0.425 ± 0.008 0.427 ± 0.015 97.8 ± 5.0 0.352 ± 0.038 62.05 ± 11.29 96.2 ± 4.5 gaussian Score 5 0.262 ± 0.018 0.156 ± 0.013 0.831 ± 0.013 0.192 ± 0.011 0.114 ± 0.008 0.613 ± 0.033 477.4 ± 33.3 0.352 ± 0.038 551.91 ± 10.87 519.8 ± 34.8 gaussian NoGAM 5 0.267 ± 0.016 0.159 ± 0.011 0.853 ± 0.013 0.202 ± 0.012 0.120 ± 0.008 0.648 ± 0.029 475.8 ± 36.3 0.323 ± 0.028 735.47 ± 27.69 523.6 ± 38.2 gaussian Avici 5 0.570 ± 0.042 0.965 ± 0.013 0.409 ± 0.040 0.525 ± 0.037 0.891 ± 0.028 0.376 ± 0.035 62.2 ± 7.5 0.318 ± 0.034 13.97 ± 1.08 39.8 ± 1.7 gaussian NOTEARS 5 0.558 ± 0.016 0.472 ± 0.017 0.685 ± 0.025 0.352 ± 0.014 0.298 ± 0.017 0.431 ± 0.010 128.6 ± 8.3 0.369 ± 0.025 2583.87 ± 397.43 139.4 ± 7.6 gaussian DAGMA 5 0.575 ± 0.038 0.647 ± 0.027 0.531 ± 0.054 0.278 ± 0.037 0.312 ± 0.036 0.257 ± 0.038 100.0 ± 7.4 0.498 ± 0.033 2571.24 ± 111.03 78.0 ± 6.9 gaussian DiffAN 5 0.282 ± 0.016 0.174 ± 0.012 0.773 ± 0.025 0.205 ± 0.012 0.126 ± 0.009 0.562 ± 0.027 403.4 ± 32.9 0.370 ± 0.031 496.94 ± 5.37 435.0 ± 35.1 gaussian SciNO 5 0.256 ± 0.010 0.154 ± 0.007 0.777 ± 0.020 0.156 ± 0.013 0.094 ± 0.008 0.473 ± 0.031 461.4 ± 17.7 0.446 ± 0.033 865.90 ± 19.27 485.8 ± 21.6 gaussian Caps 5 0.457 ± 0.028 0.349 ± 0.029 0.675 ± 0.031 0.354 ± 0.017 0.270 ± 0.019 0.524 ± 0.023 170.4 ± 14.0 0.354 ± 0.033 283.75 ± 32.91 188.8 ± 14.8 gaussian PC 5 0.768 ± 0.011 0.968 ± 0.013 0.638 ± 0.019 0.546 ± 0.021 0.688 ± 0.021 0.454 ± 0.023 55.0 ± 5.1 0.333 ± 0.017 9421.18 ± 1487.78 63.0 ± 2.4 gaussian FCI 5 0.477 ± 0.053 1.000 0.320 ± 0.049 0.409 ± 0.065 0.842 ± 0.044 0.276 ± 0.055 70.4 ± 8.4 0.338 ± 0.026 10110.79 ± 1723.04 30.2 ± 3.7 multimodal Spade-Hist 5 0.798 ± 0.026 0.980 ± 0.006 0.676 ± 0.035 0.798 ± 0.026 0.980 ± 0.006 0.676 ± 0.035 33.2 ± 5.6 0.067 ± 0.012 2.10 ± 0.17 65.8 ± 3.2 multimodal Spade-Gauss 5 0.737 ± 0.031 0.940 ± 0.014 0.610 ± 0.039 0.666 ± 0.037 0.849 ± 0.037 0.551 ± 0.040 47.0 ± 5.6 0.146 ± 0.029 0.69 ± 0.08 61.8 ± 3.0 multimodal Topic 5 0.708 ± 0.014 0.817 ± 0.009 0.627 ± 0.022 0.622 ± 0.024 0.717 ± 0.025 0.550 ± 0.026 57.0 ± 5.0 0.169 ± 0.034 122.70 ± 6.58 73.2 ± 2.4 multimodal Cam 5 0.232 ± 0.008 0.133 ± 0.005 0.917 ± 0.021 0.220 ± 0.008 0.126 ± 0.005 0.868 ± 0.015 586.0 ± 16.3 0.059 ± 0.011 30820.45 ± 92.99 661.2 ± 19.5 multimodal CauScale 5 0.771 ± 0.038 0.931 ± 0.012 0.662 ± 0.048 0.699 ± 0.046 0.841 ± 0.026 0.601 ± 0.052 44.0 ± 8.4 0.154 ± 0.017 5.84 ± 0.09 67.4 ± 3.3 multimodal SDCD 5 0.644 ± 0.031 0.718 ± 0.024 0.591 ± 0.044 0.489 ± 0.032 0.543 ± 0.023 0.449 ± 0.042 76.0 ± 7.9 0.290 ± 0.030 77.87 ± 0.89 79.0 ± 6.4 multimodal Globe 5 0.485 ± 0.028 0.866 ± 0.023 0.341 ± 0.029 0.449 ± 0.025 0.805 ± 0.036 0.315 ± 0.024 71.6 ± 6.0 0.369 ± 0.027 956.90 ± 96.66 37.8 ± 3.9 multimodal Das 5 0.590 ± 0.019 0.620 ± 0.016 0.565 ± 0.027 0.476 ± 0.031 0.501 ± 0.032 0.455 ± 0.032 86.4 ± 8.5 0.309 ± 0.037 44.71 ± 0.53 87.2 ± 5.0 multimodal Score 5 0.260 ± 0.013 0.154 ± 0.010 0.844 ± 0.025 0.198 ± 0.010 0.117 ± 0.007 0.644 ± 0.030 483.4 ± 28.4 0.309 ± 0.037 558.61 ± 18.24 530.0 ± 32.3 multimodal NoGAM 5 0.261 ± 0.015 0.156 ± 0.011 0.833 ± 0.030 0.205 ± 0.010 0.122 ± 0.008 0.655 ± 0.034 474.2 ± 35.5 0.296 ± 0.040 770.75 ± 10.55 521.0 ± 40.8 multimodal Avici 5 0.448 ± 0.025 0.915 ± 0.014 0.298 ± 0.022 0.366 ± 0.028 0.745 ± 0.018 0.244 ± 0.023 75.6 ± 5.6 0.403 ± 0.037 11.67 ± 0.53 31.0 ± 2.0 multimodal NOTEARS 5 0.608 ± 0.022 0.553 ± 0.038 0.689 ± 0.031 0.373 ± 0.017 0.340 ± 0.029 0.420 ± 0.013 111.4 ± 8.7 0.382 ± 0.024 2699.00 ± 263.33 121.4 ± 10.1 multimodal DAGMA 5 0.611 ± 0.030 0.690 ± 0.009 0.553 ± 0.044 0.285 ± 0.029 0.323 ± 0.029 0.258 ± 0.032 94.8 ± 5.1 0.511 ± 0.027 2316.61 ± 84.29 76.8 ± 6.7 multimodal DiffAN 5 0.295 ± 0.019 0.183 ± 0.015 0.791 ± 0.017 0.211 ± 0.015 0.131 ± 0.012 0.566 ± 0.014 393.4 ± 37.6 0.369 ± 0.020 487.97 ± 9.28 427.4 ± 39.7 multimodal SciNO 5 0.258 ± 0.011 0.155 ± 0.007 0.762 ± 0.031 0.153 ± 0.012 0.092 ± 0.007 0.453 ± 0.036 453.8 ± 35.7 0.441 ± 0.031 869.44 ± 12.09 473.6 ± 34.9 multimodal Caps 5 0.424 ± 0.014 0.323 ± 0.017 0.631 ± 0.034 0.340 ± 0.011 0.259 ± 0.015 0.505 ± 0.023 175.2 ± 6.7 0.331 ± 0.032 264.55 ± 23.19 187.8 ± 11.2 multimodal PC 5 0.674 ± 0.029 0.954 ± 0.019 0.526 ± 0.035 0.510 ± 0.026 0.726 ± 0.045 0.397 ± 0.024 60.8 ± 5.4 0.346 ± 0.020 8009.41 ± 1122.37 52.2 ± 1.4 multimodal FCI 5 0.424 ± 0.039 0.982 ± 0.011 0.273 ± 0.031 0.348 ± 0.040 0.800 ± 0.031 0.225 ± 0.031 75.0 ± 5.8 0.364 ± 0.040 8771.61 ± 1079.90 26.6 ± 3.2 unimodal_asymmetric Spade-Hist 5 0.782 ± 0.018 0.945 ± 0.012 0.668 ± 0.025 0.782 ± 0.018 0.945 ± 0.012 0.668 ± 0.025 36.2 ± 4.6 0.067 ± 0.016 2.34 ± 0.18 67.6 ± 3.1 unimodal_asymmetric Spade-Gauss 5 0.720 ± 0.027 0.894 ± 0.031 0.604 ± 0.027 0.656 ± 0.036 0.815 ± 0.044 0.550 ± 0.033 50.2 ± 5.6 0.164 ± 0.032 0.66 ± 0.05 64.6 ± 2.8 unimodal_asymmetric Topic 5 0.694 ± 0.029 0.773 ± 0.031 0.630 ± 0.030 0.608 ± 0.034 0.678 ± 0.037 0.553 ± 0.034 60.8 ± 5.8 0.157 ± 0.023 125.94 ± 7.85 78.0 ± 3.0 unimodal_asymmetric Cam 5 0.230 ± 0.015 0.132 ± 0.010 0.943 ± 0.014 0.204 ± 0.019 0.117 ± 0.012 0.830 ± 0.032 617.4 ± 21.6 0.137 ± 0.035 30902.11 ± 250.80 692.4 ± 19.4 unimodal_asymmetric CauScale 5 0.758 ± 0.015 0.899 ± 0.029 0.657 ± 0.012 0.622 ± 0.026 0.738 ± 0.041 0.538 ± 0.018 51.8 ± 4.6 0.225 ± 0.015 5.75 ± 0.05 70.2 ± 3.3 unimodal_asymmetric SDCD 5 0.630 ± 0.019 0.608 ± 0.025 0.655 ± 0.015 0.469 ± 0.021 0.453 ± 0.024 0.488 ± 0.019 90.0 ± 6.2 0.310 ± 0.020 72.88 ± 1.67 103.8 ± 6.3 unimodal_asymmetric Globe 5 0.505 ± 0.025 0.835 ± 0.038 0.364 ± 0.022 0.427 ± 0.025 0.706 ± 0.043 0.307 ± 0.020 74.4 ± 7.3 0.397 ± 0.008 860.79 ± 67.95 42.0 ± 3.3 unimodal_asymmetric Das 5 0.586 ± 0.024 0.578 ± 0.030 0.595 ± 0.017 0.459 ± 0.032 0.453 ± 0.037 0.465 ± 0.027 92.2 ± 2.6 0.310 ± 0.033 48.67 ± 1.26 99.0 ± 4.6 unimodal_asymmetric Score 5 0.239 ± 0.016 0.139 ± 0.011 0.885 ± 0.006 0.173 ± 0.019 0.101 ± 0.012 0.634 ± 0.031 568.2 ± 28.4 0.310 ± 0.033 549.04 ± 9.03 618.6 ± 27.4 unimodal_asymmetric NoGAM 5 0.224 ± 0.014 0.128 ± 0.010 0.901 ± 0.009 0.163 ± 0.020 0.093 ± 0.012 0.647 ± 0.040 625.4 ± 22.0 0.300 ± 0.044 771.70 ± 14.89 678.8 ± 19.7 unimodal_asymmetric Avici 5 0.496 ± 0.040 0.950 ± 0.014 0.339 ± 0.037 0.398 ± 0.043 0.757 ± 0.031 0.273 ± 0.037 72.0 ± 7.4 0.385 ± 0.027 12.46 ± 0.92 33.8 ± 2.9 unimodal_asymmetric NOTEARS 5 0.488 ± 0.020 0.370 ± 0.027 0.736 ± 0.017 0.291 ± 0.028 0.222 ± 0.027 0.434 ± 0.027 178.2 ± 16.0 0.386 ± 0.036 2022.42 ± 219.24 194.2 ± 15.5 unimodal_asymmetric DAGMA 5 0.566 ± 0.012 0.551 ± 0.019 0.589 ± 0.028 0.234 ± 0.015 0.228 ± 0.019 0.242 ± 0.015 119.4 ± 7.0 0.576 ± 0.021 2254.37 ± 115.71 102.2 ± 4.9 unimodal_asymmetric DiffAN 5 0.238 ± 0.015 0.138 ± 0.010 0.872 ± 0.011 0.173 ± 0.018 0.100 ± 0.011 0.627 ± 0.033 562.6 ± 27.1 0.310 ± 0.035 489.23 ± 9.79 611.4 ± 26.7 unimodal_asymmetric SciNO 5 0.240 ± 0.016 0.140 ± 0.011 0.847 ± 0.014 0.130 ± 0.009 0.076 ± 0.006 0.463 ± 0.024 555.4 ± 18.3 0.472 ± 0.017 951.02 ± 44.50 584.6 ± 18.5 unimodal_asymmetric Caps 5 0.349 ± 0.021 0.234 ± 0.020 0.708 ± 0.022 0.262 ± 0.019 0.175 ± 0.016 0.532 ± 0.031 276.2 ± 27.2 0.331 ± 0.034 284.22 ± 18.19 300.2 ± 31.8 unimodal_asymmetric PC 5 0.721 ± 0.044 0.973 ± 0.012 0.582 ± 0.055 0.532 ± 0.040 0.716 ± 0.026 0.429 ± 0.043 55.8 ± 2.0 0.342 ± 0.023 11820.36 ± 2308.97 58.4 ± 7.8 unimodal_asymmetric FCI 5 0.495 ± 0.031 0.995 ± 0.005 0.332 ± 0.028 0.449 ± 0.026 0.905 ± 0.021 0.301 ± 0.024 67.6 ± 5.5 0.301 ± 0.023 13306.57 ± 2713.66 32.0 ± 2.9 unimodal_symmetric Spade-Hist 5 0.769 ± 0.030 0.980 ± 0.004 0.636 ± 0.039 0.742 ± 0.032 0.945 ± 0.014 0.613 ± 0.041 39.0 ± 5.6 0.111 ± 0.033 2.30 ± 0.07 61.8 ± 3.3 unimodal_symmetric Spade-Gauss 5 0.727 ± 0.027 0.937 ± 0.011 0.597 ± 0.035 0.693 ± 0.033 0.891 ± 0.019 0.569 ± 0.039 45.8 ± 5.9 0.109 ± 0.031 0.69 ± 0.06 60.8 ± 3.1 unimodal_symmetric Topic 5 0.714 ± 0.032 0.833 ± 0.021 0.627 ± 0.038 0.659 ± 0.032 0.769 ± 0.026 0.578 ± 0.036 52.8 ± 6.2 0.142 ± 0.009 120.89 ± 7.57 71.6 ± 3.2 unimodal_symmetric Cam 5 0.225 ± 0.009 0.129 ± 0.006 0.914 ± 0.019 0.208 ± 0.007 0.119 ± 0.005 0.847 ± 0.027 609.4 ± 11.2 0.104 ± 0.023 29972.42 ± 1127.28 681.8 ± 11.1 unimodal_symmetric CauScale 5 0.755 ± 0.017 0.909 ± 0.019 0.648 ± 0.025 0.613 ± 0.030 0.736 ± 0.024 0.527 ± 0.034 52.2 ± 5.1 0.240 ± 0.024 6.02 ± 0.05 68.0 ± 2.1 unimodal_symmetric SDCD 5 0.599 ± 0.022 0.628 ± 0.014 0.575 ± 0.034 0.442 ± 0.037 0.463 ± 0.035 0.425 ± 0.040 88.2 ± 7.1 0.326 ± 0.032 78.67 ± 1.96 87.6 ± 5.8 unimodal_symmetric Globe 5 0.458 ± 0.045 0.849 ± 0.009 0.319 ± 0.042 0.427 ± 0.043 0.793 ± 0.018 0.298 ± 0.040 73.6 ± 7.7 0.400 ± 0.038 752.01 ± 40.82 35.0 ± 2.5 unimodal_symmetric Das 5 0.561 ± 0.034 0.567 ± 0.021 0.557 ± 0.047 0.426 ± 0.039 0.430 ± 0.029 0.425 ± 0.048 96.2 ± 8.9 0.372 ± 0.031 51.32 ± 6.46 93.6 ± 6.2 unimodal_symmetric Score 5 0.249 ± 0.012 0.146 ± 0.009 0.869 ± 0.017 0.166 ± 0.004 0.097 ± 0.003 0.582 ± 0.031 539.4 ± 48.5 0.372 ± 0.031 564.57 ± 12.58 582.6 ± 52.5 unimodal_symmetric NoGAM 5 0.249 ± 0.012 0.145 ± 0.008 0.875 ± 0.014 0.174 ± 0.005 0.102 ± 0.004 0.615 ± 0.022 538.0 ± 39.3 0.337 ± 0.013 775.79 ± 2.59 585.0 ± 43.1 unimodal_symmetric Avici 5 0.577 ± 0.040 0.957 ± 0.017 0.417 ± 0.039 0.546 ± 0.049 0.902 ± 0.039 0.395 ± 0.044 61.0 ± 8.2 0.265 ± 0.034 11.20 ± 0.24 41.0 ± 1.6 unimodal_symmetric NOTEARS 5 0.515 ± 0.019 0.399 ± 0.019 0.731 ± 0.019 0.330 ± 0.040 0.256 ± 0.033 0.466 ± 0.050 157.8 ± 12.5 0.371 ± 0.054 2976.79 ± 467.34 176.4 ± 8.5 unimodal_symmetric DAGMA 5 0.584 ± 0.012 0.619 ± 0.015 0.559 ± 0.027 0.278 ± 0.030 0.297 ± 0.037 0.264 ± 0.029 103.8 ± 5.8 0.464 ± 0.051 2321.62 ± 56.06 86.8 ± 6.6 unimodal_symmetric DiffAN 5 0.258 ± 0.008 0.153 ± 0.006 0.841 ± 0.033 0.185 ± 0.007 0.109 ± 0.004 0.605 ± 0.035 486.4 ± 25.1 0.326 ± 0.023 493.30 ± 6.46 529.4 ± 31.8 unimodal_symmetric SciNO 5 0.248 ± 0.008 0.147 ± 0.005 0.806 ± 0.018 0.127 ± 0.002 0.075 ± 0.002 0.415 ± 0.016 509.2 ± 35.1 0.516 ± 0.011 854.75 ± 16.27 530.6 ± 37.9 unimodal_symmetric Caps 5 0.379 ± 0.025 0.264 ± 0.024 0.686 ± 0.021 0.273 ± 0.017 0.189 ± 0.015 0.498 ± 0.033 241.8 ± 28.0 0.380 ± 0.030 267.45 ± 11.32 258.6 ± 27.7 unimodal_symmetric PC 5 0.672 ± 0.027 0.947 ± 0.015 0.522 ± 0.031 0.498 ± 0.039 0.702 ± 0.052 0.387 ± 0.034 62.2 ± 6.7 0.339 ± 0.031 8630.87 ± 1304.28 52.6 ± 2.5 unimodal_symmetric FCI 5 0.437 ± 0.043 0.986 ± 0.014 0.284 ± 0.034 0.388 ± 0.049 0.875 ± 0.062 0.253 ± 0.037 73.0 ± 8.0 0.342 ± 0.033 9038.15 ± 1331.88 27.0 ± 2.1 Table 28: Qualitative comparison without marginalization (n=1000n=1000, d=50d=50, degree 4, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.559 ± 0.013 0.910 ± 0.011 0.404 ± 0.014 0.494 ± 0.019 0.804 ± 0.021 0.357 ± 0.017 65.6 ± 4.2 0.341 ± 0.030 1.53 ± 0.04 42.6 ± 2.5 gaussian Spade-Gauss 5 0.628 ± 0.012 0.919 ± 0.010 0.477 ± 0.015 0.612 ± 0.020 0.895 ± 0.020 0.466 ± 0.020 55.4 ± 3.6 0.130 ± 0.019 0.48 ± 0.03 49.8 ± 2.6 gaussian Topic 5 0.606 ± 0.014 0.755 ± 0.014 0.507 ± 0.020 0.555 ± 0.015 0.693 ± 0.013 0.465 ± 0.020 67.2 ± 3.9 0.184 ± 0.016 98.07 ± 7.86 64.2 ± 3.2 gaussian Cam 5 0.230 ± 0.012 0.132 ± 0.008 0.893 ± 0.015 0.204 ± 0.015 0.117 ± 0.010 0.788 ± 0.013 583.4 ± 8.6 0.142 ± 0.022 28380.95 ± 50.48 649.0 ± 6.5 gaussian CauScale 5 0.703 ± 0.012 0.913 ± 0.013 0.573 ± 0.017 0.539 ± 0.013 0.701 ± 0.026 0.439 ± 0.012 59.2 ± 3.2 0.299 ± 0.017 5.86 ± 0.06 60.2 ± 3.5 gaussian SDCD 5 0.611 ± 0.015 0.699 ± 0.027 0.546 ± 0.019 0.493 ± 0.013 0.565 ± 0.025 0.440 ± 0.014 76.8 ± 5.5 0.314 ± 0.014 73.21 ± 2.44 75.2 ± 5.2 gaussian Globe 5 0.365 ± 0.044 0.858 ± 0.016 0.237 ± 0.035 0.327 ± 0.031 0.782 ± 0.036 0.211 ± 0.026 79.8 ± 5.3 0.490 ± 0.033 688.32 ± 66.15 26.8 ± 4.3 gaussian Das 5 0.515 ± 0.028 0.511 ± 0.048 0.533 ± 0.029 0.383 ± 0.018 0.380 ± 0.036 0.396 ± 0.017 109.0 ± 6.2 0.291 ± 0.014 46.23 ± 1.08 101.8 ± 7.9 gaussian Score 5 0.265 ± 0.018 0.157 ± 0.013 0.860 ± 0.006 0.191 ± 0.011 0.114 ± 0.008 0.625 ± 0.014 486.8 ± 29.6 0.291 ± 0.014 536.62 ± 16.90 533.0 ± 29.8 gaussian NoGAM 5 0.263 ± 0.020 0.155 ± 0.013 0.865 ± 0.016 0.192 ± 0.012 0.114 ± 0.008 0.637 ± 0.022 495.0 ± 30.9 0.301 ± 0.024 799.72 ± 3.18 543.2 ± 31.3 gaussian Avici 5 0.467 ± 0.036 0.936 ± 0.011 0.314 ± 0.032 0.422 ± 0.028 0.850 ± 0.017 0.284 ± 0.026 71.4 ± 6.5 0.432 ± 0.022 11.65 ± 0.40 31.6 ± 2.1 gaussian NOTEARS 5 0.538 ± 0.021 0.437 ± 0.033 0.719 ± 0.025 0.409 ± 0.015 0.332 ± 0.024 0.549 ± 0.025 134.6 ± 5.9 0.304 ± 0.038 2194.90 ± 309.86 159.6 ± 8.7 gaussian DAGMA 5 0.567 ± 0.018 0.581 ± 0.042 0.564 ± 0.022 0.305 ± 0.025 0.312 ± 0.030 0.303 ± 0.026 108.2 ± 9.5 0.552 ± 0.014 2186.91 ± 102.19 94.4 ± 7.3 gaussian DiffAN 5 0.285 ± 0.022 0.176 ± 0.017 0.777 ± 0.013 0.195 ± 0.013 0.120 ± 0.010 0.532 ± 0.009 404.6 ± 30.8 0.352 ± 0.015 493.36 ± 8.79 434.2 ± 30.4 gaussian SciNO 5 0.262 ± 0.012 0.158 ± 0.009 0.767 ± 0.011 0.134 ± 0.007 0.081 ± 0.004 0.397 ± 0.034 452.0 ± 17.8 0.499 ± 0.031 866.66 ± 21.41 467.2 ± 19.0 gaussian Caps 5 0.370 ± 0.034 0.257 ± 0.030 0.673 ± 0.024 0.269 ± 0.022 0.187 ± 0.020 0.491 ± 0.019 246.4 ± 29.6 0.316 ± 0.020 337.43 ± 11.55 261.6 ± 27.6 gaussian PC 5 0.749 ± 0.016 0.961 ± 0.012 0.615 ± 0.024 0.498 ± 0.024 0.639 ± 0.022 0.410 ± 0.026 59.4 ± 5.0 0.397 ± 0.017 6782.82 ± 987.70 61.6 ± 4.2 gaussian FCI 5 0.365 ± 0.043 0.982 ± 0.011 0.227 ± 0.032 0.321 ± 0.038 0.869 ± 0.038 0.200 ± 0.029 77.6 ± 6.4 0.447 ± 0.028 8175.53 ± 1299.26 22.0 ± 2.7 multimodal Spade-Hist 5 0.581 ± 0.047 0.961 ± 0.007 0.424 ± 0.050 0.576 ± 0.043 0.953 ± 0.009 0.419 ± 0.047 58.0 ± 6.5 0.225 ± 0.018 1.67 ± 0.13 41.8 ± 3.9 multimodal Spade-Gauss 5 0.621 ± 0.042 0.937 ± 0.003 0.471 ± 0.050 0.562 ± 0.043 0.848 ± 0.028 0.426 ± 0.047 58.6 ± 6.3 0.161 ± 0.022 0.53 ± 0.04 47.6 ± 3.7 multimodal Topic 5 0.579 ± 0.039 0.718 ± 0.029 0.490 ± 0.048 0.481 ± 0.040 0.599 ± 0.046 0.406 ± 0.042 75.4 ± 6.4 0.239 ± 0.049 102.22 ± 7.62 64.4 ± 2.8 multimodal Cam 5 0.265 ± 0.017 0.156 ± 0.012 0.893 ± 0.018 0.236 ± 0.017 0.139 ± 0.012 0.794 ± 0.029 486.6 ± 20.4 0.141 ± 0.023 29267.53 ± 627.17 553.0 ± 20.8 multimodal CauScale 5 0.716 ± 0.035 0.910 ± 0.017 0.595 ± 0.044 0.625 ± 0.035 0.794 ± 0.024 0.520 ± 0.042 52.2 ± 6.0 0.195 ± 0.020 7.75 ± 1.18 62.0 ± 2.8 multimodal SDCD 5 0.636 ± 0.025 0.727 ± 0.036 0.572 ± 0.035 0.511 ± 0.023 0.586 ± 0.037 0.458 ± 0.024 73.2 ± 4.7 0.328 ± 0.031 78.62 ± 0.66 76.0 ± 6.6 multimodal Globe 5 0.407 ± 0.051 0.890 ± 0.013 0.271 ± 0.046 0.380 ± 0.049 0.831 ± 0.018 0.253 ± 0.044 75.8 ± 7.7 0.468 ± 0.040 816.04 ± 120.21 28.4 ± 3.7 multimodal Das 5 0.545 ± 0.020 0.569 ± 0.017 0.527 ± 0.029 0.453 ± 0.025 0.471 ± 0.017 0.438 ± 0.034 92.8 ± 7.3 0.233 ± 0.023 49.01 ± 1.25 89.0 ± 6.3 multimodal Score 5 0.293 ± 0.015 0.178 ± 0.011 0.848 ± 0.020 0.235 ± 0.012 0.143 ± 0.009 0.682 ± 0.028 412.0 ± 27.3 0.233 ± 0.023 547.92 ± 2.94 462.6 ± 30.4 multimodal NoGAM 5 0.276 ± 0.016 0.166 ± 0.012 0.828 ± 0.029 0.214 ± 0.014 0.129 ± 0.010 0.640 ± 0.023 439.2 ± 32.1 0.258 ± 0.013 787.65 ± 5.47 484.2 ± 35.7 multimodal Avici 5 0.414 ± 0.049 0.933 ± 0.019 0.272 ± 0.043 0.332 ± 0.049 0.743 ± 0.031 0.219 ± 0.041 77.6 ± 7.4 0.446 ± 0.023 11.18 ± 0.18 27.6 ± 3.5 multimodal NOTEARS 5 0.602 ± 0.026 0.549 ± 0.049 0.691 ± 0.040 0.403 ± 0.038 0.370 ± 0.050 0.459 ± 0.038 111.6 ± 14.8 0.386 ± 0.033 1764.28 ± 215.17 125.6 ± 16.2 multimodal DAGMA 5 0.609 ± 0.042 0.710 ± 0.058 0.542 ± 0.050 0.346 ± 0.033 0.406 ± 0.048 0.306 ± 0.030 89.2 ± 8.6 0.539 ± 0.030 2343.68 ± 155.80 73.8 ± 5.9 multimodal DiffAN 5 0.311 ± 0.018 0.199 ± 0.016 0.734 ± 0.028 0.218 ± 0.018 0.140 ± 0.014 0.513 ± 0.019 339.8 ± 32.1 0.334 ± 0.030 495.07 ± 9.75 362.8 ± 33.8 multimodal SciNO 5 0.294 ± 0.013 0.183 ± 0.010 0.768 ± 0.035 0.160 ± 0.007 0.099 ± 0.006 0.417 ± 0.021 388.4 ± 26.0 0.466 ± 0.022 964.06 ± 34.42 406.2 ± 29.8 multimodal Caps 5 0.428 ± 0.028 0.329 ± 0.031 0.631 ± 0.028 0.330 ± 0.017 0.254 ± 0.021 0.489 ± 0.025 178.8 ± 17.9 0.264 ± 0.017 338.43 ± 33.53 190.2 ± 20.6 multimodal PC 5 0.743 ± 0.043 0.945 ± 0.021 0.617 ± 0.052 0.427 ± 0.036 0.541 ± 0.026 0.356 ± 0.040 65.4 ± 6.0 0.443 ± 0.023 8653.50 ± 1849.34 62.2 ± 4.6 multimodal FCI 5 0.384 ± 0.065 1.000 0.246 ± 0.054 0.320 ± 0.066 0.814 ± 0.049 0.207 ± 0.053 77.4 ± 9.5 0.427 ± 0.045 10021.27 ± 2234.23 22.6 ± 3.8 unimodal_asymmetric Spade-Hist 5 0.561 ± 0.025 0.957 ± 0.020 0.399 ± 0.024 0.539 ± 0.019 0.921 ± 0.031 0.383 ± 0.019 61.2 ± 4.0 0.254 ± 0.033 1.70 ± 0.06 40.0 ± 3.1 unimodal_asymmetric Spade-Gauss 5 0.593 ± 0.023 0.835 ± 0.016 0.462 ± 0.027 0.491 ± 0.027 0.693 ± 0.035 0.382 ± 0.025 68.2 ± 4.9 0.289 ± 0.030 0.51 ± 0.03 52.6 ± 2.1 unimodal_asymmetric Topic 5 0.554 ± 0.021 0.677 ± 0.020 0.471 ± 0.027 0.439 ± 0.022 0.534 ± 0.013 0.374 ± 0.027 82.0 ± 5.3 0.301 ± 0.052 101.71 ± 4.98 66.4 ± 2.5 unimodal_asymmetric Cam 5 0.244 ± 0.015 0.141 ± 0.010 0.920 ± 0.008 0.190 ± 0.016 0.110 ± 0.010 0.714 ± 0.023 569.8 ± 22.3 0.246 ± 0.021 28252.86 ± 76.20 631.0 ± 21.4 unimodal_asymmetric CauScale 5 0.713 ± 0.018 0.920 ± 0.008 0.585 ± 0.026 0.547 ± 0.015 0.706 ± 0.007 0.449 ± 0.021 58.2 ± 4.8 0.278 ± 0.026 5.78 ± 0.05 60.6 ± 2.4 unimodal_asymmetric SDCD 5 0.580 ± 0.019 0.631 ± 0.040 0.544 ± 0.021 0.412 ± 0.033 0.448 ± 0.044 0.385 ± 0.030 90.2 ± 4.5 0.383 ± 0.032 73.01 ± 1.98 83.0 ± 2.8 unimodal_asymmetric Globe 5 0.340 ± 0.046 0.832 ± 0.024 0.218 ± 0.036 0.237 ± 0.042 0.566 ± 0.042 0.153 ± 0.032 86.0 ± 7.3 0.568 ± 0.013 671.65 ± 80.43 24.8 ± 3.7 unimodal_asymmetric Das 5 0.506 ± 0.017 0.496 ± 0.035 0.526 ± 0.018 0.405 ± 0.017 0.398 ± 0.031 0.421 ± 0.014 108.4 ± 5.6 0.289 ± 0.016 47.83 ± 1.92 103.0 ± 7.0 unimodal_asymmetric Score 5 0.246 ± 0.019 0.143 ± 0.013 0.902 ± 0.008 0.179 ± 0.018 0.104 ± 0.012 0.655 ± 0.020 561.8 ± 37.0 0.289 ± 0.016 566.57 ± 9.45 615.4 ± 35.6 unimodal_asymmetric NoGAM 5 0.243 ± 0.018 0.141 ± 0.012 0.914 ± 0.012 0.177 ± 0.015 0.103 ± 0.010 0.664 ± 0.015 576.0 ± 31.7 0.287 ± 0.013 796.88 ± 1.46 631.2 ± 30.1 unimodal_asymmetric Avici 5 0.488 ± 0.032 0.980 ± 0.013 0.327 ± 0.027 0.385 ± 0.037 0.768 ± 0.035 0.258 ± 0.030 72.4 ± 7.1 0.451 ± 0.041 12.03 ± 0.59 31.6 ± 1.7 unimodal_asymmetric NOTEARS 5 0.428 ± 0.027 0.302 ± 0.025 0.751 ± 0.014 0.258 ± 0.032 0.182 ± 0.026 0.449 ± 0.038 223.8 ± 17.9 0.449 ± 0.056 1856.10 ± 102.00 243.0 ± 15.5 unimodal_asymmetric DAGMA 5 0.547 ± 0.012 0.516 ± 0.032 0.593 ± 0.024 0.318 ± 0.014 0.301 ± 0.025 0.344 ± 0.012 118.2 ± 6.9 0.493 ± 0.015 2329.23 ± 199.36 111.4 ± 7.7 unimodal_asymmetric DiffAN 5 0.246 ± 0.014 0.145 ± 0.010 0.845 ± 0.023 0.171 ± 0.015 0.101 ± 0.010 0.585 ± 0.031 522.6 ± 21.3 0.327 ± 0.026 494.39 ± 6.37 563.2 ± 21.4 unimodal_asymmetric SciNO 5 0.274 ± 0.015 0.166 ± 0.012 0.803 ± 0.022 0.151 ± 0.010 0.092 ± 0.008 0.443 ± 0.011 446.4 ± 27.0 0.456 ± 0.013 916.54 ± 53.01 469.4 ± 28.8 unimodal_asymmetric Caps 5 0.320 ± 0.026 0.206 ± 0.022 0.749 ± 0.021 0.239 ± 0.025 0.154 ± 0.020 0.556 ± 0.019 328.2 ± 24.3 0.298 ± 0.013 350.14 ± 11.30 357.4 ± 24.3 unimodal_asymmetric PC 5 0.726 ± 0.011 0.942 ± 0.026 0.593 ± 0.016 0.534 ± 0.025 0.690 ± 0.024 0.437 ± 0.027 58.0 ± 4.3 0.335 ± 0.024 8648.60 ± 1726.96 60.4 ± 2.7 unimodal_asymmetric FCI 5 0.407 ± 0.038 0.981 ± 0.013 0.260 ± 0.030 0.360 ± 0.036 0.869 ± 0.038 0.229 ± 0.028 74.8 ± 6.4 0.395 ± 0.031 9119.50 ± 1883.55 25.2 ± 2.6 unimodal_symmetric Spade-Hist 5 0.534 ± 0.043 0.933 ± 0.023 0.380 ± 0.045 0.477 ± 0.048 0.830 ± 0.041 0.339 ± 0.047 67.0 ± 8.1 0.335 ± 0.051 1.61 ± 0.07 38.8 ± 3.8 unimodal_symmetric Spade-Gauss 5 0.562 ± 0.032 0.910 ± 0.023 0.410 ± 0.033 0.522 ± 0.026 0.847 ± 0.029 0.380 ± 0.027 64.0 ± 6.1 0.305 ± 0.041 0.44 ± 0.01 43.4 ± 4.3 unimodal_symmetric Topic 5 0.559 ± 0.036 0.706 ± 0.030 0.466 ± 0.043 0.486 ± 0.044 0.614 ± 0.050 0.405 ± 0.045 75.6 ± 6.0 0.297 ± 0.041 91.46 ± 4.38 62.6 ± 2.9 unimodal_symmetric Cam 5 0.229 ± 0.015 0.132 ± 0.010 0.883 ± 0.030 0.190 ± 0.019 0.110 ± 0.011 0.732 ± 0.055 586.8 ± 20.4 0.207 ± 0.044 29774.77 ± 626.05 646.4 ± 22.9 unimodal_symmetric CauScale 5 0.671 ± 0.034 0.892 ± 0.017 0.541 ± 0.041 0.496 ± 0.046 0.656 ± 0.037 0.401 ± 0.047 64.6 ± 8.3 0.330 ± 0.043 5.95 ± 0.11 57.6 ± 2.8 unimodal_symmetric SDCD 5 0.574 ± 0.036 0.700 ± 0.036 0.493 ± 0.049 0.419 ± 0.039 0.509 ± 0.031 0.362 ± 0.048 82.0 ± 6.8 0.405 ± 0.059 71.58 ± 1.51 67.6 ± 6.4 unimodal_symmetric Globe 5 0.337 ± 0.033 0.803 ± 0.029 0.215 ± 0.025 0.274 ± 0.036 0.648 ± 0.050 0.175 ± 0.026 84.6 ± 6.9 0.552 ± 0.029 682.12 ± 35.13 25.4 ± 2.3 unimodal_symmetric Das 5 0.499 ± 0.033 0.525 ± 0.032 0.484 ± 0.048 0.377 ± 0.037 0.396 ± 0.037 0.366 ± 0.045 103.4 ± 8.6 0.278 ± 0.034 46.24 ± 3.31 89.0 ± 10.1 unimodal_symmetric Score 5 0.250 ± 0.009 0.148 ± 0.006 0.831 ± 0.032 0.184 ± 0.014 0.108 ± 0.008 0.611 ± 0.046 500.6 ± 33.9 0.278 ± 0.034 542.22 ± 18.87 544.2 ± 41.8 unimodal_symmetric NoGAM 5 0.252 ± 0.011 0.148 ± 0.007 0.842 ± 0.035 0.188 ± 0.014 0.111 ± 0.009 0.630 ± 0.044 500.6 ± 25.8 0.283 ± 0.027 795.75 ± 4.06 546.8 ± 33.0 unimodal_symmetric Avici 5 0.466 ± 0.043 0.941 ± 0.013 0.315 ± 0.041 0.383 ± 0.032 0.777 ± 0.041 0.258 ± 0.029 73.2 ± 4.7 0.467 ± 0.023 11.32 ± 0.14 31.8 ± 3.4 unimodal_symmetric NOTEARS 5 0.464 ± 0.024 0.349 ± 0.017 0.697 ± 0.053 0.324 ± 0.032 0.243 ± 0.022 0.488 ± 0.059 174.6 ± 15.4 0.391 ± 0.054 2440.94 ± 328.50 192.4 ± 17.8 unimodal_symmetric DAGMA 5 0.543 ± 0.040 0.560 ± 0.033 0.533 ± 0.052 0.290 ± 0.014 0.300 ± 0.017 0.283 ± 0.017 110.4 ± 11.9 0.551 ± 0.023 2450.24 ± 148.79 92.2 ± 10.5 unimodal_symmetric DiffAN 5 0.254 ± 0.008 0.154 ± 0.005 0.740 ± 0.050 0.175 ± 0.012 0.106 ± 0.007 0.513 ± 0.053 437.6 ± 33.6 0.332 ± 0.022 496.39 ± 9.26 462.8 ± 43.0 unimodal_symmetric SciNO 5 0.253 ± 0.014 0.154 ± 0.010 0.723 ± 0.045 0.134 ± 0.011 0.081 ± 0.007 0.385 ± 0.041 439.4 ± 14.9 0.464 ± 0.016 910.75 ± 14.28 449.8 ± 21.7 unimodal_symmetric Caps 5 0.319 ± 0.008 0.212 ± 0.003 0.652 ± 0.042 0.238 ± 0.017 0.158 ± 0.010 0.488 ± 0.048 281.0 ± 18.3 0.283 ± 0.028 345.67 ± 15.07 295.4 ± 26.3 unimodal_symmetric PC 5 0.757 ± 0.039 0.976 ± 0.013 0.624 ± 0.054 0.510 ± 0.049 0.658 ± 0.058 0.421 ± 0.048 58.2 ± 8.5 0.407 ± 0.044 7894.88 ± 1628.62 60.8 ± 3.8 unimodal_symmetric FCI 5 0.416 ± 0.065 0.989 ± 0.011 0.272 ± 0.053 0.366 ± 0.060 0.866 ± 0.039 0.239 ± 0.048 74.4 ± 9.2 0.421 ± 0.033 8717.59 ± 1891.88 25.2 ± 3.5 Table 29: Qualitative comparison without marginalization (n=2000n=2000, d=20d=20, degree 2, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.887 ± 0.030 0.947 ± 0.024 0.851 ± 0.066 0.830 ± 0.029 0.890 ± 0.043 0.794 ± 0.055 4.6 ± 1.0 0.067 ± 0.032 0.58 ± 0.01 14.8 ± 0.6 gaussian Spade-Gauss 5 0.907 ± 0.033 0.974 ± 0.016 0.860 ± 0.061 0.907 ± 0.033 0.974 ± 0.016 0.860 ± 0.061 3.2 ± 1.2 0.009 ± 0.009 0.10 ± 0.00 14.6 ± 0.7 gaussian Topic 5 0.762 ± 0.022 0.697 ± 0.029 0.860 ± 0.061 0.762 ± 0.022 0.697 ± 0.029 0.860 ± 0.061 9.0 ± 1.2 0.047 ± 0.030 17.80 ± 0.97 20.4 ± 0.7 gaussian Cam 5 0.685 ± 0.037 0.542 ± 0.040 0.943 ± 0.035 0.678 ± 0.040 0.536 ± 0.042 0.934 ± 0.041 15.2 ± 2.8 0.009 ± 0.009 333.86 ± 6.53 29.6 ± 2.6 gaussian CauScale 5 0.810 ± 0.045 0.882 ± 0.037 0.768 ± 0.076 0.438 ± 0.043 0.472 ± 0.024 0.419 ± 0.063 12.0 ± 1.9 0.467 ± 0.054 5.91 ± 0.07 14.4 ± 1.1 gaussian SDCD 5 0.741 ± 0.021 0.902 ± 0.043 0.634 ± 0.030 0.548 ± 0.025 0.666 ± 0.036 0.469 ± 0.029 10.2 ± 1.0 0.336 ± 0.065 132.10 ± 3.87 12.0 ± 1.4 gaussian Globe 5 0.837 ± 0.034 0.842 ± 0.016 0.845 ± 0.067 0.807 ± 0.044 0.814 ± 0.043 0.812 ± 0.066 6.0 ± 1.3 0.078 ± 0.046 286.59 ± 28.79 16.6 ± 1.0 gaussian Das 5 0.595 ± 0.016 0.555 ± 0.021 0.647 ± 0.029 0.433 ± 0.016 0.404 ± 0.014 0.472 ± 0.031 18.0 ± 2.2 0.281 ± 0.022 70.55 ± 3.25 20.2 ± 3.2 gaussian Score 5 0.682 ± 0.017 0.548 ± 0.014 0.910 ± 0.047 0.518 ± 0.008 0.417 ± 0.013 0.691 ± 0.026 18.2 ± 2.3 0.281 ± 0.022 52.04 ± 1.24 28.0 ± 2.8 gaussian NoGAM 5 0.693 ± 0.026 0.569 ± 0.025 0.900 ± 0.056 0.524 ± 0.018 0.432 ± 0.026 0.677 ± 0.024 17.4 ± 2.5 0.295 ± 0.024 208.46 ± 5.75 26.8 ± 3.0 gaussian Avici 5 0.846 ± 0.056 0.971 ± 0.018 0.764 ± 0.085 0.773 ± 0.039 0.893 ± 0.030 0.696 ± 0.064 6.0 ± 1.8 0.145 ± 0.048 13.65 ± 1.00 12.8 ± 0.6 gaussian NOTEARS 5 0.757 ± 0.030 0.768 ± 0.058 0.765 ± 0.046 0.481 ± 0.053 0.498 ± 0.082 0.477 ± 0.034 13.2 ± 2.2 0.362 ± 0.077 408.04 ± 112.52 17.2 ± 2.2 gaussian DAGMA 5 0.734 ± 0.031 0.772 ± 0.026 0.710 ± 0.053 0.411 ± 0.063 0.433 ± 0.067 0.397 ± 0.066 13.6 ± 1.1 0.407 ± 0.071 1774.72 ± 99.55 15.6 ± 1.9 gaussian DiffAN 5 0.744 ± 0.029 0.669 ± 0.047 0.868 ± 0.062 0.390 ± 0.023 0.351 ± 0.031 0.453 ± 0.037 17.2 ± 2.3 0.519 ± 0.046 23.89 ± 0.20 22.2 ± 2.7 gaussian SciNO 5 0.796 ± 0.034 0.750 ± 0.050 0.854 ± 0.030 0.494 ± 0.076 0.469 ± 0.082 0.525 ± 0.070 13.4 ± 2.4 0.428 ± 0.077 756.41 ± 25.06 19.6 ± 2.2 gaussian Caps 5 0.771 ± 0.043 0.775 ± 0.056 0.780 ± 0.052 0.563 ± 0.029 0.570 ± 0.049 0.567 ± 0.022 11.8 ± 2.2 0.338 ± 0.038 9.69 ± 0.23 17.4 ± 2.4 gaussian PC 5 0.676 ± 0.050 0.902 ± 0.043 0.556 ± 0.065 0.499 ± 0.091 0.660 ± 0.100 0.410 ± 0.082 11.6 ± 2.5 0.417 ± 0.062 2603.67 ± 402.62 10.6 ± 2.0 gaussian FCI 5 0.396 ± 0.066 0.933 ± 0.067 0.256 ± 0.049 0.396 ± 0.066 0.933 ± 0.067 0.256 ± 0.049 13.0 ± 2.0 0.361 ± 0.050 2566.99 ± 403.68 4.4 ± 0.7 multimodal Spade-Hist 5 0.922 ± 0.012 0.973 ± 0.016 0.878 ± 0.023 0.922 ± 0.012 0.973 ± 0.016 0.878 ± 0.023 2.6 ± 0.5 0.035 ± 0.021 0.59 ± 0.02 15.2 ± 1.3 multimodal Spade-Gauss 5 0.892 ± 0.022 0.945 ± 0.025 0.847 ± 0.033 0.792 ± 0.052 0.839 ± 0.056 0.752 ± 0.053 5.2 ± 1.3 0.124 ± 0.046 0.11 ± 0.01 15.0 ± 0.9 multimodal Topic 5 0.773 ± 0.027 0.692 ± 0.041 0.887 ± 0.024 0.663 ± 0.052 0.594 ± 0.055 0.758 ± 0.052 10.8 ± 1.4 0.168 ± 0.054 19.41 ± 2.74 21.6 ± 1.4 multimodal Cam 5 0.738 ± 0.034 0.611 ± 0.045 0.943 ± 0.016 0.672 ± 0.051 0.558 ± 0.055 0.855 ± 0.041 12.4 ± 1.3 0.106 ± 0.028 326.19 ± 2.91 26.0 ± 1.1 multimodal CauScale 5 0.797 ± 0.043 0.846 ± 0.051 0.761 ± 0.055 0.498 ± 0.054 0.529 ± 0.059 0.476 ± 0.059 11.4 ± 1.6 0.357 ± 0.060 6.20 ± 0.12 15.0 ± 0.8 multimodal SDCD 5 0.813 ± 0.030 0.930 ± 0.043 0.729 ± 0.039 0.539 ± 0.063 0.614 ± 0.067 0.486 ± 0.066 9.8 ± 1.4 0.360 ± 0.057 132.39 ± 5.49 13.6 ± 2.2 multimodal Globe 5 0.838 ± 0.030 0.906 ± 0.025 0.782 ± 0.043 0.816 ± 0.042 0.882 ± 0.035 0.763 ± 0.053 5.8 ± 1.6 0.092 ± 0.040 224.52 ± 13.34 14.4 ± 0.9 multimodal Das 5 0.763 ± 0.049 0.717 ± 0.053 0.820 ± 0.053 0.661 ± 0.047 0.622 ± 0.052 0.709 ± 0.047 10.2 ± 1.5 0.150 ± 0.027 70.03 ± 0.46 19.4 ± 1.7 multimodal Score 5 0.756 ± 0.034 0.643 ± 0.048 0.930 ± 0.021 0.668 ± 0.041 0.569 ± 0.052 0.819 ± 0.020 11.6 ± 1.0 0.150 ± 0.027 52.12 ± 1.63 24.4 ± 1.2 multimodal NoGAM 5 0.792 ± 0.025 0.686 ± 0.037 0.943 ± 0.016 0.699 ± 0.041 0.607 ± 0.050 0.829 ± 0.026 10.0 ± 1.1 0.126 ± 0.042 213.60 ± 4.71 23.2 ± 1.7 multimodal Avici 5 0.691 ± 0.048 0.850 ± 0.032 0.592 ± 0.059 0.431 ± 0.042 0.519 ± 0.034 0.373 ± 0.044 12.8 ± 1.9 0.420 ± 0.051 13.75 ± 1.00 11.8 ± 1.1 multimodal NOTEARS 5 0.786 ± 0.021 0.779 ± 0.013 0.798 ± 0.040 0.557 ± 0.048 0.552 ± 0.048 0.564 ± 0.054 11.2 ± 1.4 0.301 ± 0.063 708.22 ± 153.55 17.2 ± 1.4 multimodal DAGMA 5 0.729 ± 0.049 0.717 ± 0.061 0.743 ± 0.036 0.433 ± 0.088 0.429 ± 0.091 0.439 ± 0.085 15.6 ± 4.2 0.416 ± 0.086 1593.92 ± 122.69 18.2 ± 2.8 multimodal DiffAN 5 0.827 ± 0.035 0.753 ± 0.064 0.933 ± 0.018 0.551 ± 0.066 0.508 ± 0.084 0.613 ± 0.044 13.0 ± 3.0 0.378 ± 0.042 25.63 ± 0.68 22.0 ± 3.3 multimodal SciNO 5 0.822 ± 0.013 0.745 ± 0.023 0.920 ± 0.021 0.436 ± 0.063 0.398 ± 0.063 0.483 ± 0.061 13.6 ± 0.7 0.469 ± 0.056 863.77 ± 100.83 20.8 ± 1.7 multimodal Caps 5 0.830 ± 0.016 0.828 ± 0.024 0.836 ± 0.028 0.687 ± 0.038 0.684 ± 0.031 0.695 ± 0.050 8.4 ± 1.4 0.201 ± 0.023 9.23 ± 0.25 17.4 ± 2.5 multimodal PC 5 0.672 ± 0.059 0.913 ± 0.065 0.533 ± 0.053 0.550 ± 0.088 0.745 ± 0.113 0.438 ± 0.072 9.8 ± 0.9 0.346 ± 0.047 3079.13 ± 1117.85 10.0 ± 1.4 multimodal FCI 5 0.454 ± 0.067 0.933 ± 0.067 0.306 ± 0.054 0.429 ± 0.054 0.893 ± 0.069 0.288 ± 0.043 12.2 ± 1.2 0.356 ± 0.058 3210.40 ± 1111.98 5.6 ± 1.2 unimodal_asymmetric Spade-Hist 5 0.894 ± 0.030 0.973 ± 0.016 0.831 ± 0.049 0.894 ± 0.030 0.973 ± 0.016 0.831 ± 0.049 3.6 ± 1.1 0.020 ± 0.020 0.58 ± 0.02 14.2 ± 0.8 unimodal_asymmetric Spade-Gauss 5 0.844 ± 0.046 0.900 ± 0.039 0.803 ± 0.064 0.725 ± 0.074 0.772 ± 0.069 0.691 ± 0.084 7.4 ± 2.2 0.176 ± 0.053 0.10 ± 0.00 14.8 ± 0.9 unimodal_asymmetric Topic 5 0.753 ± 0.030 0.714 ± 0.019 0.819 ± 0.076 0.656 ± 0.048 0.622 ± 0.038 0.713 ± 0.082 10.8 ± 1.7 0.169 ± 0.051 17.50 ± 1.82 18.8 ± 0.8 unimodal_asymmetric Cam 5 0.633 ± 0.014 0.476 ± 0.011 0.949 ± 0.025 0.540 ± 0.042 0.405 ± 0.031 0.810 ± 0.067 21.4 ± 3.4 0.150 ± 0.058 325.46 ± 5.27 33.8 ± 3.3 unimodal_asymmetric CauScale 5 0.737 ± 0.019 0.765 ± 0.049 0.731 ± 0.050 0.351 ± 0.055 0.362 ± 0.058 0.350 ± 0.058 15.2 ± 2.0 0.546 ± 0.051 6.00 ± 0.10 16.2 ± 1.5 unimodal_asymmetric SDCD 5 0.795 ± 0.034 0.841 ± 0.025 0.762 ± 0.055 0.550 ± 0.052 0.585 ± 0.060 0.524 ± 0.051 10.6 ± 1.6 0.402 ± 0.045 132.52 ± 4.28 15.0 ± 0.8 unimodal_asymmetric Globe 5 0.795 ± 0.040 0.861 ± 0.039 0.751 ± 0.065 0.710 ± 0.063 0.765 ± 0.052 0.673 ± 0.086 8.0 ± 1.8 0.161 ± 0.054 224.73 ± 25.68 14.4 ± 0.5 unimodal_asymmetric Das 5 0.637 ± 0.037 0.522 ± 0.042 0.832 ± 0.052 0.365 ± 0.049 0.300 ± 0.042 0.476 ± 0.068 22.2 ± 2.8 0.366 ± 0.069 69.98 ± 2.42 27.0 ± 2.2 unimodal_asymmetric Score 5 0.623 ± 0.017 0.467 ± 0.024 0.949 ± 0.025 0.387 ± 0.044 0.291 ± 0.038 0.583 ± 0.051 25.4 ± 2.5 0.366 ± 0.069 51.63 ± 1.10 34.4 ± 2.9 unimodal_asymmetric NoGAM 5 0.626 ± 0.017 0.470 ± 0.021 0.949 ± 0.025 0.401 ± 0.021 0.300 ± 0.017 0.609 ± 0.036 24.8 ± 2.0 0.340 ± 0.040 207.44 ± 2.67 34.2 ± 3.0 unimodal_asymmetric Avici 5 0.819 ± 0.034 0.935 ± 0.016 0.735 ± 0.054 0.595 ± 0.056 0.683 ± 0.069 0.532 ± 0.056 8.8 ± 1.3 0.304 ± 0.049 14.35 ± 1.01 13.0 ± 0.7 unimodal_asymmetric NOTEARS 5 0.793 ± 0.046 0.741 ± 0.063 0.864 ± 0.044 0.597 ± 0.077 0.564 ± 0.092 0.642 ± 0.065 12.4 ± 3.7 0.294 ± 0.058 621.96 ± 94.69 20.4 ± 3.0 unimodal_asymmetric DAGMA 5 0.766 ± 0.041 0.712 ± 0.057 0.843 ± 0.051 0.452 ± 0.101 0.428 ± 0.105 0.486 ± 0.098 15.2 ± 3.8 0.417 ± 0.109 1537.76 ± 59.18 20.2 ± 2.2 unimodal_asymmetric DiffAN 5 0.739 ± 0.007 0.634 ± 0.019 0.898 ± 0.045 0.390 ± 0.029 0.336 ± 0.033 0.469 ± 0.029 17.6 ± 1.1 0.475 ± 0.056 24.29 ± 0.71 23.6 ± 1.3 unimodal_asymmetric SciNO 5 0.802 ± 0.044 0.715 ± 0.052 0.920 ± 0.038 0.447 ± 0.073 0.401 ± 0.073 0.508 ± 0.073 15.8 ± 3.8 0.443 ± 0.076 831.69 ± 28.93 22.4 ± 3.0 unimodal_asymmetric Caps 5 0.738 ± 0.053 0.664 ± 0.051 0.838 ± 0.065 0.385 ± 0.058 0.347 ± 0.056 0.434 ± 0.062 17.6 ± 3.3 0.478 ± 0.045 10.32 ± 0.38 21.4 ± 2.1 unimodal_asymmetric PC 5 0.653 ± 0.034 0.965 ± 0.021 0.501 ± 0.043 0.593 ± 0.025 0.884 ± 0.052 0.453 ± 0.030 9.8 ± 1.3 0.341 ± 0.034 2787.54 ± 863.53 8.8 ± 1.2 unimodal_asymmetric FCI 5 0.501 ± 0.052 1.000 0.340 ± 0.045 0.462 ± 0.063 0.905 ± 0.066 0.314 ± 0.049 11.6 ± 1.3 0.346 ± 0.064 2983.52 ± 987.70 5.8 ± 1.0 unimodal_symmetric Spade-Hist 5 0.907 ± 0.026 0.968 ± 0.021 0.857 ± 0.041 0.897 ± 0.031 0.958 ± 0.031 0.847 ± 0.043 3.4 ± 1.1 0.033 ± 0.021 0.59 ± 0.01 15.0 ± 1.6 unimodal_symmetric Spade-Gauss 5 0.899 ± 0.029 0.950 ± 0.032 0.857 ± 0.041 0.850 ± 0.057 0.900 ± 0.063 0.811 ± 0.060 4.6 ± 1.8 0.068 ± 0.042 0.10 ± 0.00 15.4 ± 1.9 unimodal_symmetric Topic 5 0.761 ± 0.027 0.666 ± 0.041 0.899 ± 0.034 0.717 ± 0.008 0.626 ± 0.017 0.851 ± 0.044 10.2 ± 0.4 0.070 ± 0.034 17.43 ± 1.10 22.6 ± 1.0 unimodal_symmetric Cam 5 0.661 ± 0.030 0.508 ± 0.034 0.962 ± 0.025 0.636 ± 0.031 0.489 ± 0.035 0.924 ± 0.025 17.4 ± 2.5 0.063 ± 0.022 337.18 ± 7.53 32.4 ± 3.6 unimodal_symmetric CauScale 5 0.808 ± 0.025 0.853 ± 0.040 0.778 ± 0.043 0.462 ± 0.056 0.480 ± 0.045 0.451 ± 0.068 12.2 ± 2.2 0.433 ± 0.052 5.90 ± 0.06 15.6 ± 2.1 unimodal_symmetric SDCD 5 0.829 ± 0.032 0.859 ± 0.025 0.814 ± 0.064 0.627 ± 0.029 0.649 ± 0.018 0.617 ± 0.056 8.8 ± 0.7 0.236 ± 0.032 139.31 ± 2.17 16.6 ± 3.2 unimodal_symmetric Globe 5 0.840 ± 0.044 0.899 ± 0.029 0.794 ± 0.062 0.809 ± 0.037 0.867 ± 0.028 0.764 ± 0.053 5.8 ± 1.5 0.072 ± 0.022 240.57 ± 18.34 14.8 ± 1.5 unimodal_symmetric Das 5 0.677 ± 0.043 0.622 ± 0.032 0.748 ± 0.065 0.579 ± 0.033 0.532 ± 0.020 0.640 ± 0.055 13.8 ± 1.8 0.178 ± 0.012 68.23 ± 1.41 20.6 ± 3.1 unimodal_symmetric Score 5 0.668 ± 0.014 0.514 ± 0.018 0.962 ± 0.025 0.563 ± 0.019 0.433 ± 0.021 0.809 ± 0.022 18.6 ± 1.6 0.178 ± 0.012 50.52 ± 0.85 31.8 ± 3.4 unimodal_symmetric NoGAM 5 0.673 ± 0.024 0.520 ± 0.027 0.962 ± 0.025 0.568 ± 0.023 0.439 ± 0.025 0.810 ± 0.017 18.4 ± 2.3 0.176 ± 0.020 205.16 ± 1.43 31.6 ± 3.6 unimodal_symmetric Avici 5 0.822 ± 0.037 0.956 ± 0.018 0.725 ± 0.051 0.762 ± 0.052 0.888 ± 0.057 0.671 ± 0.055 6.4 ± 1.4 0.110 ± 0.069 12.55 ± 0.07 12.6 ± 0.8 unimodal_symmetric NOTEARS 5 0.760 ± 0.021 0.733 ± 0.052 0.818 ± 0.051 0.606 ± 0.072 0.596 ± 0.095 0.637 ± 0.049 12.4 ± 3.7 0.233 ± 0.076 442.34 ± 35.71 20.4 ± 4.9 unimodal_symmetric DAGMA 5 0.774 ± 0.042 0.784 ± 0.062 0.782 ± 0.048 0.470 ± 0.066 0.482 ± 0.082 0.468 ± 0.055 13.6 ± 3.4 0.381 ± 0.088 1878.13 ± 40.98 17.8 ± 3.5 unimodal_symmetric DiffAN 5 0.742 ± 0.026 0.664 ± 0.046 0.868 ± 0.054 0.513 ± 0.067 0.458 ± 0.061 0.602 ± 0.089 14.4 ± 2.1 0.372 ± 0.081 24.20 ± 0.44 23.2 ± 4.4 unimodal_symmetric SciNO 5 0.766 ± 0.023 0.671 ± 0.013 0.897 ± 0.043 0.522 ± 0.076 0.454 ± 0.061 0.615 ± 0.100 13.4 ± 1.0 0.346 ± 0.082 737.66 ± 5.67 22.8 ± 2.9 unimodal_symmetric Caps 5 0.753 ± 0.027 0.744 ± 0.069 0.786 ± 0.042 0.628 ± 0.025 0.619 ± 0.054 0.657 ± 0.043 11.4 ± 2.2 0.234 ± 0.038 9.52 ± 0.47 19.0 ± 3.6 unimodal_symmetric PC 5 0.668 ± 0.031 0.872 ± 0.066 0.549 ± 0.028 0.569 ± 0.052 0.748 ± 0.094 0.465 ± 0.036 10.8 ± 1.7 0.288 ± 0.031 3795.06 ± 1846.37 11.0 ± 1.4 unimodal_symmetric FCI 5 0.494 ± 0.012 1.000 0.329 ± 0.011 0.463 ± 0.040 0.933 ± 0.067 0.309 ± 0.028 11.8 ± 1.4 0.250 ± 0.050 3619.76 ± 1595.57 5.6 ± 0.7 Table 30: Qualitative comparison without marginalization (n=2000n=2000, d=20d=20, degree 2, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.805 ± 0.031 0.957 ± 0.043 0.715 ± 0.065 0.707 ± 0.074 0.836 ± 0.082 0.629 ± 0.080 7.2 ± 1.6 0.278 ± 0.036 0.56 ± 0.01 12.4 ± 0.9 gaussian Spade-Gauss 5 0.829 ± 0.033 0.969 ± 0.031 0.740 ± 0.061 0.813 ± 0.030 0.954 ± 0.046 0.723 ± 0.050 5.4 ± 1.1 0.101 ± 0.021 0.09 ± 0.01 12.6 ± 0.7 gaussian Topic 5 0.724 ± 0.038 0.674 ± 0.028 0.800 ± 0.074 0.664 ± 0.034 0.619 ± 0.028 0.731 ± 0.059 11.2 ± 1.5 0.167 ± 0.042 16.83 ± 2.01 19.6 ± 1.4 gaussian Cam 5 0.697 ± 0.032 0.561 ± 0.038 0.934 ± 0.032 0.673 ± 0.041 0.540 ± 0.040 0.904 ± 0.057 14.6 ± 2.5 0.063 ± 0.048 322.19 ± 3.28 28.6 ± 3.5 gaussian CauScale 5 0.737 ± 0.052 0.812 ± 0.065 0.697 ± 0.080 0.430 ± 0.060 0.466 ± 0.052 0.411 ± 0.070 13.2 ± 2.1 0.422 ± 0.052 6.14 ± 0.22 14.4 ± 1.6 gaussian SDCD 5 0.683 ± 0.072 0.827 ± 0.056 0.602 ± 0.083 0.499 ± 0.082 0.595 ± 0.073 0.443 ± 0.084 12.2 ± 2.6 0.435 ± 0.057 139.74 ± 3.88 12.8 ± 2.7 gaussian Globe 5 0.745 ± 0.040 0.879 ± 0.036 0.657 ± 0.058 0.691 ± 0.041 0.816 ± 0.049 0.607 ± 0.051 8.6 ± 1.7 0.181 ± 0.035 227.01 ± 19.41 12.4 ± 0.9 gaussian Das 5 0.582 ± 0.024 0.563 ± 0.065 0.640 ± 0.047 0.455 ± 0.036 0.447 ± 0.071 0.492 ± 0.016 18.4 ± 3.2 0.224 ± 0.025 69.73 ± 1.01 20.8 ± 4.4 gaussian Score 5 0.690 ± 0.043 0.558 ± 0.059 0.938 ± 0.037 0.564 ± 0.060 0.461 ± 0.071 0.757 ± 0.031 18.6 ± 5.0 0.224 ± 0.025 52.29 ± 0.54 30.2 ± 5.7 gaussian NoGAM 5 0.709 ± 0.041 0.575 ± 0.052 0.951 ± 0.039 0.535 ± 0.041 0.435 ± 0.047 0.717 ± 0.038 18.0 ± 4.0 0.264 ± 0.035 207.62 ± 3.25 29.2 ± 5.0 gaussian Avici 5 0.800 ± 0.059 0.969 ± 0.031 0.691 ± 0.077 0.725 ± 0.049 0.883 ± 0.041 0.624 ± 0.063 7.2 ± 1.9 0.160 ± 0.036 13.18 ± 0.35 11.6 ± 0.4 gaussian NOTEARS 5 0.747 ± 0.036 0.808 ± 0.055 0.701 ± 0.032 0.535 ± 0.067 0.575 ± 0.074 0.506 ± 0.065 11.8 ± 2.4 0.400 ± 0.077 289.85 ± 52.42 15.2 ± 2.5 gaussian DAGMA 5 0.752 ± 0.048 0.858 ± 0.071 0.702 ± 0.067 0.465 ± 0.077 0.521 ± 0.078 0.436 ± 0.075 12.8 ± 3.2 0.438 ± 0.055 1622.72 ± 108.95 15.2 ± 3.8 gaussian DiffAN 5 0.745 ± 0.034 0.659 ± 0.052 0.888 ± 0.055 0.396 ± 0.074 0.359 ± 0.074 0.454 ± 0.073 17.8 ± 3.4 0.515 ± 0.087 24.11 ± 0.29 23.4 ± 3.4 gaussian SciNO 5 0.792 ± 0.051 0.755 ± 0.081 0.861 ± 0.048 0.466 ± 0.078 0.453 ± 0.089 0.495 ± 0.073 15.0 ± 4.5 0.458 ± 0.085 845.80 ± 90.25 20.4 ± 3.7 gaussian Caps 5 0.786 ± 0.030 0.781 ± 0.062 0.806 ± 0.028 0.487 ± 0.016 0.481 ± 0.027 0.501 ± 0.027 13.0 ± 2.2 0.365 ± 0.017 9.87 ± 0.36 18.0 ± 2.7 gaussian PC 5 0.765 ± 0.040 0.897 ± 0.034 0.671 ± 0.050 0.658 ± 0.039 0.771 ± 0.032 0.578 ± 0.049 8.4 ± 1.0 0.250 ± 0.041 2782.49 ± 1098.46 12.8 ± 1.7 gaussian FCI 5 0.482 ± 0.057 1.000 0.324 ± 0.045 0.445 ± 0.048 0.933 ± 0.041 0.299 ± 0.038 12.0 ± 1.6 0.263 ± 0.036 2598.01 ± 1003.81 5.4 ± 0.9 multimodal Spade-Hist 5 0.783 ± 0.047 0.941 ± 0.026 0.679 ± 0.063 0.783 ± 0.047 0.941 ± 0.026 0.679 ± 0.063 6.6 ± 1.8 0.064 ± 0.031 0.55 ± 0.01 12.0 ± 1.1 multimodal Spade-Gauss 5 0.822 ± 0.056 0.913 ± 0.012 0.765 ± 0.090 0.647 ± 0.066 0.715 ± 0.042 0.603 ± 0.088 8.4 ± 2.1 0.262 ± 0.053 0.09 ± 0.00 13.6 ± 0.6 multimodal Topic 5 0.700 ± 0.024 0.638 ± 0.021 0.790 ± 0.062 0.518 ± 0.037 0.469 ± 0.021 0.589 ± 0.067 14.8 ± 1.7 0.277 ± 0.049 15.71 ± 0.96 20.4 ± 0.5 multimodal Cam 5 0.706 ± 0.031 0.551 ± 0.035 0.991 ± 0.009 0.620 ± 0.050 0.486 ± 0.048 0.868 ± 0.045 16.4 ± 2.9 0.123 ± 0.043 329.93 ± 8.45 30.8 ± 3.4 multimodal CauScale 5 0.767 ± 0.034 0.858 ± 0.040 0.708 ± 0.058 0.506 ± 0.065 0.564 ± 0.067 0.468 ± 0.072 11.2 ± 1.6 0.339 ± 0.055 5.95 ± 0.11 13.8 ± 1.2 multimodal SDCD 5 0.791 ± 0.059 0.904 ± 0.042 0.708 ± 0.070 0.622 ± 0.068 0.714 ± 0.070 0.555 ± 0.071 8.8 ± 1.9 0.304 ± 0.052 135.78 ± 3.04 13.0 ± 1.0 multimodal Globe 5 0.778 ± 0.036 0.881 ± 0.030 0.702 ± 0.050 0.721 ± 0.069 0.811 ± 0.060 0.654 ± 0.077 8.0 ± 2.1 0.169 ± 0.057 226.84 ± 11.11 13.2 ± 0.6 multimodal Das 5 0.711 ± 0.025 0.665 ± 0.039 0.774 ± 0.030 0.608 ± 0.016 0.568 ± 0.030 0.661 ± 0.026 12.4 ± 1.2 0.160 ± 0.028 69.28 ± 1.39 19.8 ± 2.0 multimodal Score 5 0.731 ± 0.027 0.586 ± 0.032 0.978 ± 0.013 0.620 ± 0.027 0.497 ± 0.028 0.831 ± 0.031 14.6 ± 1.5 0.160 ± 0.028 47.25 ± 1.52 28.2 ± 2.1 multimodal NoGAM 5 0.758 ± 0.026 0.622 ± 0.035 0.978 ± 0.013 0.718 ± 0.039 0.589 ± 0.045 0.924 ± 0.023 11.8 ± 2.2 0.067 ± 0.018 208.34 ± 7.15 27.0 ± 3.1 multimodal Avici 5 0.698 ± 0.050 0.912 ± 0.027 0.584 ± 0.067 0.583 ± 0.062 0.750 ± 0.019 0.491 ± 0.072 10.0 ± 1.7 0.288 ± 0.046 12.59 ± 0.06 10.6 ± 1.2 multimodal NOTEARS 5 0.801 ± 0.046 0.766 ± 0.050 0.841 ± 0.044 0.662 ± 0.064 0.633 ± 0.064 0.695 ± 0.066 10.0 ± 3.0 0.217 ± 0.072 402.82 ± 61.67 18.8 ± 2.1 multimodal DAGMA 5 0.754 ± 0.044 0.767 ± 0.069 0.750 ± 0.031 0.495 ± 0.036 0.505 ± 0.055 0.491 ± 0.025 12.8 ± 2.2 0.337 ± 0.049 1632.44 ± 102.62 16.8 ± 1.8 multimodal DiffAN 5 0.812 ± 0.013 0.728 ± 0.023 0.930 ± 0.046 0.457 ± 0.052 0.413 ± 0.056 0.518 ± 0.055 14.2 ± 1.8 0.473 ± 0.057 25.37 ± 0.73 21.4 ± 1.6 multimodal SciNO 5 0.818 ± 0.027 0.781 ± 0.043 0.881 ± 0.061 0.506 ± 0.058 0.490 ± 0.070 0.536 ± 0.049 12.4 ± 2.1 0.455 ± 0.052 757.86 ± 32.54 19.8 ± 3.1 multimodal Caps 5 0.860 ± 0.039 0.837 ± 0.045 0.885 ± 0.037 0.738 ± 0.054 0.719 ± 0.058 0.759 ± 0.051 7.6 ± 2.2 0.167 ± 0.028 10.14 ± 0.34 18.0 ± 1.8 multimodal PC 5 0.721 ± 0.033 0.978 ± 0.022 0.574 ± 0.040 0.624 ± 0.044 0.848 ± 0.055 0.497 ± 0.043 8.8 ± 1.3 0.298 ± 0.030 3022.42 ± 1092.98 10.0 ± 1.2 multimodal FCI 5 0.448 ± 0.045 1.000 0.293 ± 0.037 0.413 ± 0.054 0.917 ± 0.053 0.271 ± 0.042 12.6 ± 1.9 0.260 ± 0.030 2856.82 ± 984.98 4.8 ± 0.5 unimodal_asymmetric Spade-Hist 5 0.776 ± 0.048 0.965 ± 0.021 0.656 ± 0.062 0.776 ± 0.048 0.965 ± 0.021 0.656 ± 0.062 6.4 ± 1.4 0.165 ± 0.048 0.53 ± 0.01 11.4 ± 1.3 unimodal_asymmetric Spade-Gauss 5 0.825 ± 0.066 0.870 ± 0.044 0.798 ± 0.088 0.692 ± 0.046 0.733 ± 0.034 0.668 ± 0.065 7.8 ± 1.8 0.227 ± 0.034 0.10 ± 0.00 15.2 ± 1.5 unimodal_asymmetric Topic 5 0.719 ± 0.038 0.645 ± 0.023 0.819 ± 0.065 0.594 ± 0.034 0.534 ± 0.028 0.675 ± 0.048 13.0 ± 1.4 0.245 ± 0.024 17.66 ± 1.68 21.2 ± 1.8 unimodal_asymmetric Cam 5 0.610 ± 0.012 0.446 ± 0.012 0.967 ± 0.021 0.456 ± 0.032 0.333 ± 0.024 0.724 ± 0.056 25.0 ± 2.4 0.243 ± 0.048 333.28 ± 2.48 36.6 ± 3.1 unimodal_asymmetric CauScale 5 0.719 ± 0.042 0.806 ± 0.028 0.659 ± 0.062 0.447 ± 0.035 0.507 ± 0.054 0.406 ± 0.034 12.8 ± 1.5 0.389 ± 0.044 5.90 ± 0.15 13.4 ± 0.4 unimodal_asymmetric SDCD 5 0.791 ± 0.022 0.880 ± 0.044 0.725 ± 0.033 0.456 ± 0.058 0.499 ± 0.054 0.423 ± 0.062 11.8 ± 1.7 0.416 ± 0.082 137.05 ± 2.55 14.2 ± 1.9 unimodal_asymmetric Globe 5 0.635 ± 0.044 0.730 ± 0.070 0.580 ± 0.056 0.540 ± 0.056 0.611 ± 0.055 0.499 ± 0.074 12.6 ± 1.9 0.249 ± 0.086 208.33 ± 12.41 13.4 ± 1.1 unimodal_asymmetric Das 5 0.600 ± 0.036 0.517 ± 0.035 0.729 ± 0.054 0.437 ± 0.051 0.378 ± 0.048 0.527 ± 0.062 20.4 ± 3.6 0.310 ± 0.037 68.67 ± 0.67 24.4 ± 3.7 unimodal_asymmetric Score 5 0.606 ± 0.009 0.453 ± 0.017 0.933 ± 0.042 0.429 ± 0.028 0.321 ± 0.025 0.657 ± 0.043 25.2 ± 3.0 0.310 ± 0.037 51.45 ± 1.31 35.2 ± 4.4 unimodal_asymmetric NoGAM 5 0.618 ± 0.014 0.463 ± 0.020 0.943 ± 0.035 0.410 ± 0.036 0.308 ± 0.031 0.622 ± 0.049 25.4 ± 3.4 0.354 ± 0.042 208.37 ± 7.40 35.0 ± 4.6 unimodal_asymmetric Avici 5 0.768 ± 0.052 0.962 ± 0.023 0.647 ± 0.064 0.626 ± 0.071 0.779 ± 0.067 0.528 ± 0.071 8.4 ± 1.7 0.273 ± 0.041 12.85 ± 0.17 11.2 ± 1.2 unimodal_asymmetric NOTEARS 5 0.759 ± 0.027 0.656 ± 0.027 0.906 ± 0.042 0.586 ± 0.021 0.506 ± 0.022 0.699 ± 0.029 13.4 ± 1.7 0.230 ± 0.031 288.95 ± 16.47 23.4 ± 2.2 unimodal_asymmetric DAGMA 5 0.764 ± 0.037 0.725 ± 0.048 0.817 ± 0.043 0.440 ± 0.063 0.420 ± 0.064 0.467 ± 0.066 15.2 ± 3.8 0.429 ± 0.080 1775.41 ± 86.63 19.4 ± 2.6 unimodal_asymmetric DiffAN 5 0.730 ± 0.015 0.613 ± 0.018 0.908 ± 0.035 0.436 ± 0.025 0.366 ± 0.020 0.543 ± 0.040 17.6 ± 2.1 0.398 ± 0.023 23.50 ± 0.30 25.2 ± 3.0 unimodal_asymmetric SciNO 5 0.775 ± 0.032 0.674 ± 0.032 0.915 ± 0.039 0.528 ± 0.059 0.461 ± 0.057 0.620 ± 0.060 14.4 ± 2.4 0.337 ± 0.057 833.38 ± 32.08 23.2 ± 2.6 unimodal_asymmetric Caps 5 0.768 ± 0.043 0.677 ± 0.049 0.893 ± 0.042 0.495 ± 0.054 0.438 ± 0.056 0.572 ± 0.049 15.2 ± 2.7 0.405 ± 0.043 10.84 ± 0.17 22.8 ± 2.8 unimodal_asymmetric PC 5 0.692 ± 0.066 0.916 ± 0.052 0.563 ± 0.073 0.546 ± 0.106 0.722 ± 0.130 0.446 ± 0.095 10.2 ± 2.3 0.326 ± 0.067 3067.33 ± 1191.80 10.4 ± 1.4 unimodal_asymmetric FCI 5 0.456 ± 0.077 1.000 0.308 ± 0.065 0.421 ± 0.081 0.917 ± 0.053 0.286 ± 0.067 12.0 ± 1.5 0.283 ± 0.055 3227.13 ± 1166.98 5.4 ± 1.5 unimodal_symmetric Spade-Hist 5 0.832 ± 0.061 0.983 ± 0.017 0.737 ± 0.091 0.832 ± 0.061 0.983 ± 0.017 0.737 ± 0.091 4.8 ± 1.7 0.088 ± 0.050 0.61 ± 0.06 12.6 ± 1.9 unimodal_symmetric Spade-Gauss 5 0.828 ± 0.048 0.934 ± 0.030 0.755 ± 0.073 0.767 ± 0.060 0.863 ± 0.041 0.701 ± 0.082 6.4 ± 1.7 0.126 ± 0.037 0.10 ± 0.01 13.6 ± 1.7 unimodal_symmetric Topic 5 0.728 ± 0.032 0.673 ± 0.054 0.814 ± 0.051 0.624 ± 0.051 0.573 ± 0.052 0.704 ± 0.083 12.0 ± 1.5 0.178 ± 0.061 16.79 ± 1.42 20.2 ± 0.4 unimodal_symmetric Cam 5 0.627 ± 0.019 0.466 ± 0.025 0.968 ± 0.021 0.548 ± 0.026 0.408 ± 0.025 0.847 ± 0.043 21.6 ± 2.3 0.143 ± 0.041 328.81 ± 4.06 35.2 ± 3.2 unimodal_symmetric CauScale 5 0.738 ± 0.030 0.789 ± 0.027 0.698 ± 0.042 0.551 ± 0.048 0.590 ± 0.052 0.521 ± 0.050 11.8 ± 2.3 0.278 ± 0.051 6.18 ± 0.17 15.2 ± 2.1 unimodal_symmetric SDCD 5 0.758 ± 0.065 0.846 ± 0.042 0.698 ± 0.082 0.528 ± 0.058 0.592 ± 0.054 0.486 ± 0.068 11.0 ± 1.8 0.329 ± 0.007 135.81 ± 5.83 14.0 ± 2.2 unimodal_symmetric Globe 5 0.718 ± 0.029 0.878 ± 0.039 0.627 ± 0.060 0.691 ± 0.019 0.851 ± 0.054 0.601 ± 0.047 8.6 ± 1.0 0.141 ± 0.021 213.71 ± 27.71 12.4 ± 2.4 unimodal_symmetric Das 5 0.594 ± 0.022 0.519 ± 0.021 0.709 ± 0.053 0.471 ± 0.032 0.410 ± 0.026 0.562 ± 0.053 18.4 ± 1.1 0.204 ± 0.025 67.59 ± 1.12 23.0 ± 2.4 unimodal_symmetric Score 5 0.653 ± 0.035 0.516 ± 0.045 0.910 ± 0.034 0.540 ± 0.036 0.427 ± 0.039 0.754 ± 0.041 18.8 ± 1.8 0.204 ± 0.025 52.43 ± 0.90 30.0 ± 2.3 unimodal_symmetric NoGAM 5 0.662 ± 0.036 0.525 ± 0.051 0.926 ± 0.024 0.578 ± 0.039 0.458 ± 0.046 0.809 ± 0.041 18.0 ± 2.2 0.149 ± 0.037 199.80 ± 17.99 30.2 ± 2.4 unimodal_symmetric Avici 5 0.808 ± 0.051 0.973 ± 0.017 0.704 ± 0.069 0.732 ± 0.034 0.887 ± 0.034 0.636 ± 0.053 6.6 ± 1.2 0.180 ± 0.042 14.85 ± 0.82 12.2 ± 1.8 unimodal_symmetric NOTEARS 5 0.769 ± 0.052 0.742 ± 0.049 0.802 ± 0.063 0.662 ± 0.050 0.640 ± 0.052 0.688 ± 0.053 9.8 ± 1.5 0.192 ± 0.044 621.20 ± 126.74 18.2 ± 1.8 unimodal_symmetric DAGMA 5 0.717 ± 0.038 0.688 ± 0.036 0.758 ± 0.058 0.387 ± 0.058 0.380 ± 0.067 0.400 ± 0.050 16.4 ± 2.9 0.455 ± 0.071 1860.08 ± 148.38 18.8 ± 2.6 unimodal_symmetric DiffAN 5 0.781 ± 0.030 0.686 ± 0.039 0.913 ± 0.035 0.538 ± 0.042 0.473 ± 0.039 0.629 ± 0.054 13.6 ± 1.9 0.341 ± 0.058 25.16 ± 0.70 23.0 ± 3.3 unimodal_symmetric SciNO 5 0.762 ± 0.025 0.673 ± 0.028 0.883 ± 0.040 0.509 ± 0.057 0.447 ± 0.044 0.594 ± 0.079 14.0 ± 1.3 0.324 ± 0.047 762.90 ± 30.92 22.2 ± 2.3 unimodal_symmetric Caps 5 0.789 ± 0.017 0.744 ± 0.027 0.848 ± 0.037 0.633 ± 0.043 0.595 ± 0.040 0.683 ± 0.062 10.4 ± 1.0 0.226 ± 0.042 9.65 ± 0.28 19.0 ± 0.9 unimodal_symmetric PC 5 0.716 ± 0.039 0.968 ± 0.020 0.578 ± 0.059 0.635 ± 0.056 0.857 ± 0.060 0.513 ± 0.065 8.8 ± 1.7 0.261 ± 0.042 3366.36 ± 1404.29 10.2 ± 1.4 unimodal_symmetric FCI 5 0.431 ± 0.048 1.000 0.280 ± 0.038 0.414 ± 0.064 0.933 ± 0.067 0.270 ± 0.047 12.6 ± 1.9 0.257 ± 0.044 3309.96 ± 1392.37 4.6 ± 0.6 Table 31: Qualitative comparison without marginalization (n=2000n=2000, d=20d=20, degree 3, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.851 ± 0.028 0.969 ± 0.022 0.762 ± 0.039 0.833 ± 0.034 0.949 ± 0.034 0.746 ± 0.041 7.2 ± 1.5 0.083 ± 0.025 0.64 ± 0.02 19.8 ± 1.1 gaussian Spade-Gauss 5 0.853 ± 0.030 0.976 ± 0.014 0.760 ± 0.043 0.853 ± 0.030 0.976 ± 0.014 0.760 ± 0.043 6.6 ± 1.4 0.033 ± 0.017 0.12 ± 0.01 19.6 ± 1.2 gaussian Topic 5 0.778 ± 0.030 0.800 ± 0.029 0.762 ± 0.039 0.770 ± 0.021 0.791 ± 0.025 0.753 ± 0.031 11.2 ± 1.5 0.087 ± 0.023 22.19 ± 1.21 24.0 ± 1.1 gaussian Cam 5 0.780 ± 0.034 0.692 ± 0.039 0.896 ± 0.028 0.780 ± 0.034 0.692 ± 0.039 0.896 ± 0.028 12.6 ± 1.7 0.009 ± 0.009 324.27 ± 1.43 32.8 ± 1.2 gaussian CauScale 5 0.782 ± 0.042 0.894 ± 0.048 0.703 ± 0.050 0.488 ± 0.032 0.563 ± 0.052 0.435 ± 0.027 16.6 ± 1.9 0.338 ± 0.028 5.96 ± 0.07 19.8 ± 1.2 gaussian SDCD 5 0.728 ± 0.039 0.843 ± 0.033 0.648 ± 0.054 0.521 ± 0.051 0.602 ± 0.051 0.465 ± 0.055 17.0 ± 2.8 0.316 ± 0.053 137.57 ± 4.04 19.4 ± 1.6 gaussian Globe 5 0.748 ± 0.047 0.825 ± 0.053 0.684 ± 0.042 0.738 ± 0.051 0.814 ± 0.057 0.676 ± 0.047 12.4 ± 3.2 0.104 ± 0.035 308.00 ± 31.68 21.2 ± 1.9 gaussian Das 5 0.687 ± 0.039 0.658 ± 0.038 0.719 ± 0.044 0.538 ± 0.056 0.517 ± 0.057 0.562 ± 0.056 20.0 ± 1.3 0.231 ± 0.037 60.11 ± 0.62 27.6 ± 1.6 gaussian Score 5 0.724 ± 0.025 0.616 ± 0.028 0.880 ± 0.019 0.590 ± 0.035 0.503 ± 0.036 0.716 ± 0.029 20.8 ± 1.0 0.231 ± 0.037 44.75 ± 0.57 36.2 ± 1.6 gaussian NoGAM 5 0.745 ± 0.029 0.651 ± 0.033 0.874 ± 0.020 0.607 ± 0.045 0.531 ± 0.045 0.711 ± 0.043 19.2 ± 2.3 0.251 ± 0.045 186.81 ± 1.57 34.2 ± 2.3 gaussian Avici 5 0.702 ± 0.019 0.955 ± 0.020 0.558 ± 0.027 0.645 ± 0.030 0.877 ± 0.034 0.513 ± 0.033 13.2 ± 1.5 0.189 ± 0.049 12.95 ± 0.30 15.0 ± 1.6 gaussian NOTEARS 5 0.721 ± 0.029 0.750 ± 0.017 0.698 ± 0.046 0.567 ± 0.047 0.588 ± 0.038 0.552 ± 0.059 17.6 ± 2.7 0.238 ± 0.062 645.65 ± 211.44 23.6 ± 2.1 gaussian DAGMA 5 0.711 ± 0.051 0.768 ± 0.045 0.669 ± 0.063 0.494 ± 0.039 0.534 ± 0.036 0.465 ± 0.046 19.0 ± 3.0 0.305 ± 0.047 1581.15 ± 79.69 21.8 ± 1.5 gaussian DiffAN 5 0.747 ± 0.020 0.705 ± 0.020 0.803 ± 0.042 0.493 ± 0.050 0.466 ± 0.049 0.528 ± 0.057 20.6 ± 2.1 0.405 ± 0.056 24.51 ± 0.54 28.8 ± 2.0 gaussian SciNO 5 0.743 ± 0.022 0.687 ± 0.039 0.821 ± 0.027 0.484 ± 0.035 0.449 ± 0.042 0.532 ± 0.031 21.6 ± 2.0 0.390 ± 0.044 799.09 ± 34.96 30.6 ± 2.4 gaussian Caps 5 0.730 ± 0.032 0.746 ± 0.043 0.718 ± 0.028 0.532 ± 0.055 0.544 ± 0.059 0.523 ± 0.053 18.2 ± 1.9 0.342 ± 0.050 8.37 ± 0.31 24.4 ± 1.2 gaussian PC 5 0.623 ± 0.034 0.940 ± 0.031 0.471 ± 0.036 0.528 ± 0.031 0.809 ± 0.075 0.396 ± 0.023 16.0 ± 0.5 0.310 ± 0.057 4400.78 ± 810.66 12.8 ± 1.4 gaussian FCI 5 0.468 ± 0.047 0.960 ± 0.040 0.312 ± 0.037 0.420 ± 0.050 0.858 ± 0.075 0.279 ± 0.037 18.4 ± 1.3 0.300 ± 0.066 4225.70 ± 803.67 8.2 ± 1.0 multimodal Spade-Hist 5 0.925 ± 0.009 1.000 0.861 ± 0.016 0.925 ± 0.009 1.000 0.861 ± 0.016 3.6 ± 0.6 0.015 ± 0.009 0.62 ± 0.02 21.8 ± 1.4 multimodal Spade-Gauss 5 0.857 ± 0.016 0.945 ± 0.023 0.785 ± 0.019 0.763 ± 0.034 0.842 ± 0.039 0.699 ± 0.034 9.0 ± 1.6 0.145 ± 0.030 0.13 ± 0.01 21.0 ± 1.1 multimodal Topic 5 0.802 ± 0.017 0.773 ± 0.007 0.835 ± 0.030 0.712 ± 0.032 0.686 ± 0.028 0.741 ± 0.041 13.0 ± 1.9 0.165 ± 0.016 28.01 ± 2.06 27.2 ± 1.3 multimodal Cam 5 0.793 ± 0.022 0.681 ± 0.030 0.951 ± 0.010 0.731 ± 0.041 0.629 ± 0.043 0.876 ± 0.036 14.2 ± 1.5 0.101 ± 0.042 349.51 ± 1.32 35.4 ± 1.7 multimodal CauScale 5 0.804 ± 0.031 0.902 ± 0.025 0.730 ± 0.046 0.592 ± 0.053 0.665 ± 0.060 0.537 ± 0.052 13.8 ± 2.1 0.270 ± 0.058 6.05 ± 0.12 20.4 ± 1.3 multimodal SDCD 5 0.739 ± 0.019 0.856 ± 0.032 0.655 ± 0.029 0.580 ± 0.035 0.675 ± 0.056 0.512 ± 0.030 15.2 ± 1.0 0.295 ± 0.040 136.01 ± 1.47 19.4 ± 1.3 multimodal Globe 5 0.834 ± 0.022 0.918 ± 0.023 0.767 ± 0.032 0.810 ± 0.023 0.891 ± 0.017 0.745 ± 0.035 8.4 ± 1.3 0.118 ± 0.038 319.05 ± 28.31 21.2 ± 1.6 multimodal Das 5 0.701 ± 0.050 0.664 ± 0.042 0.746 ± 0.066 0.615 ± 0.061 0.584 ± 0.057 0.654 ± 0.072 18.0 ± 2.5 0.182 ± 0.044 63.38 ± 1.05 28.4 ± 2.1 multimodal Score 5 0.753 ± 0.035 0.654 ± 0.041 0.892 ± 0.029 0.659 ± 0.052 0.573 ± 0.054 0.778 ± 0.046 17.4 ± 2.3 0.182 ± 0.044 48.94 ± 0.50 34.8 ± 2.2 multimodal NoGAM 5 0.767 ± 0.039 0.666 ± 0.049 0.909 ± 0.019 0.717 ± 0.051 0.624 ± 0.058 0.850 ± 0.040 15.2 ± 2.3 0.119 ± 0.045 216.79 ± 2.17 34.8 ± 1.5 multimodal Avici 5 0.661 ± 0.039 0.900 ± 0.030 0.527 ± 0.043 0.448 ± 0.045 0.610 ± 0.054 0.357 ± 0.041 17.8 ± 1.7 0.406 ± 0.058 12.95 ± 0.30 14.6 ± 0.7 multimodal NOTEARS 5 0.792 ± 0.019 0.775 ± 0.032 0.814 ± 0.023 0.550 ± 0.073 0.541 ± 0.078 0.562 ± 0.070 17.0 ± 2.1 0.337 ± 0.087 617.41 ± 100.42 26.6 ± 1.3 multimodal DAGMA 5 0.741 ± 0.024 0.771 ± 0.027 0.714 ± 0.022 0.381 ± 0.033 0.396 ± 0.036 0.367 ± 0.031 21.8 ± 2.9 0.502 ± 0.047 1502.92 ± 142.78 23.6 ± 1.9 multimodal DiffAN 5 0.833 ± 0.032 0.794 ± 0.047 0.882 ± 0.025 0.658 ± 0.057 0.630 ± 0.067 0.693 ± 0.045 14.0 ± 3.1 0.266 ± 0.050 21.31 ± 0.41 28.6 ± 2.8 multimodal SciNO 5 0.805 ± 0.025 0.760 ± 0.039 0.861 ± 0.023 0.598 ± 0.049 0.567 ± 0.057 0.636 ± 0.041 16.0 ± 1.8 0.292 ± 0.039 189.03 ± 3.43 29.0 ± 2.3 multimodal Caps 5 0.767 ± 0.023 0.813 ± 0.050 0.736 ± 0.027 0.700 ± 0.038 0.744 ± 0.063 0.670 ± 0.030 12.8 ± 1.4 0.181 ± 0.038 7.41 ± 0.15 23.4 ± 2.2 multimodal PC 5 0.751 ± 0.047 0.957 ± 0.018 0.624 ± 0.056 0.557 ± 0.043 0.708 ± 0.030 0.463 ± 0.047 14.2 ± 1.6 0.367 ± 0.026 7056.93 ± 2045.90 16.6 ± 2.2 multimodal FCI 5 0.476 ± 0.058 1.000 0.320 ± 0.050 0.403 ± 0.065 0.831 ± 0.037 0.272 ± 0.053 18.2 ± 1.0 0.356 ± 0.063 8259.99 ± 2896.80 8.4 ± 1.9 unimodal_asymmetric Spade-Hist 5 0.909 ± 0.025 0.972 ± 0.012 0.857 ± 0.041 0.873 ± 0.036 0.932 ± 0.026 0.823 ± 0.049 5.4 ± 1.6 0.064 ± 0.035 0.64 ± 0.01 22.2 ± 1.2 unimodal_asymmetric Spade-Gauss 5 0.856 ± 0.030 0.945 ± 0.032 0.785 ± 0.034 0.746 ± 0.061 0.823 ± 0.066 0.683 ± 0.058 9.4 ± 2.3 0.158 ± 0.036 0.12 ± 0.01 21.0 ± 1.1 unimodal_asymmetric Topic 5 0.793 ± 0.029 0.770 ± 0.044 0.823 ± 0.030 0.671 ± 0.054 0.654 ± 0.065 0.694 ± 0.048 14.0 ± 2.2 0.175 ± 0.045 26.86 ± 2.34 27.0 ± 0.9 unimodal_asymmetric Cam 5 0.711 ± 0.014 0.573 ± 0.010 0.940 ± 0.029 0.589 ± 0.018 0.474 ± 0.012 0.778 ± 0.034 23.4 ± 1.7 0.194 ± 0.029 346.51 ± 2.97 41.6 ± 2.8 unimodal_asymmetric CauScale 5 0.784 ± 0.023 0.908 ± 0.041 0.703 ± 0.049 0.500 ± 0.045 0.585 ± 0.070 0.445 ± 0.041 16.0 ± 1.0 0.349 ± 0.052 5.96 ± 0.11 19.6 ± 1.5 unimodal_asymmetric SDCD 5 0.849 ± 0.024 0.904 ± 0.026 0.806 ± 0.038 0.585 ± 0.061 0.627 ± 0.075 0.551 ± 0.050 13.6 ± 2.2 0.351 ± 0.063 140.39 ± 5.34 22.6 ± 1.6 unimodal_asymmetric Globe 5 0.777 ± 0.047 0.896 ± 0.027 0.692 ± 0.061 0.613 ± 0.053 0.710 ± 0.059 0.545 ± 0.057 13.8 ± 2.5 0.291 ± 0.033 315.54 ± 34.39 19.2 ± 0.6 unimodal_asymmetric Das 5 0.693 ± 0.026 0.611 ± 0.040 0.807 ± 0.014 0.484 ± 0.081 0.433 ± 0.085 0.554 ± 0.076 24.6 ± 4.0 0.345 ± 0.071 62.94 ± 0.43 33.8 ± 2.6 unimodal_asymmetric Score 5 0.686 ± 0.034 0.543 ± 0.036 0.937 ± 0.023 0.464 ± 0.070 0.370 ± 0.062 0.625 ± 0.079 29.8 ± 4.3 0.345 ± 0.071 48.63 ± 0.16 44.2 ± 3.1 unimodal_asymmetric NoGAM 5 0.699 ± 0.016 0.557 ± 0.017 0.944 ± 0.021 0.487 ± 0.048 0.387 ± 0.038 0.657 ± 0.067 27.6 ± 1.7 0.319 ± 0.058 222.56 ± 4.60 43.0 ± 2.7 unimodal_asymmetric Avici 5 0.749 ± 0.042 0.963 ± 0.024 0.619 ± 0.052 0.543 ± 0.041 0.700 ± 0.045 0.448 ± 0.045 14.8 ± 2.1 0.359 ± 0.038 12.85 ± 0.26 16.0 ± 0.6 unimodal_asymmetric NOTEARS 5 0.766 ± 0.026 0.723 ± 0.040 0.824 ± 0.027 0.524 ± 0.046 0.497 ± 0.053 0.561 ± 0.039 19.8 ± 3.1 0.318 ± 0.061 493.55 ± 67.56 29.2 ± 2.4 unimodal_asymmetric DAGMA 5 0.734 ± 0.018 0.723 ± 0.029 0.756 ± 0.040 0.384 ± 0.052 0.378 ± 0.052 0.394 ± 0.053 23.2 ± 2.9 0.508 ± 0.063 1491.50 ± 84.12 26.6 ± 2.1 unimodal_asymmetric DiffAN 5 0.731 ± 0.025 0.626 ± 0.023 0.882 ± 0.032 0.443 ± 0.038 0.379 ± 0.032 0.534 ± 0.047 25.4 ± 3.2 0.429 ± 0.044 21.28 ± 0.38 36.0 ± 3.4 unimodal_asymmetric SciNO 5 0.829 ± 0.031 0.756 ± 0.048 0.928 ± 0.023 0.586 ± 0.075 0.537 ± 0.075 0.652 ± 0.074 16.6 ± 3.2 0.317 ± 0.071 191.94 ± 2.66 31.6 ± 2.7 unimodal_asymmetric Caps 5 0.769 ± 0.018 0.719 ± 0.036 0.836 ± 0.022 0.545 ± 0.091 0.518 ± 0.095 0.579 ± 0.083 19.2 ± 3.8 0.344 ± 0.075 8.59 ± 0.23 29.8 ± 2.6 unimodal_asymmetric PC 5 0.744 ± 0.030 0.939 ± 0.020 0.618 ± 0.035 0.602 ± 0.030 0.761 ± 0.037 0.499 ± 0.028 13.8 ± 1.6 0.322 ± 0.019 7593.29 ± 1835.82 16.8 ± 1.8 unimodal_asymmetric FCI 5 0.482 ± 0.072 0.982 ± 0.018 0.332 ± 0.060 0.421 ± 0.077 0.830 ± 0.046 0.291 ± 0.061 18.0 ± 1.4 0.300 ± 0.054 8653.47 ± 2148.03 8.8 ± 1.9 unimodal_symmetric Spade-Hist 5 0.851 ± 0.016 0.955 ± 0.020 0.770 ± 0.026 0.831 ± 0.013 0.932 ± 0.010 0.752 ± 0.026 7.4 ± 1.0 0.057 ± 0.017 0.62 ± 0.01 20.4 ± 1.2 unimodal_symmetric Spade-Gauss 5 0.875 ± 0.035 0.967 ± 0.020 0.804 ± 0.051 0.817 ± 0.046 0.904 ± 0.043 0.751 ± 0.057 7.4 ± 2.2 0.081 ± 0.034 0.13 ± 0.01 21.0 ± 1.5 unimodal_symmetric Topic 5 0.798 ± 0.011 0.825 ± 0.027 0.778 ± 0.025 0.712 ± 0.027 0.735 ± 0.030 0.694 ± 0.035 12.2 ± 1.4 0.130 ± 0.040 23.75 ± 1.12 23.8 ± 1.0 unimodal_symmetric Cam 5 0.693 ± 0.022 0.556 ± 0.031 0.933 ± 0.033 0.631 ± 0.040 0.506 ± 0.035 0.852 ± 0.062 23.4 ± 3.3 0.099 ± 0.044 340.21 ± 2.27 43.0 ± 4.1 unimodal_symmetric CauScale 5 0.720 ± 0.042 0.811 ± 0.025 0.655 ± 0.058 0.470 ± 0.046 0.527 ± 0.039 0.429 ± 0.053 18.4 ± 2.0 0.360 ± 0.032 5.89 ± 0.10 20.2 ± 1.4 unimodal_symmetric SDCD 5 0.763 ± 0.032 0.897 ± 0.027 0.676 ± 0.054 0.628 ± 0.017 0.744 ± 0.043 0.554 ± 0.033 13.4 ± 0.9 0.239 ± 0.020 138.42 ± 3.65 19.0 ± 1.7 unimodal_symmetric Globe 5 0.744 ± 0.043 0.812 ± 0.041 0.689 ± 0.049 0.695 ± 0.058 0.757 ± 0.055 0.645 ± 0.064 13.6 ± 3.4 0.127 ± 0.033 302.00 ± 24.24 21.4 ± 1.4 unimodal_symmetric Das 5 0.630 ± 0.026 0.558 ± 0.022 0.726 ± 0.040 0.499 ± 0.036 0.442 ± 0.030 0.576 ± 0.050 25.0 ± 0.9 0.259 ± 0.032 63.92 ± 0.99 33.0 ± 2.5 unimodal_symmetric Score 5 0.679 ± 0.017 0.549 ± 0.026 0.904 ± 0.033 0.539 ± 0.014 0.435 ± 0.017 0.719 ± 0.037 26.4 ± 2.1 0.259 ± 0.032 50.33 ± 1.09 42.2 ± 4.0 unimodal_symmetric NoGAM 5 0.700 ± 0.017 0.576 ± 0.032 0.913 ± 0.032 0.565 ± 0.035 0.465 ± 0.042 0.733 ± 0.036 25.0 ± 3.8 0.236 ± 0.034 224.08 ± 7.68 41.2 ± 5.0 unimodal_symmetric Avici 5 0.802 ± 0.041 0.968 ± 0.014 0.695 ± 0.060 0.724 ± 0.047 0.879 ± 0.058 0.625 ± 0.055 10.2 ± 1.7 0.138 ± 0.034 13.06 ± 0.26 18.0 ± 1.5 unimodal_symmetric NOTEARS 5 0.809 ± 0.034 0.812 ± 0.057 0.818 ± 0.040 0.651 ± 0.072 0.659 ± 0.097 0.653 ± 0.057 14.8 ± 4.0 0.212 ± 0.045 985.37 ± 242.10 26.2 ± 3.1 unimodal_symmetric DAGMA 5 0.761 ± 0.047 0.797 ± 0.066 0.733 ± 0.039 0.479 ± 0.112 0.510 ± 0.131 0.456 ± 0.098 19.8 ± 5.3 0.425 ± 0.098 1775.19 ± 151.94 24.0 ± 2.9 unimodal_symmetric DiffAN 5 0.708 ± 0.034 0.622 ± 0.052 0.837 ± 0.019 0.456 ± 0.048 0.404 ± 0.058 0.535 ± 0.033 25.4 ± 3.7 0.396 ± 0.051 22.99 ± 0.43 35.0 ± 3.7 unimodal_symmetric SciNO 5 0.785 ± 0.011 0.730 ± 0.029 0.863 ± 0.044 0.586 ± 0.047 0.548 ± 0.053 0.640 ± 0.049 17.6 ± 2.2 0.330 ± 0.048 195.81 ± 8.00 30.2 ± 2.7 unimodal_symmetric Caps 5 0.754 ± 0.021 0.784 ± 0.016 0.736 ± 0.047 0.609 ± 0.045 0.636 ± 0.053 0.591 ± 0.049 15.6 ± 1.7 0.301 ± 0.043 7.57 ± 0.38 23.8 ± 2.0 unimodal_symmetric PC 5 0.715 ± 0.061 1.000 0.570 ± 0.074 0.610 ± 0.039 0.865 ± 0.043 0.484 ± 0.052 13.0 ± 1.3 0.285 ± 0.037 6839.62 ± 1850.39 14.8 ± 2.7 unimodal_symmetric FCI 5 0.629 ± 0.041 1.000 0.464 ± 0.045 0.561 ± 0.050 0.888 ± 0.029 0.415 ± 0.049 14.8 ± 1.5 0.218 ± 0.039 7132.96 ± 1918.93 11.8 ± 1.4 Table 32: Qualitative comparison without marginalization (n=2000n=2000, d=20d=20, degree 3, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.757 ± 0.043 0.987 ± 0.013 0.619 ± 0.049 0.691 ± 0.051 0.901 ± 0.050 0.564 ± 0.050 11.6 ± 2.4 0.257 ± 0.048 0.65 ± 0.03 15.6 ± 0.6 gaussian Spade-Gauss 5 0.785 ± 0.034 0.965 ± 0.014 0.666 ± 0.043 0.767 ± 0.042 0.941 ± 0.026 0.651 ± 0.049 9.8 ± 2.1 0.103 ± 0.047 0.11 ± 0.00 17.2 ± 0.2 gaussian Topic 5 0.742 ± 0.018 0.784 ± 0.047 0.713 ± 0.026 0.696 ± 0.020 0.731 ± 0.022 0.672 ± 0.041 13.6 ± 0.7 0.126 ± 0.038 19.50 ± 0.55 23.0 ± 0.5 gaussian Cam 5 0.701 ± 0.021 0.562 ± 0.027 0.938 ± 0.008 0.669 ± 0.033 0.536 ± 0.036 0.893 ± 0.025 21.6 ± 2.8 0.054 ± 0.026 323.28 ± 2.87 42.6 ± 3.2 gaussian CauScale 5 0.719 ± 0.047 0.874 ± 0.012 0.621 ± 0.065 0.496 ± 0.054 0.610 ± 0.070 0.426 ± 0.054 16.8 ± 1.7 0.376 ± 0.073 6.07 ± 0.08 17.6 ± 1.0 gaussian SDCD 5 0.724 ± 0.025 0.813 ± 0.012 0.655 ± 0.033 0.563 ± 0.037 0.632 ± 0.036 0.508 ± 0.039 16.4 ± 1.9 0.324 ± 0.061 136.11 ± 1.61 20.2 ± 0.8 gaussian Globe 5 0.642 ± 0.061 0.851 ± 0.035 0.522 ± 0.064 0.602 ± 0.059 0.798 ± 0.040 0.490 ± 0.061 15.6 ± 3.0 0.382 ± 0.053 239.45 ± 13.86 15.0 ± 0.5 gaussian Das 5 0.678 ± 0.034 0.626 ± 0.036 0.742 ± 0.035 0.544 ± 0.038 0.503 ± 0.042 0.594 ± 0.036 21.6 ± 2.4 0.166 ± 0.036 62.22 ± 1.20 30.2 ± 2.5 gaussian Score 5 0.721 ± 0.030 0.596 ± 0.042 0.923 ± 0.009 0.607 ± 0.042 0.503 ± 0.050 0.774 ± 0.027 22.0 ± 2.8 0.166 ± 0.036 44.20 ± 0.90 39.8 ± 3.3 gaussian NoGAM 5 0.715 ± 0.024 0.588 ± 0.028 0.915 ± 0.014 0.572 ± 0.034 0.471 ± 0.034 0.730 ± 0.031 23.2 ± 2.4 0.233 ± 0.035 186.47 ± 1.64 39.6 ± 2.8 gaussian Avici 5 0.728 ± 0.033 1.000 0.577 ± 0.040 0.694 ± 0.045 0.952 ± 0.034 0.550 ± 0.048 11.6 ± 1.9 0.296 ± 0.054 14.31 ± 1.04 14.6 ± 1.2 gaussian NOTEARS 5 0.734 ± 0.035 0.718 ± 0.031 0.753 ± 0.044 0.587 ± 0.061 0.573 ± 0.056 0.603 ± 0.068 18.0 ± 3.2 0.268 ± 0.045 393.11 ± 68.25 26.6 ± 2.0 gaussian DAGMA 5 0.701 ± 0.026 0.775 ± 0.045 0.642 ± 0.015 0.451 ± 0.064 0.502 ± 0.080 0.410 ± 0.052 20.4 ± 3.7 0.466 ± 0.065 1648.93 ± 152.15 21.4 ± 2.3 gaussian DiffAN 5 0.804 ± 0.019 0.798 ± 0.032 0.814 ± 0.028 0.604 ± 0.059 0.603 ± 0.071 0.609 ± 0.053 15.4 ± 2.8 0.314 ± 0.043 24.46 ± 0.30 26.0 ± 2.2 gaussian SciNO 5 0.789 ± 0.028 0.745 ± 0.048 0.848 ± 0.021 0.507 ± 0.040 0.481 ± 0.053 0.541 ± 0.027 19.2 ± 2.1 0.371 ± 0.043 825.16 ± 28.68 29.0 ± 1.8 gaussian Caps 5 0.801 ± 0.027 0.811 ± 0.053 0.799 ± 0.022 0.645 ± 0.053 0.659 ± 0.078 0.638 ± 0.029 13.8 ± 1.7 0.249 ± 0.049 9.43 ± 0.45 25.0 ± 0.7 gaussian PC 5 0.790 ± 0.032 0.937 ± 0.017 0.693 ± 0.056 0.437 ± 0.055 0.532 ± 0.086 0.375 ± 0.038 17.0 ± 1.3 0.477 ± 0.084 4596.08 ± 891.38 18.8 ± 2.0 gaussian FCI 5 0.433 ± 0.039 0.958 ± 0.026 0.284 ± 0.034 0.350 ± 0.060 0.778 ± 0.101 0.229 ± 0.044 20.0 ± 2.0 0.392 ± 0.079 4908.00 ± 1310.93 7.6 ± 1.1 multimodal Spade-Hist 5 0.756 ± 0.023 0.989 ± 0.011 0.613 ± 0.027 0.737 ± 0.031 0.963 ± 0.023 0.598 ± 0.033 10.6 ± 1.7 0.127 ± 0.025 0.59 ± 0.01 15.6 ± 0.7 multimodal Spade-Gauss 5 0.769 ± 0.043 0.922 ± 0.065 0.663 ± 0.036 0.668 ± 0.067 0.807 ± 0.098 0.573 ± 0.051 13.2 ± 4.0 0.258 ± 0.064 0.11 ± 0.01 18.6 ± 2.1 multimodal Topic 5 0.723 ± 0.035 0.734 ± 0.036 0.717 ± 0.042 0.603 ± 0.050 0.617 ± 0.061 0.594 ± 0.043 17.6 ± 3.7 0.216 ± 0.046 24.68 ± 3.10 25.0 ± 2.6 multimodal Cam 5 0.712 ± 0.015 0.588 ± 0.026 0.914 ± 0.038 0.600 ± 0.043 0.496 ± 0.042 0.770 ± 0.062 22.6 ± 3.2 0.181 ± 0.041 350.97 ± 3.74 39.4 ± 2.5 multimodal CauScale 5 0.728 ± 0.028 0.902 ± 0.029 0.613 ± 0.034 0.548 ± 0.027 0.680 ± 0.038 0.460 ± 0.025 15.4 ± 1.1 0.309 ± 0.055 5.82 ± 0.04 17.2 ± 1.2 multimodal SDCD 5 0.643 ± 0.028 0.858 ± 0.030 0.515 ± 0.027 0.442 ± 0.062 0.587 ± 0.076 0.355 ± 0.053 18.4 ± 1.7 0.427 ± 0.065 136.84 ± 1.81 15.4 ± 1.7 multimodal Globe 5 0.651 ± 0.044 0.870 ± 0.070 0.521 ± 0.032 0.590 ± 0.068 0.790 ± 0.098 0.471 ± 0.052 16.2 ± 3.9 0.420 ± 0.076 249.45 ± 38.35 15.4 ± 1.5 multimodal Das 5 0.674 ± 0.017 0.634 ± 0.029 0.732 ± 0.041 0.611 ± 0.025 0.576 ± 0.038 0.661 ± 0.031 19.8 ± 2.3 0.104 ± 0.028 64.14 ± 0.59 29.4 ± 2.6 multimodal Score 5 0.737 ± 0.018 0.634 ± 0.020 0.891 ± 0.046 0.670 ± 0.033 0.577 ± 0.035 0.807 ± 0.047 18.4 ± 2.7 0.104 ± 0.028 47.58 ± 0.48 35.6 ± 2.7 multimodal NoGAM 5 0.733 ± 0.027 0.630 ± 0.027 0.882 ± 0.045 0.694 ± 0.027 0.597 ± 0.026 0.836 ± 0.045 17.6 ± 2.2 0.076 ± 0.014 208.04 ± 3.07 35.4 ± 2.3 multimodal Avici 5 0.621 ± 0.032 0.920 ± 0.029 0.471 ± 0.031 0.474 ± 0.032 0.706 ± 0.049 0.358 ± 0.025 17.2 ± 1.1 0.403 ± 0.042 14.71 ± 1.26 13.0 ± 1.3 multimodal NOTEARS 5 0.744 ± 0.010 0.778 ± 0.037 0.719 ± 0.018 0.552 ± 0.015 0.576 ± 0.018 0.536 ± 0.028 17.4 ± 1.7 0.330 ± 0.051 521.96 ± 84.90 24.2 ± 3.4 multimodal DAGMA 5 0.666 ± 0.024 0.791 ± 0.026 0.584 ± 0.040 0.419 ± 0.050 0.497 ± 0.050 0.368 ± 0.051 20.2 ± 2.1 0.443 ± 0.054 1482.37 ± 19.46 18.6 ± 1.2 multimodal DiffAN 5 0.814 ± 0.041 0.822 ± 0.050 0.811 ± 0.045 0.593 ± 0.078 0.600 ± 0.081 0.590 ± 0.080 15.8 ± 4.4 0.293 ± 0.060 22.28 ± 0.31 25.4 ± 2.8 multimodal SciNO 5 0.800 ± 0.027 0.813 ± 0.025 0.797 ± 0.053 0.567 ± 0.060 0.582 ± 0.072 0.562 ± 0.063 16.0 ± 2.3 0.280 ± 0.046 191.96 ± 2.69 24.8 ± 2.1 multimodal Caps 5 0.787 ± 0.032 0.837 ± 0.022 0.752 ± 0.057 0.677 ± 0.012 0.724 ± 0.033 0.644 ± 0.028 13.0 ± 1.6 0.197 ± 0.025 7.50 ± 0.29 22.6 ± 1.6 multimodal PC 5 0.797 ± 0.036 0.926 ± 0.013 0.707 ± 0.052 0.544 ± 0.043 0.630 ± 0.032 0.483 ± 0.049 14.8 ± 2.0 0.378 ± 0.034 6948.06 ± 2365.21 19.2 ± 1.5 multimodal FCI 5 0.513 ± 0.035 1.000 0.348 ± 0.034 0.436 ± 0.064 0.839 ± 0.085 0.297 ± 0.051 17.8 ± 1.6 0.349 ± 0.044 7726.12 ± 2443.11 8.8 ± 0.9 unimodal_asymmetric Spade-Hist 5 0.715 ± 0.024 0.970 ± 0.018 0.567 ± 0.028 0.660 ± 0.052 0.896 ± 0.066 0.524 ± 0.046 12.2 ± 0.8 0.243 ± 0.025 0.55 ± 0.01 15.0 ± 1.7 unimodal_asymmetric Spade-Gauss 5 0.805 ± 0.011 0.922 ± 0.050 0.725 ± 0.026 0.664 ± 0.032 0.763 ± 0.062 0.595 ± 0.020 12.0 ± 1.0 0.220 ± 0.031 0.11 ± 0.01 20.2 ± 1.9 unimodal_asymmetric Topic 5 0.740 ± 0.015 0.745 ± 0.027 0.741 ± 0.030 0.605 ± 0.025 0.610 ± 0.036 0.605 ± 0.025 16.4 ± 0.7 0.232 ± 0.028 21.65 ± 1.44 25.2 ± 1.5 unimodal_asymmetric Cam 5 0.691 ± 0.026 0.540 ± 0.029 0.969 ± 0.022 0.530 ± 0.055 0.414 ± 0.047 0.740 ± 0.067 28.0 ± 3.6 0.236 ± 0.061 349.28 ± 5.13 46.0 ± 3.7 unimodal_asymmetric CauScale 5 0.703 ± 0.037 0.823 ± 0.059 0.616 ± 0.028 0.367 ± 0.039 0.432 ± 0.052 0.321 ± 0.032 20.6 ± 1.7 0.519 ± 0.035 6.08 ± 0.12 19.0 ± 0.6 unimodal_asymmetric SDCD 5 0.740 ± 0.025 0.775 ± 0.045 0.714 ± 0.017 0.534 ± 0.049 0.556 ± 0.054 0.517 ± 0.051 17.4 ± 0.7 0.350 ± 0.066 137.02 ± 1.31 23.6 ± 1.7 unimodal_asymmetric Globe 5 0.631 ± 0.012 0.782 ± 0.026 0.534 ± 0.024 0.554 ± 0.018 0.689 ± 0.041 0.468 ± 0.021 17.4 ± 1.1 0.332 ± 0.037 232.66 ± 19.46 17.2 ± 0.9 unimodal_asymmetric Das 5 0.591 ± 0.025 0.552 ± 0.029 0.642 ± 0.031 0.457 ± 0.053 0.429 ± 0.054 0.493 ± 0.053 25.8 ± 1.7 0.302 ± 0.040 62.84 ± 1.10 29.6 ± 2.2 unimodal_asymmetric Score 5 0.684 ± 0.013 0.552 ± 0.016 0.903 ± 0.019 0.499 ± 0.036 0.403 ± 0.032 0.658 ± 0.042 27.0 ± 1.1 0.302 ± 0.040 46.77 ± 0.31 41.6 ± 2.9 unimodal_asymmetric NoGAM 5 0.699 ± 0.019 0.563 ± 0.025 0.928 ± 0.020 0.514 ± 0.046 0.416 ± 0.043 0.676 ± 0.045 26.4 ± 2.5 0.293 ± 0.043 210.73 ± 1.58 42.0 ± 3.1 unimodal_asymmetric Avici 5 0.706 ± 0.043 0.966 ± 0.021 0.561 ± 0.048 0.540 ± 0.038 0.722 ± 0.029 0.433 ± 0.040 15.4 ± 1.9 0.375 ± 0.031 12.92 ± 0.36 15.0 ± 1.0 unimodal_asymmetric NOTEARS 5 0.754 ± 0.020 0.707 ± 0.025 0.813 ± 0.033 0.536 ± 0.045 0.503 ± 0.047 0.578 ± 0.050 19.4 ± 1.8 0.339 ± 0.051 527.63 ± 79.36 29.4 ± 2.7 unimodal_asymmetric DAGMA 5 0.719 ± 0.042 0.715 ± 0.056 0.732 ± 0.037 0.383 ± 0.051 0.383 ± 0.058 0.385 ± 0.045 23.0 ± 2.4 0.455 ± 0.051 1654.98 ± 76.61 26.4 ± 2.4 unimodal_asymmetric DiffAN 5 0.694 ± 0.008 0.610 ± 0.026 0.815 ± 0.028 0.435 ± 0.028 0.384 ± 0.033 0.509 ± 0.028 26.6 ± 3.9 0.444 ± 0.031 21.87 ± 0.57 34.8 ± 4.7 unimodal_asymmetric SciNO 5 0.752 ± 0.030 0.669 ± 0.041 0.870 ± 0.032 0.563 ± 0.043 0.504 ± 0.051 0.645 ± 0.025 20.4 ± 2.9 0.282 ± 0.030 189.70 ± 2.38 33.6 ± 3.4 unimodal_asymmetric Caps 5 0.752 ± 0.018 0.719 ± 0.022 0.796 ± 0.036 0.494 ± 0.045 0.475 ± 0.052 0.518 ± 0.041 20.0 ± 1.3 0.366 ± 0.045 8.45 ± 0.13 28.2 ± 2.5 unimodal_asymmetric PC 5 0.779 ± 0.070 0.940 ± 0.032 0.681 ± 0.087 0.572 ± 0.077 0.688 ± 0.070 0.500 ± 0.080 13.6 ± 2.4 0.348 ± 0.072 8139.80 ± 3667.78 18.4 ± 3.0 unimodal_asymmetric FCI 5 0.540 ± 0.054 0.975 ± 0.025 0.379 ± 0.052 0.456 ± 0.073 0.812 ± 0.065 0.323 ± 0.065 17.4 ± 2.0 0.291 ± 0.071 8191.14 ± 3350.50 9.8 ± 1.3 unimodal_symmetric Spade-Hist 5 0.689 ± 0.033 0.931 ± 0.031 0.554 ± 0.044 0.609 ± 0.040 0.820 ± 0.034 0.490 ± 0.046 14.2 ± 1.8 0.291 ± 0.059 0.59 ± 0.02 15.2 ± 1.7 unimodal_symmetric Spade-Gauss 5 0.782 ± 0.042 0.907 ± 0.033 0.691 ± 0.049 0.668 ± 0.056 0.778 ± 0.068 0.588 ± 0.052 12.4 ± 2.4 0.217 ± 0.037 0.11 ± 0.00 19.0 ± 0.4 unimodal_symmetric Topic 5 0.721 ± 0.042 0.748 ± 0.040 0.700 ± 0.052 0.621 ± 0.065 0.646 ± 0.068 0.602 ± 0.066 16.4 ± 3.3 0.213 ± 0.059 20.40 ± 0.63 23.4 ± 0.7 unimodal_symmetric Cam 5 0.626 ± 0.020 0.474 ± 0.023 0.929 ± 0.024 0.490 ± 0.056 0.373 ± 0.047 0.722 ± 0.071 33.4 ± 3.8 0.227 ± 0.060 353.41 ± 4.16 50.0 ± 3.9 unimodal_symmetric CauScale 5 0.655 ± 0.027 0.763 ± 0.045 0.585 ± 0.040 0.381 ± 0.032 0.448 ± 0.050 0.339 ± 0.032 21.6 ± 1.2 0.451 ± 0.058 5.94 ± 0.14 19.4 ± 1.2 unimodal_symmetric SDCD 5 0.684 ± 0.032 0.836 ± 0.029 0.584 ± 0.040 0.530 ± 0.027 0.646 ± 0.013 0.453 ± 0.035 17.0 ± 1.9 0.389 ± 0.039 139.24 ± 0.79 17.6 ± 1.2 unimodal_symmetric Globe 5 0.675 ± 0.022 0.859 ± 0.042 0.558 ± 0.022 0.618 ± 0.024 0.787 ± 0.039 0.512 ± 0.024 14.8 ± 1.3 0.292 ± 0.021 258.82 ± 27.24 16.6 ± 1.4 unimodal_symmetric Das 5 0.597 ± 0.024 0.529 ± 0.026 0.688 ± 0.029 0.506 ± 0.037 0.449 ± 0.038 0.582 ± 0.038 25.8 ± 0.8 0.193 ± 0.039 62.99 ± 0.44 33.0 ± 2.1 unimodal_symmetric Score 5 0.669 ± 0.031 0.532 ± 0.030 0.909 ± 0.041 0.564 ± 0.047 0.449 ± 0.041 0.763 ± 0.057 26.2 ± 2.6 0.193 ± 0.039 49.41 ± 1.49 43.6 ± 3.2 unimodal_symmetric NoGAM 5 0.663 ± 0.035 0.532 ± 0.039 0.896 ± 0.043 0.540 ± 0.061 0.436 ± 0.057 0.721 ± 0.066 27.2 ± 3.7 0.221 ± 0.050 219.11 ± 9.81 43.2 ± 3.6 unimodal_symmetric Avici 5 0.766 ± 0.030 0.944 ± 0.026 0.649 ± 0.041 0.671 ± 0.057 0.820 ± 0.039 0.572 ± 0.064 11.8 ± 1.5 0.204 ± 0.051 13.06 ± 0.31 17.6 ± 1.9 unimodal_symmetric NOTEARS 5 0.737 ± 0.035 0.707 ± 0.035 0.774 ± 0.047 0.560 ± 0.064 0.535 ± 0.059 0.590 ± 0.073 19.2 ± 3.9 0.303 ± 0.051 505.22 ± 68.82 27.8 ± 2.3 unimodal_symmetric DAGMA 5 0.727 ± 0.040 0.750 ± 0.057 0.710 ± 0.038 0.484 ± 0.090 0.505 ± 0.105 0.469 ± 0.081 20.4 ± 4.4 0.371 ± 0.092 1850.55 ± 78.78 24.4 ± 2.4 unimodal_symmetric DiffAN 5 0.699 ± 0.026 0.597 ± 0.027 0.848 ± 0.039 0.469 ± 0.059 0.402 ± 0.053 0.566 ± 0.070 25.8 ± 3.7 0.370 ± 0.066 22.21 ± 0.36 36.0 ± 2.3 unimodal_symmetric SciNO 5 0.758 ± 0.011 0.670 ± 0.023 0.882 ± 0.028 0.541 ± 0.038 0.480 ± 0.043 0.625 ± 0.030 20.6 ± 1.8 0.290 ± 0.035 192.08 ± 2.35 33.6 ± 2.8 unimodal_symmetric Caps 5 0.738 ± 0.025 0.701 ± 0.020 0.787 ± 0.051 0.609 ± 0.041 0.581 ± 0.044 0.647 ± 0.049 17.2 ± 0.9 0.237 ± 0.055 8.03 ± 0.55 28.4 ± 2.2 unimodal_symmetric PC 5 0.750 ± 0.061 0.942 ± 0.027 0.635 ± 0.080 0.590 ± 0.042 0.754 ± 0.061 0.494 ± 0.046 13.6 ± 0.9 0.325 ± 0.052 7050.44 ± 2889.29 17.2 ± 2.7 unimodal_symmetric FCI 5 0.384 ± 0.069 1.000 0.247 ± 0.052 0.384 ± 0.069 1.000 0.247 ± 0.052 19.0 ± 1.4 0.369 ± 0.055 7299.94 ± 3234.99 6.4 ± 1.4 Table 33: Qualitative comparison without marginalization (n=2000n=2000, d=20d=20, degree 4, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.789 ± 0.034 0.925 ± 0.048 0.689 ± 0.030 0.718 ± 0.058 0.844 ± 0.073 0.627 ± 0.050 16.0 ± 4.3 0.139 ± 0.058 0.73 ± 0.04 26.4 ± 2.4 gaussian Spade-Gauss 5 0.830 ± 0.018 0.992 ± 0.008 0.716 ± 0.025 0.830 ± 0.018 0.992 ± 0.008 0.716 ± 0.025 10.2 ± 1.2 0.032 ± 0.013 0.17 ± 0.03 25.4 ± 2.0 gaussian Topic 5 0.783 ± 0.029 0.850 ± 0.018 0.728 ± 0.038 0.777 ± 0.030 0.843 ± 0.018 0.722 ± 0.040 14.2 ± 1.8 0.032 ± 0.020 32.39 ± 6.13 30.0 ± 2.2 gaussian Cam 5 0.808 ± 0.037 0.729 ± 0.039 0.910 ± 0.041 0.798 ± 0.040 0.719 ± 0.039 0.899 ± 0.046 15.8 ± 3.5 0.017 ± 0.012 335.82 ± 3.59 44.2 ± 3.7 gaussian CauScale 5 0.719 ± 0.041 0.856 ± 0.047 0.623 ± 0.042 0.493 ± 0.042 0.586 ± 0.047 0.428 ± 0.040 24.2 ± 3.2 0.306 ± 0.043 5.97 ± 0.12 25.4 ± 1.1 gaussian SDCD 5 0.729 ± 0.029 0.890 ± 0.040 0.622 ± 0.033 0.612 ± 0.041 0.748 ± 0.051 0.522 ± 0.039 20.2 ± 3.3 0.301 ± 0.018 142.69 ± 3.02 25.0 ± 3.1 gaussian Globe 5 0.685 ± 0.024 0.899 ± 0.022 0.556 ± 0.030 0.669 ± 0.025 0.879 ± 0.032 0.543 ± 0.030 18.4 ± 1.9 0.205 ± 0.021 322.35 ± 24.89 21.6 ± 1.1 gaussian Das 5 0.708 ± 0.022 0.685 ± 0.021 0.735 ± 0.032 0.582 ± 0.031 0.564 ± 0.034 0.603 ± 0.032 25.6 ± 1.6 0.223 ± 0.039 67.05 ± 1.24 37.6 ± 2.3 gaussian Score 5 0.727 ± 0.025 0.624 ± 0.023 0.877 ± 0.044 0.586 ± 0.042 0.505 ± 0.044 0.702 ± 0.040 28.6 ± 2.1 0.223 ± 0.039 46.12 ± 1.78 49.2 ± 2.9 gaussian NoGAM 5 0.750 ± 0.018 0.651 ± 0.014 0.887 ± 0.036 0.642 ± 0.028 0.558 ± 0.029 0.758 ± 0.030 25.2 ± 2.0 0.173 ± 0.032 224.90 ± 12.90 48.0 ± 3.7 gaussian Avici 5 0.681 ± 0.023 0.981 ± 0.011 0.523 ± 0.028 0.645 ± 0.037 0.928 ± 0.035 0.496 ± 0.036 18.2 ± 2.2 0.253 ± 0.058 13.13 ± 0.23 18.8 ± 1.6 gaussian NOTEARS 5 0.725 ± 0.016 0.745 ± 0.025 0.708 ± 0.021 0.510 ± 0.024 0.525 ± 0.032 0.497 ± 0.019 26.2 ± 1.7 0.325 ± 0.037 1007.48 ± 67.77 33.6 ± 2.6 gaussian DAGMA 5 0.663 ± 0.024 0.727 ± 0.027 0.612 ± 0.029 0.346 ± 0.031 0.380 ± 0.035 0.319 ± 0.028 32.2 ± 3.2 0.497 ± 0.028 1780.54 ± 132.31 29.6 ± 2.2 gaussian DiffAN 5 0.750 ± 0.030 0.713 ± 0.044 0.802 ± 0.039 0.532 ± 0.019 0.505 ± 0.024 0.571 ± 0.035 27.2 ± 3.0 0.348 ± 0.038 25.18 ± 0.57 40.4 ± 4.6 gaussian SciNO 5 0.782 ± 0.027 0.763 ± 0.022 0.803 ± 0.035 0.607 ± 0.033 0.593 ± 0.031 0.624 ± 0.038 21.8 ± 2.1 0.283 ± 0.030 753.46 ± 27.82 36.8 ± 2.0 gaussian Caps 5 0.717 ± 0.025 0.769 ± 0.042 0.677 ± 0.024 0.609 ± 0.036 0.655 ± 0.051 0.574 ± 0.031 22.2 ± 2.1 0.235 ± 0.032 9.06 ± 0.15 31.0 ± 1.9 gaussian PC 5 0.693 ± 0.066 0.945 ± 0.026 0.558 ± 0.070 0.570 ± 0.036 0.790 ± 0.035 0.456 ± 0.043 20.2 ± 2.4 0.331 ± 0.043 14366.46 ± 4325.37 20.6 ± 2.8 gaussian FCI 5 0.504 ± 0.059 0.988 ± 0.012 0.347 ± 0.052 0.467 ± 0.060 0.919 ± 0.052 0.321 ± 0.052 24.0 ± 2.4 0.259 ± 0.022 15942.50 ± 5039.54 12.6 ± 2.1 multimodal Spade-Hist 5 0.838 ± 0.037 0.949 ± 0.017 0.755 ± 0.051 0.826 ± 0.049 0.933 ± 0.027 0.746 ± 0.060 10.8 ± 3.2 0.102 ± 0.074 0.71 ± 0.02 27.6 ± 1.3 multimodal Spade-Gauss 5 0.822 ± 0.026 0.963 ± 0.019 0.719 ± 0.034 0.778 ± 0.027 0.912 ± 0.027 0.680 ± 0.031 12.4 ± 2.2 0.120 ± 0.022 0.17 ± 0.01 26.2 ± 1.9 multimodal Topic 5 0.788 ± 0.022 0.821 ± 0.026 0.762 ± 0.033 0.723 ± 0.021 0.753 ± 0.027 0.699 ± 0.028 16.6 ± 1.9 0.108 ± 0.014 35.45 ± 3.64 32.4 ± 1.1 multimodal Cam 5 0.848 ± 0.017 0.781 ± 0.032 0.933 ± 0.014 0.801 ± 0.016 0.738 ± 0.031 0.881 ± 0.010 13.4 ± 0.9 0.062 ± 0.009 328.08 ± 4.45 42.0 ± 2.1 multimodal CauScale 5 0.710 ± 0.015 0.887 ± 0.024 0.594 ± 0.024 0.474 ± 0.063 0.594 ± 0.081 0.397 ± 0.054 23.6 ± 1.5 0.321 ± 0.060 6.06 ± 0.16 23.4 ± 1.0 multimodal SDCD 5 0.719 ± 0.025 0.830 ± 0.018 0.636 ± 0.032 0.464 ± 0.046 0.534 ± 0.047 0.411 ± 0.046 25.4 ± 3.0 0.396 ± 0.034 139.48 ± 0.77 27.0 ± 2.3 multimodal Globe 5 0.739 ± 0.029 0.896 ± 0.029 0.639 ± 0.046 0.720 ± 0.025 0.875 ± 0.036 0.621 ± 0.040 16.4 ± 2.1 0.147 ± 0.051 412.06 ± 40.89 25.0 ± 2.2 multimodal Das 5 0.665 ± 0.024 0.638 ± 0.034 0.702 ± 0.036 0.561 ± 0.034 0.538 ± 0.041 0.591 ± 0.038 28.2 ± 1.1 0.174 ± 0.019 67.45 ± 3.55 38.8 ± 2.9 multimodal Score 5 0.754 ± 0.031 0.647 ± 0.044 0.913 ± 0.019 0.634 ± 0.039 0.546 ± 0.050 0.766 ± 0.019 26.0 ± 3.2 0.174 ± 0.019 48.11 ± 1.00 49.8 ± 2.2 multimodal NoGAM 5 0.796 ± 0.032 0.712 ± 0.053 0.917 ± 0.012 0.711 ± 0.045 0.638 ± 0.063 0.816 ± 0.017 19.8 ± 3.2 0.128 ± 0.017 227.01 ± 22.92 45.6 ± 2.2 multimodal Avici 5 0.612 ± 0.036 0.989 ± 0.011 0.447 ± 0.039 0.496 ± 0.054 0.803 ± 0.078 0.362 ± 0.045 22.8 ± 2.4 0.310 ± 0.033 14.01 ± 1.00 15.8 ± 1.5 multimodal NOTEARS 5 0.707 ± 0.021 0.679 ± 0.034 0.743 ± 0.015 0.465 ± 0.048 0.443 ± 0.040 0.492 ± 0.060 31.2 ± 4.5 0.339 ± 0.075 1045.68 ± 209.67 38.8 ± 3.3 multimodal DAGMA 5 0.683 ± 0.016 0.701 ± 0.015 0.669 ± 0.028 0.339 ± 0.023 0.351 ± 0.031 0.331 ± 0.020 33.8 ± 3.1 0.487 ± 0.035 1480.85 ± 51.58 33.6 ± 2.9 multimodal DiffAN 5 0.781 ± 0.029 0.720 ± 0.047 0.859 ± 0.017 0.561 ± 0.058 0.521 ± 0.070 0.613 ± 0.046 25.2 ± 3.3 0.323 ± 0.058 26.63 ± 0.97 42.0 ± 1.7 multimodal SciNO 5 0.800 ± 0.029 0.752 ± 0.036 0.856 ± 0.025 0.568 ± 0.063 0.537 ± 0.068 0.604 ± 0.057 24.0 ± 4.0 0.337 ± 0.059 829.66 ± 32.63 40.2 ± 2.8 multimodal Caps 5 0.788 ± 0.017 0.810 ± 0.024 0.767 ± 0.013 0.667 ± 0.031 0.687 ± 0.038 0.649 ± 0.025 18.4 ± 1.1 0.204 ± 0.031 9.17 ± 0.27 33.2 ± 1.7 multimodal PC 5 0.790 ± 0.028 0.984 ± 0.010 0.664 ± 0.043 0.599 ± 0.023 0.753 ± 0.049 0.501 ± 0.018 18.0 ± 1.5 0.287 ± 0.043 13214.87 ± 3755.25 23.4 ± 0.7 multimodal FCI 5 0.550 ± 0.021 0.985 ± 0.015 0.383 ± 0.021 0.485 ± 0.035 0.873 ± 0.067 0.338 ± 0.026 23.6 ± 2.2 0.288 ± 0.044 16532.67 ± 5624.96 13.6 ± 0.9 unimodal_asymmetric Spade-Hist 5 0.839 ± 0.031 0.954 ± 0.019 0.750 ± 0.038 0.839 ± 0.031 0.954 ± 0.019 0.750 ± 0.038 10.0 ± 2.0 0.012 ± 0.007 0.74 ± 0.03 27.6 ± 2.2 unimodal_asymmetric Spade-Gauss 5 0.824 ± 0.029 0.918 ± 0.015 0.752 ± 0.043 0.736 ± 0.031 0.820 ± 0.025 0.672 ± 0.042 14.2 ± 2.2 0.133 ± 0.015 0.17 ± 0.02 28.6 ± 1.8 unimodal_asymmetric Topic 5 0.819 ± 0.027 0.841 ± 0.020 0.798 ± 0.035 0.699 ± 0.024 0.718 ± 0.019 0.682 ± 0.030 16.8 ± 2.4 0.187 ± 0.005 34.81 ± 4.50 33.2 ± 1.9 unimodal_asymmetric Cam 5 0.763 ± 0.019 0.653 ± 0.019 0.920 ± 0.027 0.682 ± 0.028 0.585 ± 0.031 0.821 ± 0.021 23.4 ± 1.9 0.120 ± 0.033 331.45 ± 4.67 49.4 ± 2.6 unimodal_asymmetric CauScale 5 0.730 ± 0.029 0.882 ± 0.015 0.629 ± 0.044 0.478 ± 0.050 0.576 ± 0.054 0.413 ± 0.050 23.8 ± 2.4 0.372 ± 0.061 5.86 ± 0.09 24.8 ± 1.4 unimodal_asymmetric SDCD 5 0.734 ± 0.022 0.810 ± 0.021 0.671 ± 0.025 0.539 ± 0.020 0.595 ± 0.017 0.494 ± 0.022 23.6 ± 2.5 0.280 ± 0.029 140.63 ± 0.77 29.0 ± 1.5 unimodal_asymmetric Globe 5 0.654 ± 0.017 0.850 ± 0.046 0.539 ± 0.027 0.592 ± 0.050 0.761 ± 0.054 0.491 ± 0.051 21.4 ± 1.2 0.246 ± 0.042 408.31 ± 68.69 22.6 ± 2.3 unimodal_asymmetric Das 5 0.697 ± 0.037 0.629 ± 0.041 0.784 ± 0.038 0.520 ± 0.044 0.471 ± 0.047 0.583 ± 0.042 31.0 ± 3.6 0.300 ± 0.036 68.23 ± 0.82 43.8 ± 2.0 unimodal_asymmetric Score 5 0.688 ± 0.015 0.551 ± 0.012 0.915 ± 0.027 0.508 ± 0.024 0.408 ± 0.022 0.674 ± 0.027 37.6 ± 2.8 0.300 ± 0.036 46.35 ± 2.25 58.2 ± 3.4 unimodal_asymmetric NoGAM 5 0.707 ± 0.022 0.573 ± 0.027 0.932 ± 0.029 0.517 ± 0.024 0.419 ± 0.028 0.679 ± 0.018 36.2 ± 3.6 0.290 ± 0.028 192.28 ± 8.47 57.6 ± 4.5 unimodal_asymmetric Avici 5 0.685 ± 0.041 0.915 ± 0.015 0.552 ± 0.048 0.482 ± 0.024 0.647 ± 0.022 0.388 ± 0.029 23.6 ± 2.7 0.377 ± 0.033 12.89 ± 0.23 20.8 ± 0.9 unimodal_asymmetric NOTEARS 5 0.728 ± 0.020 0.675 ± 0.023 0.792 ± 0.025 0.472 ± 0.033 0.439 ± 0.037 0.512 ± 0.029 31.2 ± 4.2 0.359 ± 0.029 679.64 ± 48.20 41.6 ± 3.9 unimodal_asymmetric DAGMA 5 0.715 ± 0.028 0.728 ± 0.036 0.704 ± 0.024 0.369 ± 0.035 0.377 ± 0.040 0.363 ± 0.031 32.2 ± 4.1 0.490 ± 0.020 1473.49 ± 55.08 34.2 ± 2.6 unimodal_asymmetric DiffAN 5 0.736 ± 0.025 0.636 ± 0.038 0.885 ± 0.029 0.470 ± 0.035 0.409 ± 0.045 0.560 ± 0.018 34.6 ± 5.4 0.389 ± 0.035 24.14 ± 0.47 50.0 ± 5.7 unimodal_asymmetric SciNO 5 0.772 ± 0.043 0.695 ± 0.053 0.877 ± 0.036 0.542 ± 0.048 0.490 ± 0.053 0.614 ± 0.043 29.0 ± 6.2 0.318 ± 0.033 776.46 ± 33.69 45.6 ± 5.4 unimodal_asymmetric Caps 5 0.717 ± 0.036 0.690 ± 0.043 0.753 ± 0.044 0.520 ± 0.034 0.501 ± 0.040 0.546 ± 0.035 28.6 ± 4.0 0.331 ± 0.016 9.68 ± 0.31 38.6 ± 3.3 unimodal_asymmetric PC 5 0.774 ± 0.048 0.975 ± 0.015 0.652 ± 0.063 0.550 ± 0.053 0.689 ± 0.045 0.464 ± 0.055 19.8 ± 2.9 0.402 ± 0.026 19056.70 ± 4312.84 23.4 ± 2.5 unimodal_asymmetric FCI 5 0.514 ± 0.054 0.989 ± 0.011 0.356 ± 0.052 0.486 ± 0.052 0.939 ± 0.040 0.336 ± 0.050 23.8 ± 2.8 0.292 ± 0.047 20781.26 ± 4610.60 12.6 ± 2.0 unimodal_symmetric Spade-Hist 5 0.853 ± 0.030 0.993 ± 0.007 0.753 ± 0.045 0.853 ± 0.030 0.993 ± 0.007 0.753 ± 0.045 9.0 ± 2.0 0.033 ± 0.014 0.76 ± 0.04 26.6 ± 2.0 unimodal_symmetric Spade-Gauss 5 0.824 ± 0.025 0.971 ± 0.007 0.719 ± 0.039 0.798 ± 0.030 0.941 ± 0.020 0.697 ± 0.041 11.6 ± 2.0 0.052 ± 0.016 0.17 ± 0.02 26.0 ± 1.9 unimodal_symmetric Topic 5 0.791 ± 0.027 0.869 ± 0.033 0.731 ± 0.038 0.755 ± 0.019 0.830 ± 0.028 0.697 ± 0.030 14.8 ± 2.1 0.089 ± 0.015 31.59 ± 3.70 29.6 ± 2.3 unimodal_symmetric Cam 5 0.723 ± 0.024 0.619 ± 0.022 0.875 ± 0.038 0.669 ± 0.041 0.575 ± 0.041 0.805 ± 0.042 26.0 ± 3.5 0.094 ± 0.039 330.96 ± 2.91 49.6 ± 3.1 unimodal_symmetric CauScale 5 0.639 ± 0.030 0.796 ± 0.024 0.537 ± 0.035 0.419 ± 0.048 0.527 ± 0.064 0.349 ± 0.040 27.6 ± 2.4 0.374 ± 0.077 6.20 ± 0.12 23.6 ± 1.6 unimodal_symmetric SDCD 5 0.700 ± 0.038 0.816 ± 0.047 0.622 ± 0.049 0.468 ± 0.032 0.542 ± 0.028 0.417 ± 0.037 26.0 ± 3.1 0.383 ± 0.039 143.17 ± 2.24 27.6 ± 3.6 unimodal_symmetric Globe 5 0.628 ± 0.025 0.819 ± 0.020 0.511 ± 0.028 0.622 ± 0.022 0.811 ± 0.020 0.505 ± 0.024 21.6 ± 2.5 0.217 ± 0.028 365.92 ± 40.01 21.8 ± 1.2 unimodal_symmetric Das 5 0.638 ± 0.034 0.604 ± 0.038 0.680 ± 0.035 0.501 ± 0.054 0.477 ± 0.061 0.530 ± 0.049 32.8 ± 4.8 0.250 ± 0.057 69.66 ± 0.38 40.2 ± 3.8 unimodal_symmetric Score 5 0.679 ± 0.031 0.574 ± 0.038 0.839 ± 0.032 0.532 ± 0.065 0.454 ± 0.065 0.651 ± 0.063 35.4 ± 6.2 0.250 ± 0.057 46.11 ± 1.85 52.2 ± 4.7 unimodal_symmetric NoGAM 5 0.701 ± 0.025 0.597 ± 0.037 0.863 ± 0.032 0.579 ± 0.056 0.496 ± 0.061 0.705 ± 0.049 32.0 ± 5.2 0.182 ± 0.048 207.07 ± 10.45 51.6 ± 4.9 unimodal_symmetric Avici 5 0.722 ± 0.009 0.973 ± 0.011 0.575 ± 0.013 0.625 ± 0.030 0.842 ± 0.039 0.498 ± 0.027 18.2 ± 1.4 0.292 ± 0.052 13.85 ± 0.99 20.8 ± 1.5 unimodal_symmetric NOTEARS 5 0.654 ± 0.032 0.621 ± 0.048 0.698 ± 0.023 0.477 ± 0.057 0.457 ± 0.065 0.505 ± 0.051 34.4 ± 7.3 0.317 ± 0.049 754.57 ± 31.67 41.2 ± 5.8 unimodal_symmetric DAGMA 5 0.665 ± 0.043 0.719 ± 0.061 0.626 ± 0.040 0.345 ± 0.059 0.372 ± 0.062 0.325 ± 0.059 34.0 ± 6.5 0.478 ± 0.048 1584.43 ± 98.10 32.0 ± 4.8 unimodal_symmetric DiffAN 5 0.706 ± 0.030 0.636 ± 0.036 0.802 ± 0.043 0.468 ± 0.050 0.424 ± 0.054 0.528 ± 0.049 34.0 ± 5.6 0.354 ± 0.023 26.08 ± 1.13 45.0 ± 4.4 unimodal_symmetric SciNO 5 0.695 ± 0.051 0.670 ± 0.065 0.736 ± 0.053 0.499 ± 0.090 0.487 ± 0.096 0.521 ± 0.089 30.8 ± 6.7 0.347 ± 0.078 835.31 ± 17.41 39.8 ± 4.3 unimodal_symmetric Caps 5 0.727 ± 0.047 0.734 ± 0.050 0.725 ± 0.055 0.569 ± 0.065 0.576 ± 0.071 0.565 ± 0.063 25.2 ± 4.8 0.234 ± 0.039 8.97 ± 0.39 34.8 ± 2.7 unimodal_symmetric PC 5 0.699 ± 0.036 0.931 ± 0.019 0.569 ± 0.052 0.441 ± 0.045 0.589 ± 0.059 0.358 ± 0.045 24.2 ± 2.4 0.482 ± 0.038 13057.23 ± 3467.60 21.4 ± 2.2 unimodal_symmetric FCI 5 0.446 ± 0.089 1.000 0.304 ± 0.076 0.343 ± 0.109 0.685 ± 0.134 0.239 ± 0.086 26.6 ± 3.1 0.388 ± 0.080 16715.52 ± 5309.80 10.8 ± 2.8 Table 34: Qualitative comparison without marginalization (n=2000n=2000, d=20d=20, degree 4, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.738 ± 0.024 0.919 ± 0.026 0.624 ± 0.040 0.594 ± 0.019 0.741 ± 0.030 0.502 ± 0.029 19.6 ± 1.4 0.339 ± 0.016 0.66 ± 0.03 24.2 ± 2.7 gaussian Spade-Gauss 5 0.762 ± 0.040 0.912 ± 0.017 0.659 ± 0.050 0.736 ± 0.045 0.880 ± 0.028 0.636 ± 0.053 14.8 ± 2.1 0.096 ± 0.025 0.14 ± 0.02 25.6 ± 2.8 gaussian Topic 5 0.763 ± 0.042 0.819 ± 0.034 0.719 ± 0.056 0.682 ± 0.054 0.730 ± 0.046 0.644 ± 0.065 17.6 ± 1.9 0.154 ± 0.032 30.30 ± 4.87 31.0 ± 3.2 gaussian Cam 5 0.714 ± 0.032 0.581 ± 0.039 0.935 ± 0.019 0.646 ± 0.036 0.526 ± 0.040 0.845 ± 0.028 29.4 ± 2.9 0.116 ± 0.036 335.78 ± 6.66 57.2 ± 4.2 gaussian CauScale 5 0.686 ± 0.014 0.864 ± 0.035 0.572 ± 0.021 0.514 ± 0.024 0.644 ± 0.016 0.430 ± 0.029 23.4 ± 1.8 0.296 ± 0.032 6.07 ± 0.16 23.6 ± 2.4 gaussian SDCD 5 0.722 ± 0.029 0.904 ± 0.029 0.611 ± 0.047 0.597 ± 0.032 0.747 ± 0.039 0.505 ± 0.043 20.0 ± 1.9 0.275 ± 0.022 139.60 ± 3.47 24.6 ± 4.2 gaussian Globe 5 0.628 ± 0.025 0.899 ± 0.034 0.488 ± 0.035 0.608 ± 0.013 0.875 ± 0.049 0.471 ± 0.019 20.8 ± 1.7 0.297 ± 0.032 307.00 ± 39.86 19.4 ± 2.3 gaussian Das 5 0.681 ± 0.024 0.620 ± 0.024 0.760 ± 0.036 0.537 ± 0.034 0.489 ± 0.032 0.598 ± 0.040 30.2 ± 1.7 0.253 ± 0.044 71.17 ± 1.26 43.4 ± 4.0 gaussian Score 5 0.688 ± 0.020 0.556 ± 0.023 0.905 ± 0.018 0.530 ± 0.032 0.429 ± 0.030 0.696 ± 0.034 35.8 ± 1.8 0.253 ± 0.044 50.52 ± 0.79 57.4 ± 4.0 gaussian NoGAM 5 0.706 ± 0.022 0.571 ± 0.028 0.928 ± 0.014 0.543 ± 0.045 0.441 ± 0.043 0.711 ± 0.045 34.4 ± 2.8 0.249 ± 0.053 224.79 ± 8.44 57.4 ± 4.0 gaussian Avici 5 0.690 ± 0.015 0.974 ± 0.016 0.535 ± 0.017 0.573 ± 0.024 0.810 ± 0.037 0.444 ± 0.020 20.2 ± 1.9 0.268 ± 0.048 14.83 ± 1.17 19.4 ± 1.7 gaussian NOTEARS 5 0.733 ± 0.021 0.712 ± 0.029 0.772 ± 0.053 0.613 ± 0.039 0.600 ± 0.052 0.640 ± 0.044 24.2 ± 2.8 0.200 ± 0.038 647.20 ± 121.11 39.2 ± 5.6 gaussian DAGMA 5 0.686 ± 0.019 0.699 ± 0.026 0.684 ± 0.041 0.357 ± 0.013 0.368 ± 0.031 0.354 ± 0.011 33.6 ± 3.1 0.496 ± 0.017 1684.55 ± 129.31 35.0 ± 4.2 gaussian DiffAN 5 0.729 ± 0.051 0.674 ± 0.064 0.802 ± 0.043 0.525 ± 0.080 0.488 ± 0.086 0.573 ± 0.074 28.4 ± 4.3 0.386 ± 0.066 26.07 ± 1.22 42.6 ± 3.6 gaussian SciNO 5 0.771 ± 0.017 0.708 ± 0.012 0.849 ± 0.035 0.564 ± 0.044 0.520 ± 0.045 0.619 ± 0.047 25.8 ± 2.9 0.309 ± 0.050 818.37 ± 28.20 42.6 ± 4.5 gaussian Caps 5 0.694 ± 0.029 0.697 ± 0.044 0.699 ± 0.039 0.526 ± 0.042 0.532 ± 0.058 0.528 ± 0.037 28.0 ± 3.9 0.356 ± 0.033 9.19 ± 0.45 36.2 ± 4.5 gaussian PC 5 0.763 ± 0.032 0.979 ± 0.021 0.629 ± 0.041 0.538 ± 0.019 0.695 ± 0.042 0.442 ± 0.018 20.0 ± 1.1 0.367 ± 0.040 14864.24 ± 4327.91 22.4 ± 1.3 gaussian FCI 5 0.518 ± 0.032 1.000 0.352 ± 0.029 0.433 ± 0.030 0.837 ± 0.036 0.294 ± 0.025 25.0 ± 2.5 0.380 ± 0.044 17462.83 ± 5051.92 12.2 ± 0.8 multimodal Spade-Hist 5 0.702 ± 0.040 0.992 ± 0.008 0.549 ± 0.048 0.693 ± 0.040 0.979 ± 0.013 0.542 ± 0.048 16.4 ± 2.3 0.109 ± 0.019 0.74 ± 0.05 19.4 ± 1.9 multimodal Spade-Gauss 5 0.687 ± 0.051 0.884 ± 0.039 0.567 ± 0.059 0.584 ± 0.077 0.749 ± 0.075 0.485 ± 0.079 20.8 ± 3.9 0.210 ± 0.066 0.15 ± 0.02 22.4 ± 2.1 multimodal Topic 5 0.679 ± 0.046 0.761 ± 0.030 0.616 ± 0.056 0.574 ± 0.063 0.642 ± 0.056 0.522 ± 0.070 23.4 ± 3.3 0.224 ± 0.069 28.94 ± 5.91 28.2 ± 2.1 multimodal Cam 5 0.733 ± 0.018 0.615 ± 0.018 0.909 ± 0.027 0.624 ± 0.044 0.524 ± 0.040 0.772 ± 0.052 27.8 ± 2.6 0.180 ± 0.058 325.55 ± 3.28 52.0 ± 3.8 multimodal CauScale 5 0.638 ± 0.047 0.881 ± 0.031 0.506 ± 0.050 0.479 ± 0.059 0.654 ± 0.051 0.382 ± 0.055 24.6 ± 3.7 0.316 ± 0.066 6.15 ± 0.17 19.8 ± 1.2 multimodal SDCD 5 0.683 ± 0.030 0.885 ± 0.034 0.560 ± 0.036 0.536 ± 0.046 0.694 ± 0.059 0.440 ± 0.043 22.8 ± 3.4 0.355 ± 0.033 138.19 ± 2.11 22.4 ± 2.1 multimodal Globe 5 0.620 ± 0.044 0.883 ± 0.023 0.483 ± 0.047 0.598 ± 0.049 0.851 ± 0.036 0.466 ± 0.050 21.4 ± 3.2 0.293 ± 0.058 354.19 ± 31.65 18.8 ± 1.0 multimodal Das 5 0.678 ± 0.011 0.642 ± 0.021 0.723 ± 0.021 0.595 ± 0.015 0.565 ± 0.027 0.634 ± 0.012 27.2 ± 2.1 0.151 ± 0.020 70.01 ± 0.60 40.0 ± 3.9 multimodal Score 5 0.721 ± 0.011 0.608 ± 0.012 0.887 ± 0.026 0.628 ± 0.010 0.531 ± 0.016 0.773 ± 0.011 28.2 ± 2.3 0.151 ± 0.020 50.56 ± 2.27 51.6 ± 4.8 multimodal NoGAM 5 0.726 ± 0.014 0.615 ± 0.016 0.887 ± 0.026 0.658 ± 0.028 0.557 ± 0.025 0.804 ± 0.040 26.4 ± 2.0 0.119 ± 0.038 213.92 ± 4.06 51.2 ± 5.0 multimodal Avici 5 0.509 ± 0.010 0.890 ± 0.021 0.357 ± 0.010 0.356 ± 0.029 0.621 ± 0.045 0.250 ± 0.022 27.8 ± 1.3 0.450 ± 0.048 12.79 ± 0.29 14.2 ± 1.3 multimodal NOTEARS 5 0.649 ± 0.029 0.668 ± 0.010 0.636 ± 0.049 0.460 ± 0.035 0.472 ± 0.022 0.453 ± 0.048 30.0 ± 1.1 0.393 ± 0.047 520.35 ± 132.05 34.0 ± 4.6 multimodal DAGMA 5 0.634 ± 0.046 0.771 ± 0.042 0.549 ± 0.055 0.431 ± 0.060 0.519 ± 0.055 0.376 ± 0.061 28.6 ± 4.3 0.458 ± 0.063 1498.77 ± 73.36 25.6 ± 3.8 multimodal DiffAN 5 0.757 ± 0.031 0.737 ± 0.028 0.781 ± 0.042 0.560 ± 0.040 0.543 ± 0.031 0.579 ± 0.051 24.6 ± 2.7 0.316 ± 0.052 25.01 ± 0.48 37.6 ± 4.0 multimodal SciNO 5 0.750 ± 0.036 0.726 ± 0.051 0.785 ± 0.028 0.515 ± 0.057 0.500 ± 0.062 0.535 ± 0.051 27.2 ± 3.9 0.319 ± 0.062 814.09 ± 36.97 38.8 ± 3.9 multimodal Caps 5 0.784 ± 0.020 0.816 ± 0.030 0.761 ± 0.035 0.686 ± 0.016 0.713 ± 0.026 0.665 ± 0.028 18.2 ± 2.1 0.161 ± 0.035 9.52 ± 0.16 33.6 ± 4.3 multimodal PC 5 0.851 ± 0.020 0.957 ± 0.015 0.770 ± 0.037 0.524 ± 0.056 0.587 ± 0.055 0.476 ± 0.057 20.0 ± 2.6 0.361 ± 0.059 14873.63 ± 4140.83 28.2 ± 1.9 multimodal FCI 5 0.464 ± 0.044 1.000 0.306 ± 0.035 0.314 ± 0.039 0.673 ± 0.037 0.208 ± 0.030 27.8 ± 1.9 0.435 ± 0.036 21920.09 ± 7825.36 10.8 ± 1.5 unimodal_asymmetric Spade-Hist 5 0.635 ± 0.051 0.955 ± 0.013 0.485 ± 0.062 0.619 ± 0.057 0.928 ± 0.027 0.474 ± 0.066 19.6 ± 2.7 0.228 ± 0.030 0.64 ± 0.03 17.6 ± 1.8 unimodal_asymmetric Spade-Gauss 5 0.703 ± 0.050 0.868 ± 0.067 0.597 ± 0.051 0.539 ± 0.067 0.666 ± 0.081 0.457 ± 0.063 23.0 ± 4.1 0.318 ± 0.066 0.13 ± 0.01 24.2 ± 1.6 unimodal_asymmetric Topic 5 0.685 ± 0.037 0.739 ± 0.047 0.647 ± 0.051 0.501 ± 0.038 0.544 ± 0.052 0.472 ± 0.041 27.0 ± 3.2 0.334 ± 0.033 29.42 ± 2.89 30.6 ± 1.6 unimodal_asymmetric Cam 5 0.621 ± 0.036 0.473 ± 0.040 0.920 ± 0.019 0.449 ± 0.050 0.343 ± 0.048 0.660 ± 0.044 50.6 ± 7.8 0.302 ± 0.038 327.68 ± 4.86 70.6 ± 7.6 unimodal_asymmetric CauScale 5 0.623 ± 0.050 0.827 ± 0.042 0.510 ± 0.063 0.420 ± 0.052 0.553 ± 0.045 0.345 ± 0.058 27.2 ± 3.2 0.348 ± 0.043 6.15 ± 0.09 21.2 ± 1.2 unimodal_asymmetric SDCD 5 0.692 ± 0.028 0.735 ± 0.028 0.655 ± 0.033 0.504 ± 0.076 0.533 ± 0.075 0.479 ± 0.078 27.4 ± 4.9 0.356 ± 0.066 131.65 ± 3.32 31.2 ± 1.8 unimodal_asymmetric Globe 5 0.568 ± 0.043 0.877 ± 0.023 0.431 ± 0.056 0.472 ± 0.053 0.723 ± 0.049 0.360 ± 0.056 25.0 ± 2.6 0.427 ± 0.050 263.89 ± 25.21 17.0 ± 1.7 unimodal_asymmetric Das 5 0.593 ± 0.023 0.530 ± 0.041 0.689 ± 0.022 0.446 ± 0.046 0.402 ± 0.055 0.511 ± 0.037 40.0 ± 5.4 0.301 ± 0.063 65.80 ± 2.46 47.2 ± 5.8 unimodal_asymmetric Score 5 0.628 ± 0.028 0.472 ± 0.029 0.945 ± 0.011 0.446 ± 0.042 0.336 ± 0.035 0.669 ± 0.050 50.0 ± 6.4 0.301 ± 0.063 50.27 ± 0.59 71.8 ± 7.0 unimodal_asymmetric NoGAM 5 0.625 ± 0.031 0.470 ± 0.033 0.940 ± 0.009 0.416 ± 0.053 0.315 ± 0.044 0.621 ± 0.063 52.0 ± 7.6 0.349 ± 0.075 206.24 ± 2.33 72.0 ± 7.8 unimodal_asymmetric Avici 5 0.662 ± 0.055 0.955 ± 0.012 0.516 ± 0.063 0.488 ± 0.028 0.712 ± 0.032 0.378 ± 0.036 23.0 ± 2.8 0.392 ± 0.028 14.48 ± 1.00 18.4 ± 1.0 unimodal_asymmetric NOTEARS 5 0.697 ± 0.039 0.606 ± 0.049 0.828 ± 0.021 0.501 ± 0.049 0.437 ± 0.055 0.592 ± 0.037 34.6 ± 5.6 0.293 ± 0.027 479.89 ± 85.60 49.2 ± 4.9 unimodal_asymmetric DAGMA 5 0.678 ± 0.033 0.645 ± 0.037 0.720 ± 0.039 0.345 ± 0.039 0.325 ± 0.033 0.369 ± 0.049 36.8 ± 4.1 0.516 ± 0.046 1581.52 ± 96.53 39.2 ± 2.1 unimodal_asymmetric DiffAN 5 0.714 ± 0.034 0.606 ± 0.045 0.880 ± 0.022 0.446 ± 0.063 0.382 ± 0.066 0.542 ± 0.053 37.4 ± 6.0 0.390 ± 0.068 24.69 ± 0.38 52.0 ± 4.6 unimodal_asymmetric SciNO 5 0.678 ± 0.019 0.576 ± 0.027 0.829 ± 0.028 0.423 ± 0.064 0.364 ± 0.065 0.511 ± 0.064 38.8 ± 4.2 0.388 ± 0.095 725.48 ± 3.46 51.0 ± 4.5 unimodal_asymmetric Caps 5 0.658 ± 0.025 0.609 ± 0.047 0.729 ± 0.022 0.460 ± 0.043 0.427 ± 0.050 0.507 ± 0.038 35.4 ± 5.3 0.343 ± 0.058 10.17 ± 0.51 43.8 ± 5.7 unimodal_asymmetric PC 5 0.771 ± 0.040 0.991 ± 0.009 0.636 ± 0.051 0.601 ± 0.044 0.772 ± 0.031 0.497 ± 0.048 18.2 ± 2.9 0.300 ± 0.044 11914.48 ± 3237.65 22.2 ± 1.2 unimodal_asymmetric FCI 5 0.480 ± 0.058 0.985 ± 0.015 0.325 ± 0.050 0.407 ± 0.049 0.845 ± 0.053 0.275 ± 0.042 26.0 ± 3.0 0.368 ± 0.036 13629.05 ± 3441.95 11.4 ± 1.5 unimodal_symmetric Spade-Hist 5 0.654 ± 0.068 0.903 ± 0.043 0.520 ± 0.072 0.598 ± 0.071 0.823 ± 0.053 0.476 ± 0.072 20.8 ± 4.5 0.339 ± 0.049 0.63 ± 0.02 19.6 ± 1.5 unimodal_symmetric Spade-Gauss 5 0.699 ± 0.054 0.838 ± 0.037 0.611 ± 0.068 0.606 ± 0.045 0.730 ± 0.043 0.527 ± 0.053 21.0 ± 3.3 0.173 ± 0.011 0.15 ± 0.03 25.0 ± 1.4 unimodal_symmetric Topic 5 0.700 ± 0.049 0.767 ± 0.035 0.650 ± 0.063 0.574 ± 0.037 0.631 ± 0.034 0.532 ± 0.046 23.6 ± 3.6 0.247 ± 0.021 25.91 ± 4.04 29.2 ± 1.1 unimodal_symmetric Cam 5 0.621 ± 0.020 0.478 ± 0.018 0.887 ± 0.025 0.534 ± 0.023 0.412 ± 0.021 0.762 ± 0.027 43.0 ± 5.1 0.162 ± 0.017 339.00 ± 3.40 65.8 ± 6.0 unimodal_symmetric CauScale 5 0.577 ± 0.047 0.839 ± 0.066 0.454 ± 0.048 0.414 ± 0.047 0.603 ± 0.066 0.326 ± 0.044 27.4 ± 2.7 0.380 ± 0.042 5.86 ± 0.06 19.0 ± 1.8 unimodal_symmetric SDCD 5 0.662 ± 0.041 0.802 ± 0.059 0.577 ± 0.053 0.494 ± 0.049 0.608 ± 0.083 0.426 ± 0.040 26.4 ± 4.5 0.414 ± 0.036 127.84 ± 4.28 26.2 ± 3.9 unimodal_symmetric Globe 5 0.523 ± 0.053 0.946 ± 0.033 0.377 ± 0.058 0.460 ± 0.051 0.829 ± 0.037 0.331 ± 0.053 25.0 ± 3.0 0.422 ± 0.056 237.24 ± 23.71 13.8 ± 2.1 unimodal_symmetric Das 5 0.602 ± 0.011 0.531 ± 0.008 0.702 ± 0.040 0.429 ± 0.028 0.381 ± 0.032 0.497 ± 0.025 39.8 ± 3.4 0.293 ± 0.048 68.18 ± 1.29 46.6 ± 4.0 unimodal_symmetric Score 5 0.625 ± 0.011 0.480 ± 0.011 0.899 ± 0.022 0.441 ± 0.030 0.339 ± 0.026 0.632 ± 0.034 47.6 ± 4.7 0.293 ± 0.048 52.07 ± 2.11 66.2 ± 5.1 unimodal_symmetric NoGAM 5 0.629 ± 0.015 0.486 ± 0.011 0.894 ± 0.025 0.472 ± 0.027 0.365 ± 0.023 0.669 ± 0.031 45.4 ± 4.7 0.256 ± 0.048 195.00 ± 4.21 65.0 ± 5.0 unimodal_symmetric Avici 5 0.701 ± 0.040 0.972 ± 0.019 0.558 ± 0.051 0.554 ± 0.024 0.772 ± 0.036 0.439 ± 0.034 20.6 ± 2.1 0.351 ± 0.029 13.21 ± 0.28 19.8 ± 1.2 unimodal_symmetric NOTEARS 5 0.664 ± 0.022 0.626 ± 0.030 0.710 ± 0.021 0.494 ± 0.039 0.468 ± 0.045 0.526 ± 0.033 32.2 ± 4.3 0.327 ± 0.032 584.97 ± 170.60 40.6 ± 4.7 unimodal_symmetric DAGMA 5 0.659 ± 0.043 0.703 ± 0.042 0.623 ± 0.049 0.378 ± 0.020 0.404 ± 0.023 0.356 ± 0.021 32.2 ± 3.7 0.487 ± 0.029 1692.48 ± 112.77 31.0 ± 2.3 unimodal_symmetric DiffAN 5 0.706 ± 0.038 0.621 ± 0.032 0.819 ± 0.046 0.406 ± 0.040 0.357 ± 0.035 0.471 ± 0.048 36.6 ± 4.9 0.452 ± 0.050 25.82 ± 0.96 46.4 ± 3.2 unimodal_symmetric SciNO 5 0.698 ± 0.039 0.599 ± 0.033 0.838 ± 0.054 0.441 ± 0.049 0.381 ± 0.046 0.526 ± 0.052 37.2 ± 6.2 0.370 ± 0.061 871.01 ± 83.95 49.4 ± 4.1 unimodal_symmetric Caps 5 0.666 ± 0.028 0.635 ± 0.042 0.711 ± 0.039 0.469 ± 0.026 0.449 ± 0.038 0.499 ± 0.025 33.2 ± 4.3 0.358 ± 0.044 9.19 ± 0.29 40.2 ± 4.4 unimodal_symmetric PC 5 0.711 ± 0.037 0.961 ± 0.011 0.570 ± 0.050 0.546 ± 0.054 0.735 ± 0.056 0.439 ± 0.055 20.8 ± 3.1 0.376 ± 0.064 12782.07 ± 3231.36 20.8 ± 2.0 unimodal_symmetric FCI 5 0.415 ± 0.067 0.967 ± 0.033 0.272 ± 0.054 0.389 ± 0.082 0.876 ± 0.076 0.257 ± 0.063 26.6 ± 3.6 0.333 ± 0.057 13919.71 ± 3797.36 9.6 ± 1.7 Table 35: Qualitative comparison without marginalization (n=2000n=2000, d=50d=50, degree 2, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.908 ± 0.016 0.982 ± 0.014 0.846 ± 0.024 0.891 ± 0.020 0.963 ± 0.021 0.830 ± 0.025 9.0 ± 2.2 0.055 ± 0.011 1.83 ± 0.05 39.8 ± 2.3 gaussian Spade-Gauss 5 0.900 ± 0.016 0.975 ± 0.013 0.837 ± 0.023 0.895 ± 0.016 0.969 ± 0.015 0.832 ± 0.022 9.0 ± 1.8 0.030 ± 0.009 0.51 ± 0.02 39.6 ± 2.2 gaussian Topic 5 0.733 ± 0.025 0.656 ± 0.034 0.837 ± 0.026 0.725 ± 0.025 0.648 ± 0.033 0.828 ± 0.025 28.6 ± 3.0 0.039 ± 0.011 129.93 ± 6.79 59.0 ± 2.3 gaussian Cam 5 0.546 ± 0.030 0.385 ± 0.031 0.957 ± 0.005 0.544 ± 0.032 0.383 ± 0.032 0.952 ± 0.009 73.4 ± 4.4 0.005 ± 0.005 13046.86 ± 25.11 115.6 ± 2.4 gaussian CauScale 5 0.852 ± 0.016 0.890 ± 0.015 0.820 ± 0.026 0.591 ± 0.027 0.616 ± 0.023 0.569 ± 0.033 24.8 ± 2.7 0.355 ± 0.039 6.01 ± 0.17 42.4 ± 1.9 gaussian SDCD 5 0.767 ± 0.013 0.827 ± 0.016 0.717 ± 0.023 0.570 ± 0.027 0.616 ± 0.037 0.532 ± 0.024 28.8 ± 2.7 0.322 ± 0.033 137.50 ± 3.78 40.0 ± 2.2 gaussian Globe 5 0.727 ± 0.086 0.925 ± 0.015 0.633 ± 0.105 0.707 ± 0.082 0.902 ± 0.022 0.615 ± 0.101 21.6 ± 6.2 0.156 ± 0.061 1130.82 ± 116.01 30.4 ± 4.2 gaussian Das 5 0.610 ± 0.023 0.532 ± 0.028 0.718 ± 0.014 0.421 ± 0.044 0.367 ± 0.043 0.493 ± 0.047 52.6 ± 4.0 0.351 ± 0.043 213.20 ± 2.67 62.6 ± 3.5 gaussian Score 5 0.542 ± 0.007 0.386 ± 0.007 0.911 ± 0.007 0.377 ± 0.025 0.269 ± 0.018 0.633 ± 0.040 84.0 ± 5.0 0.351 ± 0.043 217.21 ± 7.35 109.2 ± 6.3 gaussian NoGAM 5 0.545 ± 0.009 0.388 ± 0.009 0.916 ± 0.013 0.394 ± 0.009 0.280 ± 0.007 0.663 ± 0.018 82.4 ± 4.4 0.314 ± 0.014 1305.33 ± 18.49 109.0 ± 5.2 gaussian Avici 5 0.669 ± 0.015 0.937 ± 0.024 0.523 ± 0.023 0.647 ± 0.018 0.907 ± 0.029 0.506 ± 0.023 24.8 ± 2.3 0.252 ± 0.023 25.00 ± 0.43 26.0 ± 2.3 gaussian NOTEARS 5 0.754 ± 0.023 0.739 ± 0.040 0.780 ± 0.037 0.528 ± 0.020 0.518 ± 0.034 0.545 ± 0.025 34.8 ± 4.1 0.309 ± 0.016 2459.16 ± 208.31 49.2 ± 3.9 gaussian DAGMA 5 0.736 ± 0.020 0.739 ± 0.017 0.739 ± 0.039 0.452 ± 0.036 0.458 ± 0.048 0.449 ± 0.028 37.4 ± 2.1 0.419 ± 0.042 4355.69 ± 214.32 45.8 ± 1.6 gaussian DiffAN 5 0.758 ± 0.011 0.672 ± 0.012 0.870 ± 0.018 0.463 ± 0.021 0.410 ± 0.015 0.533 ± 0.031 41.6 ± 3.9 0.430 ± 0.025 95.30 ± 0.99 60.0 ± 4.1 gaussian SciNO 5 0.742 ± 0.013 0.647 ± 0.019 0.872 ± 0.009 0.437 ± 0.019 0.381 ± 0.020 0.513 ± 0.019 45.0 ± 4.7 0.420 ± 0.026 921.67 ± 31.59 62.8 ± 5.6 gaussian Caps 5 0.793 ± 0.022 0.771 ± 0.028 0.818 ± 0.017 0.497 ± 0.040 0.483 ± 0.042 0.512 ± 0.038 34.4 ± 4.6 0.400 ± 0.051 46.55 ± 1.81 49.6 ± 4.7 gaussian PC 5 0.665 ± 0.026 0.910 ± 0.019 0.525 ± 0.029 0.515 ± 0.025 0.706 ± 0.023 0.407 ± 0.026 29.6 ± 1.4 0.339 ± 0.048 17152.67 ± 2172.64 27.0 ± 3.2 gaussian FCI 5 0.450 ± 0.042 0.969 ± 0.019 0.297 ± 0.036 0.389 ± 0.050 0.833 ± 0.060 0.257 ± 0.040 35.0 ± 3.3 0.358 ± 0.050 16981.89 ± 1529.95 14.0 ± 1.4 multimodal Spade-Hist 5 0.946 ± 0.009 0.995 ± 0.005 0.901 ± 0.014 0.946 ± 0.009 0.995 ± 0.005 0.901 ± 0.014 4.8 ± 0.8 0.013 ± 0.005 1.86 ± 0.07 42.0 ± 3.2 multimodal Spade-Gauss 5 0.881 ± 0.018 0.953 ± 0.019 0.822 ± 0.027 0.770 ± 0.029 0.834 ± 0.040 0.717 ± 0.025 15.4 ± 2.9 0.160 ± 0.018 0.48 ± 0.01 39.8 ± 2.3 multimodal Topic 5 0.739 ± 0.013 0.668 ± 0.016 0.830 ± 0.028 0.632 ± 0.017 0.572 ± 0.019 0.710 ± 0.027 32.8 ± 3.2 0.162 ± 0.018 125.17 ± 6.00 57.2 ± 2.7 multimodal Cam 5 0.601 ± 0.021 0.437 ± 0.022 0.966 ± 0.004 0.535 ± 0.015 0.389 ± 0.016 0.862 ± 0.020 64.0 ± 2.9 0.109 ± 0.020 13162.20 ± 43.25 102.4 ± 5.1 multimodal CauScale 5 0.872 ± 0.017 0.925 ± 0.016 0.826 ± 0.025 0.762 ± 0.029 0.809 ± 0.033 0.722 ± 0.030 16.4 ± 2.8 0.145 ± 0.022 6.01 ± 0.05 41.2 ± 2.2 multimodal SDCD 5 0.773 ± 0.012 0.801 ± 0.005 0.749 ± 0.025 0.516 ± 0.035 0.533 ± 0.027 0.501 ± 0.042 32.0 ± 3.3 0.342 ± 0.038 128.51 ± 3.35 43.2 ± 3.0 multimodal Globe 5 0.770 ± 0.038 0.886 ± 0.025 0.688 ± 0.053 0.737 ± 0.042 0.847 ± 0.027 0.659 ± 0.057 20.6 ± 4.0 0.127 ± 0.034 1455.12 ± 164.80 36.0 ± 3.7 multimodal Das 5 0.633 ± 0.025 0.594 ± 0.043 0.694 ± 0.035 0.469 ± 0.028 0.437 ± 0.028 0.517 ± 0.041 45.2 ± 2.9 0.237 ± 0.017 213.81 ± 3.41 55.4 ± 6.0 multimodal Score 5 0.604 ± 0.014 0.443 ± 0.015 0.953 ± 0.015 0.470 ± 0.016 0.345 ± 0.015 0.741 ± 0.018 67.2 ± 2.8 0.237 ± 0.017 205.09 ± 6.62 99.4 ± 4.7 multimodal NoGAM 5 0.618 ± 0.014 0.458 ± 0.016 0.953 ± 0.015 0.519 ± 0.016 0.385 ± 0.017 0.799 ± 0.008 61.6 ± 4.1 0.180 ± 0.007 1275.40 ± 38.89 96.4 ± 6.0 multimodal Avici 5 0.531 ± 0.045 0.919 ± 0.012 0.380 ± 0.045 0.357 ± 0.046 0.621 ± 0.059 0.256 ± 0.041 36.2 ± 3.4 0.462 ± 0.037 24.35 ± 0.41 19.0 ± 2.2 multimodal NOTEARS 5 0.741 ± 0.020 0.687 ± 0.034 0.812 ± 0.029 0.453 ± 0.042 0.423 ± 0.050 0.491 ± 0.032 41.4 ± 5.2 0.415 ± 0.045 2583.63 ± 255.97 55.2 ± 4.8 multimodal DAGMA 5 0.757 ± 0.013 0.759 ± 0.014 0.759 ± 0.030 0.487 ± 0.027 0.488 ± 0.031 0.488 ± 0.030 35.0 ± 2.4 0.352 ± 0.055 4280.14 ± 251.00 46.2 ± 3.1 multimodal DiffAN 5 0.815 ± 0.016 0.741 ± 0.022 0.909 ± 0.015 0.563 ± 0.023 0.511 ± 0.019 0.629 ± 0.033 31.8 ± 1.1 0.316 ± 0.036 95.32 ± 0.87 57.0 ± 4.2 multimodal SciNO 5 0.727 ± 0.022 0.617 ± 0.028 0.889 ± 0.030 0.379 ± 0.048 0.324 ± 0.048 0.459 ± 0.048 51.6 ± 7.3 0.489 ± 0.046 852.90 ± 34.83 67.6 ± 7.3 multimodal Caps 5 0.788 ± 0.021 0.773 ± 0.022 0.807 ± 0.035 0.499 ± 0.035 0.487 ± 0.026 0.513 ± 0.044 33.6 ± 2.2 0.288 ± 0.014 45.35 ± 1.47 48.6 ± 4.5 multimodal PC 5 0.586 ± 0.057 0.935 ± 0.026 0.442 ± 0.062 0.520 ± 0.046 0.837 ± 0.043 0.390 ± 0.049 29.8 ± 2.3 0.295 ± 0.016 16019.19 ± 2254.56 22.0 ± 3.5 multimodal FCI 5 0.462 ± 0.020 1.000 0.301 ± 0.017 0.433 ± 0.017 0.939 ± 0.029 0.282 ± 0.014 33.2 ± 2.1 0.337 ± 0.036 17484.12 ± 2609.01 14.0 ± 1.4 unimodal_asymmetric Spade-Hist 5 0.933 ± 0.011 0.995 ± 0.005 0.880 ± 0.019 0.923 ± 0.008 0.984 ± 0.006 0.870 ± 0.016 6.4 ± 1.2 0.040 ± 0.018 1.87 ± 0.06 40.8 ± 2.3 unimodal_asymmetric Spade-Gauss 5 0.866 ± 0.016 0.903 ± 0.016 0.836 ± 0.030 0.740 ± 0.035 0.771 ± 0.034 0.714 ± 0.041 17.6 ± 2.3 0.170 ± 0.021 0.50 ± 0.02 42.8 ± 2.8 unimodal_asymmetric Topic 5 0.712 ± 0.021 0.615 ± 0.026 0.849 ± 0.027 0.582 ± 0.036 0.504 ± 0.038 0.694 ± 0.038 38.6 ± 2.8 0.197 ± 0.020 128.15 ± 6.65 63.6 ± 1.8 unimodal_asymmetric Cam 5 0.471 ± 0.013 0.315 ± 0.012 0.941 ± 0.019 0.400 ± 0.018 0.267 ± 0.014 0.799 ± 0.034 104.2 ± 6.1 0.165 ± 0.031 13247.82 ± 64.63 138.0 ± 6.3 unimodal_asymmetric CauScale 5 0.850 ± 0.019 0.848 ± 0.030 0.858 ± 0.028 0.583 ± 0.035 0.580 ± 0.036 0.589 ± 0.042 26.2 ± 2.3 0.329 ± 0.029 5.99 ± 0.15 46.6 ± 1.6 unimodal_asymmetric SDCD 5 0.830 ± 0.024 0.850 ± 0.024 0.815 ± 0.033 0.622 ± 0.026 0.637 ± 0.032 0.609 ± 0.025 25.0 ± 2.9 0.283 ± 0.026 134.75 ± 3.09 44.8 ± 4.2 unimodal_asymmetric Globe 5 0.733 ± 0.029 0.888 ± 0.019 0.633 ± 0.049 0.651 ± 0.030 0.789 ± 0.030 0.562 ± 0.047 24.6 ± 3.1 0.250 ± 0.053 1109.54 ± 77.40 33.2 ± 3.9 unimodal_asymmetric Das 5 0.577 ± 0.022 0.484 ± 0.018 0.716 ± 0.031 0.396 ± 0.032 0.332 ± 0.026 0.492 ± 0.044 58.6 ± 3.6 0.384 ± 0.042 218.21 ± 2.72 68.4 ± 4.1 unimodal_asymmetric Score 5 0.512 ± 0.013 0.353 ± 0.012 0.936 ± 0.015 0.327 ± 0.018 0.225 ± 0.012 0.600 ± 0.037 98.6 ± 7.3 0.384 ± 0.042 214.96 ± 2.15 123.4 ± 9.1 unimodal_asymmetric NoGAM 5 0.500 ± 0.015 0.342 ± 0.013 0.936 ± 0.015 0.328 ± 0.010 0.224 ± 0.007 0.615 ± 0.026 102.4 ± 8.6 0.366 ± 0.023 1339.92 ± 19.68 127.6 ± 9.6 unimodal_asymmetric Avici 5 0.603 ± 0.064 0.924 ± 0.027 0.459 ± 0.071 0.465 ± 0.072 0.698 ± 0.051 0.357 ± 0.072 31.8 ± 4.2 0.410 ± 0.054 24.47 ± 0.40 22.6 ± 2.7 unimodal_asymmetric NOTEARS 5 0.750 ± 0.035 0.671 ± 0.044 0.856 ± 0.031 0.510 ± 0.022 0.456 ± 0.028 0.582 ± 0.022 39.6 ± 4.9 0.338 ± 0.021 2102.01 ± 225.78 59.4 ± 3.8 unimodal_asymmetric DAGMA 5 0.722 ± 0.032 0.681 ± 0.051 0.782 ± 0.029 0.335 ± 0.037 0.320 ± 0.044 0.359 ± 0.029 47.0 ± 3.5 0.552 ± 0.024 4338.63 ± 301.93 53.4 ± 2.4 unimodal_asymmetric DiffAN 5 0.698 ± 0.014 0.574 ± 0.016 0.892 ± 0.023 0.455 ± 0.009 0.375 ± 0.012 0.581 ± 0.008 50.2 ± 3.8 0.380 ± 0.006 95.56 ± 1.03 72.0 ± 4.8 unimodal_asymmetric SciNO 5 0.707 ± 0.016 0.576 ± 0.019 0.918 ± 0.014 0.384 ± 0.036 0.313 ± 0.033 0.495 ± 0.038 55.0 ± 5.7 0.452 ± 0.040 959.22 ± 100.45 74.0 ± 5.4 unimodal_asymmetric Caps 5 0.752 ± 0.018 0.698 ± 0.024 0.818 ± 0.018 0.501 ± 0.038 0.466 ± 0.042 0.542 ± 0.035 37.2 ± 2.1 0.409 ± 0.031 48.84 ± 1.82 54.0 ± 1.9 unimodal_asymmetric PC 5 0.696 ± 0.028 0.885 ± 0.036 0.583 ± 0.044 0.600 ± 0.016 0.768 ± 0.050 0.500 ± 0.026 26.8 ± 1.2 0.279 ± 0.006 18473.17 ± 2531.44 30.6 ± 2.9 unimodal_asymmetric FCI 5 0.495 ± 0.024 1.000 0.330 ± 0.021 0.484 ± 0.019 0.979 ± 0.013 0.322 ± 0.018 31.4 ± 2.4 0.289 ± 0.044 18841.39 ± 2704.95 15.4 ± 1.7 unimodal_symmetric Spade-Hist 5 0.912 ± 0.015 0.986 ± 0.010 0.850 ± 0.023 0.908 ± 0.018 0.982 ± 0.011 0.847 ± 0.026 8.0 ± 2.0 0.035 ± 0.018 1.84 ± 0.04 39.8 ± 2.3 unimodal_symmetric Spade-Gauss 5 0.879 ± 0.014 0.943 ± 0.013 0.824 ± 0.023 0.843 ± 0.024 0.904 ± 0.020 0.792 ± 0.031 12.4 ± 2.5 0.057 ± 0.021 0.52 ± 0.02 40.4 ± 2.4 unimodal_symmetric Topic 5 0.743 ± 0.015 0.661 ± 0.027 0.854 ± 0.018 0.706 ± 0.022 0.628 ± 0.030 0.811 ± 0.021 29.2 ± 2.2 0.074 ± 0.026 126.60 ± 4.88 59.6 ± 1.8 unimodal_symmetric Cam 5 0.490 ± 0.017 0.336 ± 0.017 0.923 ± 0.018 0.449 ± 0.014 0.307 ± 0.014 0.848 ± 0.029 92.2 ± 5.1 0.087 ± 0.018 13217.24 ± 99.27 127.4 ± 5.7 unimodal_symmetric CauScale 5 0.842 ± 0.014 0.882 ± 0.016 0.808 ± 0.023 0.555 ± 0.021 0.581 ± 0.023 0.532 ± 0.023 26.8 ± 2.3 0.356 ± 0.027 6.02 ± 0.14 42.2 ± 1.9 unimodal_symmetric SDCD 5 0.713 ± 0.018 0.771 ± 0.046 0.673 ± 0.026 0.501 ± 0.044 0.544 ± 0.063 0.470 ± 0.036 35.2 ± 5.2 0.326 ± 0.039 138.32 ± 0.91 41.2 ± 4.3 unimodal_symmetric Globe 5 0.767 ± 0.030 0.846 ± 0.028 0.711 ± 0.047 0.763 ± 0.034 0.840 ± 0.027 0.708 ± 0.050 20.2 ± 3.6 0.081 ± 0.037 1282.78 ± 46.94 38.4 ± 1.4 unimodal_symmetric Das 5 0.515 ± 0.024 0.448 ± 0.030 0.616 ± 0.031 0.388 ± 0.036 0.337 ± 0.035 0.465 ± 0.045 60.6 ± 4.3 0.356 ± 0.045 212.44 ± 1.72 64.6 ± 6.0 unimodal_symmetric Score 5 0.493 ± 0.019 0.344 ± 0.018 0.878 ± 0.025 0.339 ± 0.022 0.236 ± 0.017 0.606 ± 0.045 97.2 ± 9.8 0.356 ± 0.045 211.81 ± 8.68 119.6 ± 11.2 unimodal_symmetric NoGAM 5 0.503 ± 0.021 0.354 ± 0.021 0.878 ± 0.021 0.354 ± 0.023 0.249 ± 0.016 0.623 ± 0.052 93.4 ± 9.9 0.339 ± 0.050 1294.49 ± 18.49 116.6 ± 11.5 unimodal_symmetric Avici 5 0.741 ± 0.029 0.947 ± 0.023 0.612 ± 0.036 0.687 ± 0.047 0.878 ± 0.048 0.568 ± 0.047 22.0 ± 3.9 0.240 ± 0.044 25.17 ± 0.76 29.6 ± 1.4 unimodal_symmetric NOTEARS 5 0.730 ± 0.027 0.691 ± 0.047 0.789 ± 0.033 0.529 ± 0.039 0.502 ± 0.050 0.568 ± 0.034 37.0 ± 3.9 0.289 ± 0.019 2516.56 ± 386.29 53.0 ± 2.8 unimodal_symmetric DAGMA 5 0.727 ± 0.014 0.767 ± 0.022 0.692 ± 0.013 0.469 ± 0.041 0.495 ± 0.045 0.446 ± 0.038 35.4 ± 2.8 0.355 ± 0.046 4534.36 ± 287.35 42.0 ± 3.5 unimodal_symmetric DiffAN 5 0.690 ± 0.016 0.584 ± 0.022 0.847 ± 0.020 0.406 ± 0.036 0.342 ± 0.026 0.502 ± 0.055 51.2 ± 3.5 0.453 ± 0.053 96.92 ± 0.49 67.4 ± 4.9 unimodal_symmetric SciNO 5 0.684 ± 0.021 0.574 ± 0.029 0.852 ± 0.011 0.373 ± 0.026 0.313 ± 0.028 0.462 ± 0.022 55.6 ± 7.2 0.492 ± 0.029 822.13 ± 8.76 69.8 ± 6.8 unimodal_symmetric Caps 5 0.689 ± 0.024 0.652 ± 0.034 0.732 ± 0.012 0.449 ± 0.059 0.425 ± 0.058 0.478 ± 0.060 43.0 ± 6.0 0.392 ± 0.059 45.90 ± 2.80 52.6 ± 5.0 unimodal_symmetric PC 5 0.657 ± 0.031 0.806 ± 0.040 0.556 ± 0.027 0.545 ± 0.021 0.667 ± 0.018 0.461 ± 0.022 31.0 ± 1.8 0.287 ± 0.017 16775.22 ± 2073.66 32.0 ± 2.3 unimodal_symmetric FCI 5 0.465 ± 0.042 1.000 0.307 ± 0.038 0.419 ± 0.055 0.889 ± 0.051 0.278 ± 0.045 33.6 ± 3.5 0.339 ± 0.051 16505.65 ± 2009.57 14.2 ± 2.0 Table 36: Qualitative comparison without marginalization (n=2000n=2000, d=50d=50, degree 2, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.772 ± 0.028 0.939 ± 0.032 0.658 ± 0.029 0.685 ± 0.038 0.833 ± 0.047 0.583 ± 0.035 21.6 ± 3.0 0.238 ± 0.015 1.71 ± 0.04 32.2 ± 1.2 gaussian Spade-Gauss 5 0.829 ± 0.029 0.965 ± 0.017 0.729 ± 0.038 0.820 ± 0.035 0.954 ± 0.027 0.720 ± 0.042 14.4 ± 2.9 0.035 ± 0.013 0.48 ± 0.02 34.8 ± 2.2 gaussian Topic 5 0.687 ± 0.021 0.621 ± 0.029 0.776 ± 0.028 0.665 ± 0.029 0.600 ± 0.033 0.751 ± 0.039 33.6 ± 2.6 0.085 ± 0.032 122.11 ± 6.90 57.6 ± 2.0 gaussian Cam 5 0.529 ± 0.022 0.368 ± 0.023 0.959 ± 0.013 0.509 ± 0.021 0.354 ± 0.022 0.922 ± 0.015 80.4 ± 4.5 0.049 ± 0.012 13382.61 ± 218.57 121.0 ± 5.0 gaussian CauScale 5 0.806 ± 0.021 0.853 ± 0.025 0.766 ± 0.024 0.587 ± 0.029 0.621 ± 0.030 0.558 ± 0.030 26.8 ± 3.0 0.285 ± 0.029 6.17 ± 0.16 41.4 ± 2.2 gaussian SDCD 5 0.722 ± 0.027 0.811 ± 0.038 0.652 ± 0.026 0.552 ± 0.031 0.622 ± 0.046 0.497 ± 0.023 30.6 ± 3.5 0.321 ± 0.017 139.53 ± 0.83 37.2 ± 2.2 gaussian Globe 5 0.632 ± 0.025 0.887 ± 0.025 0.494 ± 0.029 0.579 ± 0.026 0.815 ± 0.039 0.452 ± 0.026 28.6 ± 3.2 0.346 ± 0.030 1016.66 ± 51.60 25.8 ± 2.0 gaussian Das 5 0.515 ± 0.025 0.473 ± 0.031 0.570 ± 0.026 0.379 ± 0.024 0.345 ± 0.017 0.423 ± 0.037 56.4 ± 2.8 0.346 ± 0.043 212.80 ± 3.00 56.2 ± 4.0 gaussian Score 5 0.545 ± 0.013 0.391 ± 0.015 0.910 ± 0.032 0.373 ± 0.017 0.267 ± 0.010 0.628 ± 0.050 83.0 ± 2.9 0.346 ± 0.043 205.20 ± 6.92 107.4 ± 4.6 gaussian NoGAM 5 0.562 ± 0.017 0.408 ± 0.021 0.914 ± 0.029 0.392 ± 0.022 0.285 ± 0.020 0.639 ± 0.040 78.0 ± 1.6 0.338 ± 0.036 1336.26 ± 9.01 103.4 ± 3.6 gaussian Avici 5 0.679 ± 0.024 0.943 ± 0.034 0.532 ± 0.024 0.652 ± 0.028 0.906 ± 0.044 0.510 ± 0.025 24.4 ± 2.8 0.286 ± 0.034 25.32 ± 0.41 26.2 ± 2.0 gaussian NOTEARS 5 0.733 ± 0.041 0.708 ± 0.074 0.775 ± 0.019 0.555 ± 0.061 0.542 ± 0.088 0.581 ± 0.035 35.4 ± 5.2 0.304 ± 0.048 2893.03 ± 703.13 52.2 ± 4.6 gaussian DAGMA 5 0.759 ± 0.011 0.808 ± 0.032 0.720 ± 0.018 0.485 ± 0.026 0.515 ± 0.031 0.461 ± 0.027 33.0 ± 1.8 0.424 ± 0.028 4230.79 ± 141.03 41.6 ± 3.5 gaussian DiffAN 5 0.750 ± 0.016 0.683 ± 0.024 0.835 ± 0.023 0.463 ± 0.027 0.422 ± 0.032 0.515 ± 0.025 40.2 ± 1.4 0.447 ± 0.038 95.67 ± 1.43 56.4 ± 2.3 gaussian SciNO 5 0.708 ± 0.013 0.602 ± 0.022 0.867 ± 0.023 0.400 ± 0.017 0.339 ± 0.015 0.490 ± 0.026 50.4 ± 2.9 0.471 ± 0.031 909.87 ± 41.56 66.8 ± 4.1 gaussian Caps 5 0.805 ± 0.014 0.795 ± 0.017 0.820 ± 0.030 0.529 ± 0.022 0.521 ± 0.010 0.541 ± 0.035 31.4 ± 3.0 0.377 ± 0.047 50.62 ± 1.22 47.6 ± 2.6 gaussian PC 5 0.770 ± 0.017 0.877 ± 0.028 0.691 ± 0.030 0.610 ± 0.017 0.696 ± 0.034 0.546 ± 0.019 25.4 ± 0.9 0.299 ± 0.033 18310.27 ± 1433.16 36.4 ± 2.3 gaussian FCI 5 0.493 ± 0.058 1.000 0.335 ± 0.055 0.446 ± 0.052 0.908 ± 0.035 0.303 ± 0.049 32.4 ± 3.7 0.355 ± 0.057 19017.50 ± 1690.51 15.4 ± 2.5 multimodal Spade-Hist 5 0.812 ± 0.028 0.983 ± 0.007 0.696 ± 0.043 0.802 ± 0.026 0.971 ± 0.008 0.687 ± 0.041 15.4 ± 2.6 0.099 ± 0.029 1.69 ± 0.05 32.6 ± 2.3 multimodal Spade-Gauss 5 0.811 ± 0.017 0.912 ± 0.016 0.733 ± 0.025 0.649 ± 0.046 0.729 ± 0.047 0.587 ± 0.046 22.4 ± 2.8 0.263 ± 0.041 0.47 ± 0.01 37.0 ± 1.7 multimodal Topic 5 0.696 ± 0.022 0.625 ± 0.028 0.787 ± 0.019 0.527 ± 0.041 0.473 ± 0.039 0.597 ± 0.045 40.2 ± 2.5 0.312 ± 0.038 117.06 ± 5.27 58.2 ± 2.3 multimodal Cam 5 0.571 ± 0.029 0.416 ± 0.033 0.931 ± 0.019 0.461 ± 0.026 0.335 ± 0.027 0.752 ± 0.035 73.0 ± 5.1 0.211 ± 0.031 13176.85 ± 175.48 104.4 ± 5.4 multimodal CauScale 5 0.825 ± 0.016 0.872 ± 0.025 0.787 ± 0.025 0.660 ± 0.022 0.697 ± 0.025 0.630 ± 0.027 22.8 ± 2.5 0.200 ± 0.019 5.96 ± 0.07 41.6 ± 1.7 multimodal SDCD 5 0.768 ± 0.018 0.857 ± 0.031 0.699 ± 0.026 0.532 ± 0.017 0.596 ± 0.035 0.483 ± 0.007 29.8 ± 3.1 0.391 ± 0.015 136.22 ± 3.08 38.0 ± 3.1 multimodal Globe 5 0.724 ± 0.030 0.915 ± 0.013 0.605 ± 0.044 0.687 ± 0.040 0.867 ± 0.033 0.574 ± 0.051 22.6 ± 3.2 0.276 ± 0.057 1275.00 ± 43.18 30.4 ± 2.3 multimodal Das 5 0.582 ± 0.022 0.539 ± 0.015 0.642 ± 0.051 0.443 ± 0.024 0.411 ± 0.018 0.489 ± 0.045 49.0 ± 2.8 0.297 ± 0.049 210.76 ± 4.86 55.4 ± 6.3 multimodal Score 5 0.583 ± 0.031 0.429 ± 0.032 0.921 ± 0.020 0.419 ± 0.042 0.309 ± 0.037 0.659 ± 0.047 73.4 ± 6.6 0.297 ± 0.049 202.71 ± 6.78 100.6 ± 7.1 multimodal NoGAM 5 0.588 ± 0.023 0.436 ± 0.024 0.913 ± 0.021 0.483 ± 0.020 0.358 ± 0.020 0.750 ± 0.025 67.4 ± 6.7 0.210 ± 0.008 1336.79 ± 19.20 97.8 ± 7.6 multimodal Avici 5 0.461 ± 0.065 0.894 ± 0.031 0.322 ± 0.056 0.319 ± 0.055 0.608 ± 0.032 0.224 ± 0.046 38.0 ± 3.9 0.476 ± 0.030 24.22 ± 0.35 16.6 ± 2.9 multimodal NOTEARS 5 0.809 ± 0.011 0.791 ± 0.021 0.833 ± 0.032 0.646 ± 0.021 0.632 ± 0.027 0.665 ± 0.032 25.6 ± 0.6 0.244 ± 0.010 2272.31 ± 403.72 48.8 ± 3.5 multimodal DAGMA 5 0.723 ± 0.034 0.775 ± 0.054 0.695 ± 0.045 0.494 ± 0.053 0.531 ± 0.062 0.473 ± 0.052 35.4 ± 6.0 0.379 ± 0.035 4408.91 ± 200.11 42.4 ± 5.2 multimodal DiffAN 5 0.789 ± 0.028 0.727 ± 0.038 0.866 ± 0.021 0.543 ± 0.040 0.499 ± 0.037 0.599 ± 0.048 34.6 ± 4.5 0.340 ± 0.040 95.32 ± 1.04 55.8 ± 5.0 multimodal SciNO 5 0.733 ± 0.031 0.656 ± 0.052 0.854 ± 0.033 0.399 ± 0.045 0.362 ± 0.053 0.457 ± 0.028 48.2 ± 8.3 0.476 ± 0.048 833.01 ± 24.43 62.6 ± 9.1 multimodal Caps 5 0.791 ± 0.023 0.763 ± 0.034 0.826 ± 0.026 0.512 ± 0.023 0.495 ± 0.032 0.534 ± 0.018 33.6 ± 2.6 0.322 ± 0.039 47.83 ± 1.14 50.4 ± 3.9 multimodal PC 5 0.725 ± 0.023 0.887 ± 0.016 0.615 ± 0.029 0.631 ± 0.031 0.775 ± 0.045 0.534 ± 0.027 25.0 ± 1.3 0.243 ± 0.058 17089.05 ± 1988.02 32.4 ± 3.6 multimodal FCI 5 0.561 ± 0.060 1.000 0.400 ± 0.061 0.490 ± 0.061 0.867 ± 0.040 0.350 ± 0.057 30.2 ± 3.5 0.306 ± 0.050 18771.95 ± 3022.76 18.4 ± 2.8 unimodal_asymmetric Spade-Hist 5 0.760 ± 0.044 0.945 ± 0.018 0.642 ± 0.055 0.707 ± 0.043 0.880 ± 0.022 0.597 ± 0.052 20.8 ± 3.7 0.215 ± 0.034 1.80 ± 0.10 31.0 ± 2.0 unimodal_asymmetric Spade-Gauss 5 0.801 ± 0.023 0.869 ± 0.012 0.749 ± 0.042 0.641 ± 0.020 0.696 ± 0.021 0.599 ± 0.032 24.2 ± 2.7 0.240 ± 0.012 0.49 ± 0.02 39.4 ± 1.2 unimodal_asymmetric Topic 5 0.671 ± 0.008 0.590 ± 0.014 0.785 ± 0.031 0.514 ± 0.012 0.452 ± 0.016 0.600 ± 0.023 43.8 ± 2.1 0.292 ± 0.005 124.33 ± 6.78 61.0 ± 1.1 unimodal_asymmetric Cam 5 0.471 ± 0.019 0.315 ± 0.018 0.948 ± 0.017 0.363 ± 0.011 0.242 ± 0.010 0.732 ± 0.028 108.4 ± 5.1 0.245 ± 0.024 13227.58 ± 89.41 139.4 ± 5.1 unimodal_asymmetric CauScale 5 0.759 ± 0.021 0.792 ± 0.047 0.740 ± 0.029 0.504 ± 0.024 0.524 ± 0.029 0.494 ± 0.036 33.0 ± 2.1 0.342 ± 0.031 6.00 ± 0.08 43.2 ± 1.5 unimodal_asymmetric SDCD 5 0.742 ± 0.038 0.799 ± 0.021 0.699 ± 0.057 0.588 ± 0.057 0.633 ± 0.054 0.554 ± 0.062 29.4 ± 5.1 0.287 ± 0.053 135.48 ± 1.71 39.8 ± 1.7 unimodal_asymmetric Globe 5 0.627 ± 0.026 0.848 ± 0.017 0.502 ± 0.036 0.524 ± 0.052 0.704 ± 0.052 0.422 ± 0.050 31.6 ± 4.3 0.378 ± 0.040 940.97 ± 77.61 27.2 ± 2.0 unimodal_asymmetric Das 5 0.593 ± 0.016 0.507 ± 0.020 0.723 ± 0.034 0.417 ± 0.007 0.356 ± 0.011 0.509 ± 0.024 55.4 ± 2.5 0.349 ± 0.021 207.13 ± 6.59 65.6 ± 1.9 unimodal_asymmetric Score 5 0.510 ± 0.015 0.351 ± 0.016 0.937 ± 0.023 0.336 ± 0.017 0.232 ± 0.015 0.616 ± 0.024 98.0 ± 5.7 0.349 ± 0.021 210.02 ± 12.21 123.2 ± 5.4 unimodal_asymmetric NoGAM 5 0.511 ± 0.019 0.352 ± 0.020 0.944 ± 0.019 0.333 ± 0.016 0.230 ± 0.015 0.615 ± 0.019 98.6 ± 4.8 0.361 ± 0.020 1311.92 ± 5.85 124.2 ± 5.1 unimodal_asymmetric Avici 5 0.634 ± 0.037 0.921 ± 0.016 0.489 ± 0.042 0.471 ± 0.050 0.683 ± 0.055 0.364 ± 0.046 31.8 ± 4.0 0.396 ± 0.050 24.58 ± 0.74 24.6 ± 2.6 unimodal_asymmetric NOTEARS 5 0.715 ± 0.025 0.641 ± 0.032 0.815 ± 0.035 0.559 ± 0.064 0.500 ± 0.060 0.640 ± 0.074 39.2 ± 7.0 0.285 ± 0.060 2150.46 ± 270.14 59.0 ± 4.0 unimodal_asymmetric DAGMA 5 0.735 ± 0.019 0.716 ± 0.029 0.761 ± 0.025 0.410 ± 0.054 0.402 ± 0.058 0.422 ± 0.050 40.8 ± 3.7 0.486 ± 0.064 4450.51 ± 286.29 49.0 ± 1.8 unimodal_asymmetric DiffAN 5 0.742 ± 0.032 0.643 ± 0.050 0.892 ± 0.027 0.531 ± 0.036 0.462 ± 0.046 0.636 ± 0.027 40.0 ± 3.1 0.313 ± 0.038 94.71 ± 0.70 64.4 ± 1.8 unimodal_asymmetric SciNO 5 0.700 ± 0.026 0.576 ± 0.029 0.899 ± 0.030 0.380 ± 0.041 0.313 ± 0.036 0.487 ± 0.051 55.2 ± 5.9 0.456 ± 0.046 1019.80 ± 96.26 72.4 ± 4.2 unimodal_asymmetric Caps 5 0.783 ± 0.015 0.714 ± 0.027 0.874 ± 0.029 0.527 ± 0.015 0.481 ± 0.022 0.588 ± 0.021 35.8 ± 3.2 0.355 ± 0.019 55.28 ± 1.90 56.6 ± 3.0 unimodal_asymmetric PC 5 0.766 ± 0.030 0.881 ± 0.019 0.687 ± 0.052 0.575 ± 0.019 0.668 ± 0.050 0.511 ± 0.019 27.0 ± 1.7 0.316 ± 0.021 18659.35 ± 1684.60 36.2 ± 3.8 unimodal_asymmetric FCI 5 0.541 ± 0.043 1.000 0.375 ± 0.041 0.458 ± 0.019 0.863 ± 0.051 0.316 ± 0.019 31.8 ± 2.7 0.314 ± 0.035 19334.48 ± 2601.27 17.2 ± 1.7 unimodal_symmetric Spade-Hist 5 0.792 ± 0.045 0.958 ± 0.023 0.679 ± 0.052 0.753 ± 0.055 0.908 ± 0.038 0.646 ± 0.059 18.6 ± 5.1 0.198 ± 0.041 1.65 ± 0.01 32.2 ± 1.1 unimodal_symmetric Spade-Gauss 5 0.790 ± 0.033 0.935 ± 0.005 0.689 ± 0.046 0.682 ± 0.040 0.806 ± 0.029 0.596 ± 0.047 21.4 ± 3.7 0.204 ± 0.048 0.45 ± 0.01 33.6 ± 1.1 unimodal_symmetric Topic 5 0.662 ± 0.016 0.601 ± 0.031 0.749 ± 0.027 0.543 ± 0.032 0.493 ± 0.037 0.614 ± 0.039 41.2 ± 2.5 0.245 ± 0.041 115.24 ± 3.48 57.4 ± 0.9 unimodal_symmetric Cam 5 0.492 ± 0.019 0.334 ± 0.017 0.942 ± 0.016 0.396 ± 0.031 0.269 ± 0.024 0.756 ± 0.045 99.0 ± 8.5 0.212 ± 0.047 13224.15 ± 75.48 130.8 ± 7.6 unimodal_symmetric CauScale 5 0.768 ± 0.005 0.786 ± 0.017 0.754 ± 0.020 0.492 ± 0.012 0.504 ± 0.018 0.482 ± 0.014 33.6 ± 2.3 0.384 ± 0.025 6.05 ± 0.06 44.2 ± 2.0 unimodal_symmetric SDCD 5 0.694 ± 0.011 0.794 ± 0.020 0.619 ± 0.014 0.498 ± 0.055 0.569 ± 0.064 0.444 ± 0.050 33.6 ± 4.5 0.439 ± 0.046 131.20 ± 7.10 36.2 ± 2.6 unimodal_symmetric Globe 5 0.639 ± 0.050 0.872 ± 0.036 0.517 ± 0.061 0.556 ± 0.047 0.762 ± 0.053 0.449 ± 0.056 29.6 ± 4.2 0.374 ± 0.019 1030.50 ± 63.52 27.0 ± 2.8 unimodal_symmetric Das 5 0.537 ± 0.018 0.461 ± 0.023 0.654 ± 0.038 0.371 ± 0.024 0.320 ± 0.029 0.449 ± 0.023 61.0 ± 3.2 0.323 ± 0.035 215.93 ± 1.74 65.2 ± 2.4 unimodal_symmetric Score 5 0.505 ± 0.012 0.350 ± 0.010 0.907 ± 0.025 0.346 ± 0.007 0.240 ± 0.005 0.623 ± 0.024 95.4 ± 5.4 0.323 ± 0.035 207.78 ± 13.93 119.6 ± 6.1 unimodal_symmetric NoGAM 5 0.508 ± 0.018 0.354 ± 0.018 0.906 ± 0.022 0.358 ± 0.009 0.249 ± 0.007 0.641 ± 0.032 94.2 ± 7.3 0.304 ± 0.024 1277.96 ± 46.68 119.0 ± 7.6 unimodal_symmetric Avici 5 0.736 ± 0.050 0.931 ± 0.024 0.618 ± 0.061 0.660 ± 0.060 0.833 ± 0.045 0.555 ± 0.065 23.2 ± 4.9 0.290 ± 0.035 26.73 ± 1.22 30.0 ± 1.6 unimodal_symmetric NOTEARS 5 0.713 ± 0.024 0.670 ± 0.015 0.763 ± 0.040 0.529 ± 0.016 0.499 ± 0.019 0.565 ± 0.020 37.8 ± 4.1 0.374 ± 0.015 2779.26 ± 345.03 52.4 ± 3.3 unimodal_symmetric DAGMA 5 0.706 ± 0.028 0.766 ± 0.038 0.657 ± 0.024 0.426 ± 0.029 0.464 ± 0.041 0.395 ± 0.021 37.6 ± 3.6 0.495 ± 0.031 5029.57 ± 115.93 39.8 ± 2.4 unimodal_symmetric DiffAN 5 0.678 ± 0.014 0.584 ± 0.023 0.817 ± 0.025 0.432 ± 0.055 0.373 ± 0.052 0.519 ± 0.060 50.2 ± 6.7 0.383 ± 0.042 96.79 ± 1.12 64.8 ± 3.8 unimodal_symmetric SciNO 5 0.653 ± 0.012 0.543 ± 0.011 0.821 ± 0.027 0.343 ± 0.027 0.285 ± 0.023 0.432 ± 0.038 58.6 ± 5.0 0.479 ± 0.036 828.80 ± 16.29 69.8 ± 4.2 unimodal_symmetric Caps 5 0.700 ± 0.016 0.634 ± 0.021 0.787 ± 0.031 0.446 ± 0.031 0.403 ± 0.030 0.502 ± 0.039 44.6 ± 4.5 0.353 ± 0.055 49.29 ± 2.69 57.2 ± 2.7 unimodal_symmetric PC 5 0.715 ± 0.029 0.848 ± 0.018 0.626 ± 0.048 0.487 ± 0.064 0.574 ± 0.070 0.427 ± 0.062 31.2 ± 1.4 0.369 ± 0.058 17242.71 ± 2649.42 34.4 ± 3.8 unimodal_symmetric FCI 5 0.482 ± 0.048 1.000 0.323 ± 0.043 0.392 ± 0.062 0.802 ± 0.056 0.264 ± 0.052 34.2 ± 3.8 0.387 ± 0.054 17792.07 ± 2782.23 14.8 ± 2.0 Table 37: Qualitative comparison without marginalization (n=2000n=2000, d=50d=50, degree 3, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.827 ± 0.032 0.966 ± 0.020 0.725 ± 0.042 0.807 ± 0.036 0.943 ± 0.028 0.708 ± 0.044 23.0 ± 4.4 0.086 ± 0.028 2.24 ± 0.15 53.6 ± 3.3 gaussian Spade-Gauss 5 0.839 ± 0.022 0.993 ± 0.004 0.729 ± 0.035 0.836 ± 0.023 0.990 ± 0.007 0.727 ± 0.036 20.2 ± 3.0 0.026 ± 0.011 0.71 ± 0.08 52.6 ± 3.6 gaussian Topic 5 0.765 ± 0.020 0.769 ± 0.019 0.763 ± 0.031 0.746 ± 0.022 0.750 ± 0.018 0.744 ± 0.035 34.8 ± 3.0 0.058 ± 0.004 177.65 ± 17.15 70.8 ± 3.0 gaussian Cam 5 0.610 ± 0.025 0.467 ± 0.029 0.889 ± 0.009 0.610 ± 0.025 0.467 ± 0.029 0.889 ± 0.009 82.0 ± 7.4 0.039 ± 0.014 12536.20 ± 52.22 137.8 ± 8.6 gaussian CauScale 5 0.784 ± 0.011 0.867 ± 0.018 0.717 ± 0.022 0.592 ± 0.019 0.654 ± 0.019 0.542 ± 0.025 41.0 ± 3.2 0.298 ± 0.018 5.98 ± 0.09 59.2 ± 2.7 gaussian SDCD 5 0.703 ± 0.017 0.803 ± 0.021 0.627 ± 0.020 0.514 ± 0.030 0.585 ± 0.031 0.459 ± 0.031 50.2 ± 5.0 0.323 ± 0.036 142.96 ± 2.48 56.4 ± 4.6 gaussian Globe 5 0.694 ± 0.036 0.906 ± 0.023 0.569 ± 0.047 0.680 ± 0.043 0.886 ± 0.027 0.558 ± 0.052 36.4 ± 4.9 0.155 ± 0.037 1613.46 ± 116.29 44.6 ± 3.0 gaussian Das 5 0.622 ± 0.010 0.576 ± 0.022 0.681 ± 0.019 0.472 ± 0.018 0.437 ± 0.019 0.517 ± 0.026 71.6 ± 6.6 0.324 ± 0.017 195.59 ± 1.74 85.6 ± 7.5 gaussian Score 5 0.537 ± 0.013 0.393 ± 0.016 0.855 ± 0.011 0.385 ± 0.006 0.281 ± 0.006 0.616 ± 0.024 123.6 ± 8.6 0.324 ± 0.017 194.53 ± 1.20 157.4 ± 11.6 gaussian NoGAM 5 0.554 ± 0.014 0.409 ± 0.017 0.867 ± 0.009 0.418 ± 0.011 0.308 ± 0.012 0.654 ± 0.014 115.4 ± 6.6 0.296 ± 0.004 1119.04 ± 3.47 152.8 ± 9.0 gaussian Avici 5 0.621 ± 0.040 0.952 ± 0.010 0.466 ± 0.047 0.591 ± 0.034 0.908 ± 0.011 0.444 ± 0.041 41.8 ± 4.3 0.261 ± 0.035 23.92 ± 0.46 34.8 ± 3.0 gaussian NOTEARS 5 0.688 ± 0.005 0.680 ± 0.026 0.704 ± 0.028 0.452 ± 0.024 0.444 ± 0.017 0.466 ± 0.040 62.8 ± 3.2 0.353 ± 0.028 4123.17 ± 775.10 74.8 ± 5.9 gaussian DAGMA 5 0.645 ± 0.023 0.704 ± 0.026 0.603 ± 0.037 0.350 ± 0.014 0.384 ± 0.023 0.326 ± 0.017 66.8 ± 3.7 0.452 ± 0.014 4268.30 ± 219.52 61.0 ± 3.0 gaussian DiffAN 5 0.719 ± 0.018 0.655 ± 0.031 0.801 ± 0.009 0.508 ± 0.021 0.463 ± 0.028 0.566 ± 0.017 63.0 ± 7.7 0.359 ± 0.014 95.23 ± 0.66 89.2 ± 8.8 gaussian SciNO 5 0.657 ± 0.022 0.566 ± 0.030 0.788 ± 0.023 0.395 ± 0.029 0.339 ± 0.025 0.477 ± 0.040 81.0 ± 2.9 0.461 ± 0.039 898.18 ± 36.51 100.6 ± 6.5 gaussian Caps 5 0.707 ± 0.022 0.704 ± 0.030 0.715 ± 0.030 0.476 ± 0.037 0.473 ± 0.038 0.482 ± 0.041 59.8 ± 7.7 0.379 ± 0.011 40.05 ± 1.25 73.6 ± 6.9 gaussian PC 5 0.722 ± 0.035 0.905 ± 0.020 0.604 ± 0.044 0.519 ± 0.031 0.653 ± 0.034 0.434 ± 0.033 45.2 ± 4.4 0.362 ± 0.041 41655.15 ± 8603.83 47.4 ± 2.8 gaussian FCI 5 0.497 ± 0.050 0.979 ± 0.013 0.338 ± 0.044 0.421 ± 0.040 0.831 ± 0.026 0.286 ± 0.036 51.8 ± 4.6 0.304 ± 0.035 42821.74 ± 8805.87 24.4 ± 2.5 multimodal Spade-Hist 5 0.867 ± 0.013 0.976 ± 0.009 0.781 ± 0.021 0.867 ± 0.013 0.976 ± 0.009 0.781 ± 0.021 17.2 ± 2.1 0.031 ± 0.007 2.27 ± 0.10 57.4 ± 3.5 multimodal Spade-Gauss 5 0.818 ± 0.028 0.957 ± 0.019 0.718 ± 0.039 0.745 ± 0.044 0.871 ± 0.042 0.654 ± 0.050 27.8 ± 5.4 0.136 ± 0.022 0.71 ± 0.05 53.6 ± 3.2 multimodal Topic 5 0.767 ± 0.020 0.751 ± 0.026 0.785 ± 0.020 0.691 ± 0.029 0.677 ± 0.033 0.707 ± 0.029 40.2 ± 4.9 0.148 ± 0.020 217.64 ± 26.96 75.0 ± 4.1 multimodal Cam 5 0.658 ± 0.020 0.520 ± 0.024 0.904 ± 0.019 0.629 ± 0.024 0.497 ± 0.023 0.865 ± 0.033 70.6 ± 6.6 0.055 ± 0.016 13894.96 ± 99.32 125.0 ± 6.5 multimodal CauScale 5 0.823 ± 0.014 0.893 ± 0.011 0.763 ± 0.020 0.724 ± 0.022 0.786 ± 0.023 0.672 ± 0.023 29.8 ± 1.7 0.141 ± 0.021 5.88 ± 0.11 61.2 ± 3.4 multimodal SDCD 5 0.713 ± 0.026 0.787 ± 0.018 0.654 ± 0.034 0.540 ± 0.043 0.595 ± 0.040 0.496 ± 0.045 48.8 ± 5.1 0.282 ± 0.040 138.31 ± 1.79 59.4 ± 3.6 multimodal Globe 5 0.721 ± 0.019 0.915 ± 0.016 0.598 ± 0.028 0.709 ± 0.023 0.900 ± 0.025 0.588 ± 0.029 34.0 ± 4.0 0.108 ± 0.020 2037.46 ± 177.07 47.0 ± 3.6 multimodal Das 5 0.612 ± 0.030 0.563 ± 0.038 0.673 ± 0.023 0.458 ± 0.053 0.423 ± 0.057 0.501 ± 0.049 74.4 ± 8.9 0.296 ± 0.045 251.24 ± 8.09 86.6 ± 6.3 multimodal Score 5 0.596 ± 0.010 0.453 ± 0.009 0.870 ± 0.014 0.438 ± 0.030 0.333 ± 0.022 0.641 ± 0.048 101.0 ± 6.1 0.296 ± 0.045 232.79 ± 3.28 137.4 ± 6.1 multimodal NoGAM 5 0.590 ± 0.010 0.445 ± 0.011 0.877 ± 0.009 0.453 ± 0.023 0.341 ± 0.017 0.674 ± 0.038 102.2 ± 6.6 0.273 ± 0.040 1330.80 ± 45.47 141.6 ± 7.9 multimodal Avici 5 0.502 ± 0.043 0.927 ± 0.030 0.353 ± 0.043 0.369 ± 0.050 0.666 ± 0.049 0.262 ± 0.043 55.4 ± 3.6 0.375 ± 0.044 24.41 ± 0.41 27.6 ± 3.7 multimodal NOTEARS 5 0.677 ± 0.032 0.656 ± 0.028 0.705 ± 0.044 0.403 ± 0.048 0.392 ± 0.050 0.418 ± 0.049 69.8 ± 10.3 0.410 ± 0.041 3597.31 ± 473.44 77.6 ± 7.0 multimodal DAGMA 5 0.639 ± 0.019 0.676 ± 0.021 0.613 ± 0.038 0.328 ± 0.028 0.348 ± 0.035 0.313 ± 0.028 71.2 ± 6.6 0.460 ± 0.041 4393.92 ± 136.66 65.0 ± 4.8 multimodal DiffAN 5 0.747 ± 0.026 0.677 ± 0.039 0.839 ± 0.016 0.540 ± 0.028 0.490 ± 0.034 0.605 ± 0.021 58.0 ± 6.2 0.315 ± 0.024 92.68 ± 0.64 90.0 ± 7.0 multimodal SciNO 5 0.735 ± 0.018 0.663 ± 0.024 0.825 ± 0.010 0.456 ± 0.039 0.412 ± 0.037 0.512 ± 0.042 66.0 ± 7.5 0.410 ± 0.047 286.90 ± 2.65 90.0 ± 7.6 multimodal Caps 5 0.712 ± 0.027 0.715 ± 0.033 0.712 ± 0.023 0.493 ± 0.050 0.495 ± 0.051 0.493 ± 0.050 57.4 ± 7.1 0.346 ± 0.048 40.62 ± 1.71 71.8 ± 4.9 multimodal PC 5 0.718 ± 0.028 0.945 ± 0.012 0.585 ± 0.038 0.562 ± 0.027 0.739 ± 0.021 0.457 ± 0.032 41.6 ± 3.2 0.289 ± 0.041 54607.94 ± 8946.30 44.8 ± 5.2 multimodal FCI 5 0.513 ± 0.052 1.000 0.351 ± 0.048 0.478 ± 0.052 0.931 ± 0.021 0.328 ± 0.047 48.6 ± 5.5 0.252 ± 0.032 55101.53 ± 9390.21 24.8 ± 2.8 unimodal_asymmetric Spade-Hist 5 0.884 ± 0.008 0.994 ± 0.004 0.796 ± 0.012 0.881 ± 0.010 0.991 ± 0.006 0.793 ± 0.013 15.4 ± 2.0 0.050 ± 0.008 2.32 ± 0.10 57.4 ± 3.3 unimodal_asymmetric Spade-Gauss 5 0.849 ± 0.008 0.946 ± 0.015 0.771 ± 0.012 0.745 ± 0.028 0.831 ± 0.036 0.677 ± 0.023 26.6 ± 3.2 0.143 ± 0.040 0.63 ± 0.03 58.4 ± 2.9 unimodal_asymmetric Topic 5 0.769 ± 0.005 0.762 ± 0.010 0.776 ± 0.011 0.665 ± 0.011 0.660 ± 0.011 0.672 ± 0.015 41.0 ± 2.6 0.161 ± 0.029 184.27 ± 14.93 72.8 ± 3.1 unimodal_asymmetric Cam 5 0.619 ± 0.010 0.462 ± 0.010 0.935 ± 0.007 0.576 ± 0.013 0.430 ± 0.012 0.870 ± 0.011 86.8 ± 2.1 0.083 ± 0.012 13675.62 ± 33.52 144.8 ± 6.3 unimodal_asymmetric CauScale 5 0.779 ± 0.020 0.849 ± 0.017 0.722 ± 0.029 0.589 ± 0.044 0.641 ± 0.042 0.547 ± 0.047 41.8 ± 4.3 0.281 ± 0.045 5.95 ± 0.08 60.8 ± 3.1 unimodal_asymmetric SDCD 5 0.733 ± 0.016 0.803 ± 0.022 0.678 ± 0.027 0.573 ± 0.025 0.627 ± 0.031 0.529 ± 0.030 45.8 ± 3.0 0.261 ± 0.022 138.32 ± 2.41 60.2 ± 2.1 unimodal_asymmetric Globe 5 0.609 ± 0.018 0.857 ± 0.020 0.475 ± 0.025 0.534 ± 0.016 0.752 ± 0.025 0.417 ± 0.022 47.8 ± 3.2 0.346 ± 0.031 1158.64 ± 71.66 39.6 ± 2.1 unimodal_asymmetric Das 5 0.611 ± 0.013 0.555 ± 0.009 0.679 ± 0.020 0.470 ± 0.014 0.427 ± 0.011 0.523 ± 0.019 73.6 ± 6.1 0.321 ± 0.021 249.34 ± 5.79 87.8 ± 5.7 unimodal_asymmetric Score 5 0.566 ± 0.017 0.415 ± 0.015 0.892 ± 0.023 0.409 ± 0.020 0.300 ± 0.017 0.644 ± 0.026 116.2 ± 8.9 0.321 ± 0.021 226.52 ± 0.72 155.0 ± 11.6 unimodal_asymmetric NoGAM 5 0.559 ± 0.014 0.409 ± 0.012 0.883 ± 0.017 0.423 ± 0.014 0.309 ± 0.011 0.668 ± 0.019 116.4 ± 10.6 0.284 ± 0.007 1323.48 ± 35.83 155.8 ± 12.1 unimodal_asymmetric Avici 5 0.531 ± 0.034 0.967 ± 0.016 0.371 ± 0.035 0.415 ± 0.031 0.756 ± 0.038 0.290 ± 0.030 52.2 ± 4.1 0.397 ± 0.022 24.01 ± 0.39 27.2 ± 2.2 unimodal_asymmetric NOTEARS 5 0.685 ± 0.014 0.635 ± 0.017 0.747 ± 0.026 0.419 ± 0.026 0.390 ± 0.031 0.454 ± 0.019 70.6 ± 6.3 0.378 ± 0.030 3018.23 ± 347.29 84.6 ± 5.8 unimodal_asymmetric DAGMA 5 0.644 ± 0.013 0.664 ± 0.014 0.629 ± 0.026 0.289 ± 0.050 0.301 ± 0.055 0.279 ± 0.046 74.4 ± 5.1 0.536 ± 0.060 4932.18 ± 190.20 67.6 ± 2.7 unimodal_asymmetric DiffAN 5 0.694 ± 0.013 0.597 ± 0.014 0.830 ± 0.014 0.462 ± 0.031 0.397 ± 0.027 0.552 ± 0.036 72.8 ± 7.3 0.381 ± 0.031 91.80 ± 1.40 100.0 ± 7.1 unimodal_asymmetric SciNO 5 0.721 ± 0.020 0.637 ± 0.023 0.833 ± 0.018 0.495 ± 0.037 0.438 ± 0.037 0.570 ± 0.037 65.2 ± 6.3 0.356 ± 0.024 280.92 ± 2.92 94.0 ± 5.6 unimodal_asymmetric Caps 5 0.719 ± 0.015 0.706 ± 0.021 0.734 ± 0.021 0.516 ± 0.027 0.508 ± 0.031 0.527 ± 0.026 56.4 ± 5.6 0.335 ± 0.022 47.75 ± 1.39 75.2 ± 6.4 unimodal_asymmetric PC 5 0.692 ± 0.048 0.945 ± 0.005 0.555 ± 0.055 0.589 ± 0.036 0.807 ± 0.021 0.472 ± 0.043 40.0 ± 2.5 0.282 ± 0.016 55552.34 ± 11515.10 42.6 ± 5.5 unimodal_asymmetric FCI 5 0.497 ± 0.038 1.000 0.334 ± 0.035 0.454 ± 0.030 0.916 ± 0.026 0.305 ± 0.027 49.6 ± 2.4 0.286 ± 0.027 54179.19 ± 11275.31 24.4 ± 3.6 unimodal_symmetric Spade-Hist 5 0.869 ± 0.024 0.974 ± 0.008 0.788 ± 0.041 0.866 ± 0.025 0.971 ± 0.009 0.786 ± 0.041 17.2 ± 3.4 0.022 ± 0.011 2.36 ± 0.09 58.0 ± 4.1 unimodal_symmetric Spade-Gauss 5 0.842 ± 0.022 0.955 ± 0.012 0.758 ± 0.040 0.817 ± 0.018 0.927 ± 0.015 0.735 ± 0.036 22.0 ± 3.1 0.047 ± 0.005 0.72 ± 0.04 57.0 ± 4.3 unimodal_symmetric Topic 5 0.755 ± 0.019 0.740 ± 0.015 0.772 ± 0.031 0.717 ± 0.020 0.703 ± 0.013 0.733 ± 0.033 38.6 ± 3.3 0.094 ± 0.016 197.59 ± 13.30 74.4 ± 3.2 unimodal_symmetric Cam 5 0.587 ± 0.013 0.432 ± 0.013 0.920 ± 0.018 0.568 ± 0.013 0.417 ± 0.013 0.890 ± 0.025 94.8 ± 5.1 0.049 ± 0.011 14064.67 ± 81.10 153.2 ± 9.9 unimodal_symmetric CauScale 5 0.802 ± 0.021 0.876 ± 0.020 0.740 ± 0.023 0.616 ± 0.049 0.673 ± 0.050 0.569 ± 0.048 38.4 ± 4.8 0.248 ± 0.035 5.94 ± 0.08 60.4 ± 2.8 unimodal_symmetric SDCD 5 0.717 ± 0.009 0.795 ± 0.025 0.657 ± 0.019 0.575 ± 0.017 0.636 ± 0.018 0.527 ± 0.026 46.6 ± 3.5 0.263 ± 0.018 134.33 ± 4.35 59.4 ± 3.7 unimodal_symmetric Globe 5 0.684 ± 0.040 0.919 ± 0.006 0.552 ± 0.053 0.665 ± 0.042 0.894 ± 0.016 0.536 ± 0.054 37.2 ± 5.3 0.221 ± 0.027 1505.69 ± 127.63 42.4 ± 3.0 unimodal_symmetric Das 5 0.586 ± 0.013 0.519 ± 0.020 0.675 ± 0.004 0.423 ± 0.024 0.374 ± 0.024 0.487 ± 0.026 82.2 ± 5.8 0.346 ± 0.044 238.51 ± 1.98 94.0 ± 6.7 unimodal_symmetric Score 5 0.564 ± 0.019 0.418 ± 0.019 0.873 ± 0.025 0.397 ± 0.028 0.294 ± 0.023 0.615 ± 0.041 115.0 ± 6.2 0.346 ± 0.044 228.34 ± 1.88 150.8 ± 10.3 unimodal_symmetric NoGAM 5 0.563 ± 0.014 0.417 ± 0.014 0.874 ± 0.033 0.411 ± 0.025 0.304 ± 0.017 0.642 ± 0.051 113.4 ± 4.1 0.319 ± 0.052 1246.14 ± 9.01 151.0 ± 10.4 unimodal_symmetric Avici 5 0.613 ± 0.050 0.963 ± 0.020 0.457 ± 0.050 0.558 ± 0.042 0.879 ± 0.017 0.416 ± 0.043 43.0 ± 3.1 0.293 ± 0.019 24.07 ± 0.41 34.2 ± 4.1 unimodal_symmetric NOTEARS 5 0.656 ± 0.021 0.637 ± 0.052 0.691 ± 0.017 0.462 ± 0.046 0.453 ± 0.062 0.482 ± 0.032 68.0 ± 8.8 0.311 ± 0.037 3819.07 ± 348.31 80.4 ± 8.7 unimodal_symmetric DAGMA 5 0.646 ± 0.026 0.668 ± 0.031 0.635 ± 0.042 0.345 ± 0.037 0.361 ± 0.044 0.335 ± 0.034 70.8 ± 5.8 0.436 ± 0.035 4624.20 ± 308.37 68.8 ± 6.3 unimodal_symmetric DiffAN 5 0.701 ± 0.022 0.603 ± 0.028 0.843 ± 0.026 0.486 ± 0.036 0.419 ± 0.036 0.582 ± 0.039 71.2 ± 9.0 0.361 ± 0.035 92.02 ± 0.59 101.6 ± 9.0 unimodal_symmetric SciNO 5 0.719 ± 0.014 0.636 ± 0.025 0.835 ± 0.021 0.520 ± 0.042 0.459 ± 0.039 0.604 ± 0.049 63.0 ± 5.0 0.330 ± 0.050 288.29 ± 5.48 95.4 ± 8.4 unimodal_symmetric Caps 5 0.687 ± 0.008 0.655 ± 0.021 0.727 ± 0.018 0.491 ± 0.025 0.469 ± 0.030 0.518 ± 0.022 62.4 ± 4.3 0.362 ± 0.048 41.23 ± 2.01 80.2 ± 6.4 unimodal_symmetric PC 5 0.729 ± 0.028 0.918 ± 0.018 0.608 ± 0.036 0.558 ± 0.030 0.704 ± 0.038 0.465 ± 0.029 42.6 ± 4.0 0.348 ± 0.032 59534.40 ± 10189.30 47.6 ± 4.2 unimodal_symmetric FCI 5 0.500 ± 0.038 0.975 ± 0.015 0.340 ± 0.034 0.423 ± 0.027 0.827 ± 0.015 0.287 ± 0.025 52.0 ± 4.4 0.349 ± 0.041 60343.97 ± 10308.58 24.8 ± 2.4 Table 38: Qualitative comparison without marginalization (n=2000n=2000, d=50d=50, degree 3, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.688 ± 0.016 0.967 ± 0.018 0.536 ± 0.021 0.642 ± 0.018 0.902 ± 0.016 0.500 ± 0.022 37.2 ± 2.3 0.252 ± 0.019 1.82 ± 0.09 40.0 ± 3.5 gaussian Spade-Gauss 5 0.713 ± 0.021 0.952 ± 0.022 0.573 ± 0.027 0.703 ± 0.023 0.938 ± 0.026 0.565 ± 0.028 33.6 ± 3.2 0.104 ± 0.011 0.53 ± 0.03 43.2 ± 3.1 gaussian Topic 5 0.650 ± 0.025 0.688 ± 0.025 0.619 ± 0.031 0.620 ± 0.028 0.656 ± 0.031 0.590 ± 0.030 49.2 ± 2.8 0.157 ± 0.024 139.67 ± 9.67 64.2 ± 2.9 gaussian Cam 5 0.574 ± 0.035 0.422 ± 0.035 0.914 ± 0.032 0.528 ± 0.031 0.388 ± 0.032 0.841 ± 0.029 104.4 ± 11.7 0.100 ± 0.017 12483.49 ± 44.34 158.4 ± 12.6 gaussian CauScale 5 0.740 ± 0.010 0.878 ± 0.013 0.641 ± 0.018 0.514 ± 0.038 0.609 ± 0.043 0.446 ± 0.036 46.0 ± 2.5 0.336 ± 0.029 5.93 ± 0.06 52.8 ± 4.9 gaussian SDCD 5 0.677 ± 0.027 0.834 ± 0.016 0.571 ± 0.030 0.548 ± 0.021 0.675 ± 0.016 0.462 ± 0.024 46.4 ± 2.4 0.341 ± 0.011 139.54 ± 6.00 49.2 ± 3.9 gaussian Globe 5 0.515 ± 0.027 0.875 ± 0.018 0.368 ± 0.027 0.479 ± 0.028 0.814 ± 0.034 0.342 ± 0.027 51.0 ± 3.0 0.396 ± 0.024 1176.57 ± 175.76 30.4 ± 3.4 gaussian Das 5 0.541 ± 0.017 0.511 ± 0.025 0.580 ± 0.022 0.396 ± 0.050 0.372 ± 0.049 0.426 ± 0.056 81.0 ± 4.4 0.328 ± 0.055 194.73 ± 2.85 82.2 ± 7.1 gaussian Score 5 0.551 ± 0.023 0.409 ± 0.026 0.853 ± 0.018 0.387 ± 0.025 0.286 ± 0.019 0.606 ± 0.052 118.0 ± 5.9 0.328 ± 0.055 184.37 ± 5.24 151.2 ± 10.3 gaussian NoGAM 5 0.547 ± 0.026 0.407 ± 0.028 0.841 ± 0.016 0.396 ± 0.027 0.294 ± 0.021 0.614 ± 0.044 117.0 ± 7.2 0.318 ± 0.050 1103.98 ± 1.20 150.2 ± 10.7 gaussian Avici 5 0.532 ± 0.031 0.979 ± 0.015 0.368 ± 0.030 0.451 ± 0.053 0.820 ± 0.054 0.314 ± 0.043 50.0 ± 4.6 0.390 ± 0.035 24.44 ± 0.55 26.8 ± 2.1 gaussian NOTEARS 5 0.661 ± 0.019 0.707 ± 0.078 0.649 ± 0.032 0.508 ± 0.031 0.547 ± 0.077 0.498 ± 0.031 59.8 ± 7.3 0.344 ± 0.018 2760.81 ± 638.60 69.8 ± 9.7 gaussian DAGMA 5 0.624 ± 0.024 0.740 ± 0.036 0.548 ± 0.037 0.358 ± 0.056 0.427 ± 0.068 0.312 ± 0.048 64.0 ± 6.5 0.493 ± 0.049 4693.37 ± 338.39 53.2 ± 4.2 gaussian DiffAN 5 0.722 ± 0.034 0.674 ± 0.044 0.783 ± 0.026 0.515 ± 0.058 0.481 ± 0.063 0.557 ± 0.055 58.4 ± 4.9 0.362 ± 0.059 95.56 ± 0.97 83.8 ± 3.8 gaussian SciNO 5 0.658 ± 0.025 0.583 ± 0.030 0.765 ± 0.039 0.413 ± 0.017 0.365 ± 0.018 0.481 ± 0.031 77.2 ± 5.1 0.424 ± 0.012 906.95 ± 29.30 94.8 ± 7.7 gaussian Caps 5 0.694 ± 0.020 0.704 ± 0.038 0.691 ± 0.021 0.490 ± 0.041 0.498 ± 0.051 0.487 ± 0.041 57.6 ± 1.0 0.340 ± 0.058 43.45 ± 1.43 70.4 ± 2.4 gaussian PC 5 0.783 ± 0.031 0.962 ± 0.014 0.664 ± 0.040 0.543 ± 0.041 0.665 ± 0.030 0.462 ± 0.044 40.6 ± 4.1 0.341 ± 0.027 31603.37 ± 5835.60 49.6 ± 4.4 gaussian FCI 5 0.407 ± 0.033 0.990 ± 0.010 0.258 ± 0.026 0.354 ± 0.037 0.860 ± 0.058 0.225 ± 0.027 56.2 ± 5.4 0.382 ± 0.047 35660.20 ± 8615.82 18.6 ± 1.7 multimodal Spade-Hist 5 0.724 ± 0.026 0.967 ± 0.015 0.581 ± 0.032 0.713 ± 0.029 0.952 ± 0.014 0.573 ± 0.035 32.4 ± 3.9 0.129 ± 0.024 1.93 ± 0.08 42.8 ± 2.2 multimodal Spade-Gauss 5 0.710 ± 0.025 0.859 ± 0.038 0.609 ± 0.031 0.552 ± 0.044 0.668 ± 0.055 0.473 ± 0.042 45.4 ± 5.2 0.289 ± 0.019 0.60 ± 0.04 50.8 ± 2.9 multimodal Topic 5 0.656 ± 0.025 0.650 ± 0.037 0.667 ± 0.024 0.490 ± 0.038 0.485 ± 0.044 0.497 ± 0.037 62.6 ± 6.7 0.297 ± 0.020 175.34 ± 17.23 73.8 ± 3.4 multimodal Cam 5 0.590 ± 0.015 0.436 ± 0.017 0.918 ± 0.015 0.469 ± 0.026 0.346 ± 0.022 0.729 ± 0.036 104.2 ± 2.2 0.226 ± 0.037 13475.54 ± 152.95 150.6 ± 3.4 multimodal CauScale 5 0.775 ± 0.017 0.905 ± 0.015 0.678 ± 0.019 0.653 ± 0.026 0.763 ± 0.032 0.571 ± 0.023 36.0 ± 3.0 0.227 ± 0.043 5.93 ± 0.06 53.6 ± 2.7 multimodal SDCD 5 0.679 ± 0.022 0.805 ± 0.018 0.589 ± 0.029 0.527 ± 0.025 0.625 ± 0.026 0.458 ± 0.027 49.0 ± 2.9 0.324 ± 0.031 140.26 ± 3.61 52.4 ± 3.2 multimodal Globe 5 0.591 ± 0.030 0.850 ± 0.030 0.459 ± 0.035 0.530 ± 0.025 0.761 ± 0.026 0.411 ± 0.029 48.8 ± 4.8 0.303 ± 0.031 1505.06 ± 145.70 38.4 ± 2.4 multimodal Das 5 0.555 ± 0.011 0.530 ± 0.026 0.587 ± 0.013 0.424 ± 0.015 0.406 ± 0.026 0.449 ± 0.012 77.2 ± 3.0 0.270 ± 0.029 238.18 ± 0.66 79.6 ± 3.5 multimodal Score 5 0.584 ± 0.011 0.433 ± 0.009 0.895 ± 0.021 0.436 ± 0.018 0.323 ± 0.012 0.669 ± 0.034 107.2 ± 3.8 0.270 ± 0.029 227.10 ± 1.08 148.0 ± 8.3 multimodal NoGAM 5 0.591 ± 0.017 0.443 ± 0.016 0.890 ± 0.021 0.452 ± 0.021 0.338 ± 0.017 0.680 ± 0.031 102.6 ± 2.8 0.262 ± 0.032 1317.11 ± 20.72 143.8 ± 5.7 multimodal Avici 5 0.419 ± 0.059 0.968 ± 0.014 0.276 ± 0.051 0.332 ± 0.060 0.748 ± 0.027 0.220 ± 0.049 57.0 ± 5.5 0.460 ± 0.044 24.54 ± 0.49 20.4 ± 3.7 multimodal NOTEARS 5 0.694 ± 0.029 0.671 ± 0.047 0.724 ± 0.012 0.471 ± 0.054 0.459 ± 0.065 0.488 ± 0.042 64.0 ± 9.0 0.392 ± 0.050 3143.57 ± 373.04 78.6 ± 6.1 multimodal DAGMA 5 0.670 ± 0.028 0.723 ± 0.030 0.629 ± 0.039 0.418 ± 0.041 0.454 ± 0.054 0.390 ± 0.036 62.0 ± 7.3 0.432 ± 0.056 4927.09 ± 349.31 62.4 ± 4.3 multimodal DiffAN 5 0.738 ± 0.022 0.684 ± 0.026 0.802 ± 0.021 0.545 ± 0.043 0.505 ± 0.043 0.592 ± 0.044 56.2 ± 6.4 0.304 ± 0.041 92.76 ± 0.66 84.2 ± 4.3 multimodal SciNO 5 0.719 ± 0.011 0.645 ± 0.017 0.815 ± 0.012 0.461 ± 0.035 0.414 ± 0.035 0.521 ± 0.035 66.2 ± 3.4 0.367 ± 0.051 280.97 ± 2.93 90.8 ± 5.3 multimodal Caps 5 0.733 ± 0.015 0.713 ± 0.018 0.754 ± 0.015 0.502 ± 0.019 0.488 ± 0.018 0.518 ± 0.023 56.2 ± 2.2 0.304 ± 0.024 39.57 ± 0.86 76.0 ± 4.8 multimodal PC 5 0.817 ± 0.031 0.938 ± 0.017 0.728 ± 0.043 0.550 ± 0.026 0.632 ± 0.029 0.489 ± 0.030 40.2 ± 3.9 0.358 ± 0.028 46714.43 ± 6777.93 55.4 ± 3.6 multimodal FCI 5 0.486 ± 0.066 1.000 0.331 ± 0.058 0.410 ± 0.063 0.834 ± 0.027 0.280 ± 0.053 52.2 ± 6.4 0.339 ± 0.033 50917.83 ± 7984.95 23.2 ± 3.5 unimodal_asymmetric Spade-Hist 5 0.666 ± 0.008 0.948 ± 0.014 0.514 ± 0.010 0.645 ± 0.009 0.917 ± 0.018 0.498 ± 0.009 38.2 ± 2.7 0.232 ± 0.024 1.81 ± 0.07 38.8 ± 1.7 unimodal_asymmetric Spade-Gauss 5 0.728 ± 0.013 0.877 ± 0.026 0.625 ± 0.019 0.599 ± 0.020 0.721 ± 0.029 0.515 ± 0.022 41.6 ± 3.6 0.267 ± 0.021 0.55 ± 0.03 51.0 ± 2.0 unimodal_asymmetric Topic 5 0.684 ± 0.025 0.705 ± 0.023 0.668 ± 0.038 0.526 ± 0.025 0.543 ± 0.027 0.514 ± 0.031 55.4 ± 5.0 0.323 ± 0.017 166.54 ± 14.36 67.4 ± 2.2 unimodal_asymmetric Cam 5 0.528 ± 0.013 0.369 ± 0.013 0.935 ± 0.013 0.408 ± 0.017 0.285 ± 0.014 0.722 ± 0.028 134.4 ± 4.9 0.247 ± 0.031 13778.80 ± 160.89 181.4 ± 6.6 unimodal_asymmetric CauScale 5 0.757 ± 0.024 0.859 ± 0.028 0.681 ± 0.034 0.554 ± 0.027 0.632 ± 0.043 0.497 ± 0.024 44.2 ± 2.6 0.284 ± 0.045 6.07 ± 0.10 56.4 ± 1.0 unimodal_asymmetric SDCD 5 0.698 ± 0.020 0.765 ± 0.038 0.644 ± 0.010 0.538 ± 0.022 0.589 ± 0.028 0.497 ± 0.021 50.4 ± 2.7 0.301 ± 0.024 132.64 ± 4.26 60.8 ± 3.9 unimodal_asymmetric Globe 5 0.517 ± 0.036 0.887 ± 0.022 0.369 ± 0.039 0.437 ± 0.046 0.744 ± 0.038 0.313 ± 0.044 53.2 ± 5.7 0.458 ± 0.031 974.60 ± 57.98 29.6 ± 2.5 unimodal_asymmetric Das 5 0.567 ± 0.023 0.524 ± 0.028 0.622 ± 0.021 0.445 ± 0.018 0.410 ± 0.020 0.488 ± 0.021 77.6 ± 4.7 0.295 ± 0.037 248.30 ± 6.12 86.0 ± 6.8 unimodal_asymmetric Score 5 0.543 ± 0.012 0.387 ± 0.013 0.917 ± 0.013 0.389 ± 0.027 0.277 ± 0.021 0.655 ± 0.034 129.0 ± 7.2 0.295 ± 0.037 226.40 ± 0.69 170.2 ± 9.3 unimodal_asymmetric NoGAM 5 0.548 ± 0.014 0.393 ± 0.014 0.910 ± 0.012 0.395 ± 0.017 0.283 ± 0.015 0.655 ± 0.015 126.2 ± 8.5 0.289 ± 0.020 1327.33 ± 33.62 166.8 ± 10.4 unimodal_asymmetric Avici 5 0.538 ± 0.032 0.978 ± 0.015 0.373 ± 0.030 0.395 ± 0.047 0.712 ± 0.049 0.275 ± 0.038 53.0 ± 5.6 0.443 ± 0.032 24.45 ± 0.56 27.0 ± 1.4 unimodal_asymmetric NOTEARS 5 0.676 ± 0.033 0.587 ± 0.047 0.812 ± 0.023 0.490 ± 0.033 0.425 ± 0.041 0.588 ± 0.030 73.6 ± 9.6 0.316 ± 0.035 3166.94 ± 712.27 101.6 ± 9.6 unimodal_asymmetric DAGMA 5 0.658 ± 0.026 0.699 ± 0.044 0.627 ± 0.028 0.405 ± 0.035 0.431 ± 0.045 0.385 ± 0.032 64.8 ± 7.1 0.442 ± 0.050 4773.23 ± 298.63 64.8 ± 4.3 unimodal_asymmetric DiffAN 5 0.703 ± 0.035 0.613 ± 0.045 0.831 ± 0.019 0.471 ± 0.032 0.411 ± 0.037 0.554 ± 0.021 71.8 ± 9.7 0.364 ± 0.014 92.11 ± 0.54 99.0 ± 8.6 unimodal_asymmetric SciNO 5 0.695 ± 0.012 0.594 ± 0.018 0.839 ± 0.013 0.489 ± 0.029 0.417 ± 0.019 0.594 ± 0.047 70.6 ± 4.3 0.293 ± 0.044 285.34 ± 3.32 101.4 ± 6.0 unimodal_asymmetric Caps 5 0.719 ± 0.028 0.675 ± 0.038 0.772 ± 0.022 0.525 ± 0.026 0.494 ± 0.034 0.563 ± 0.019 58.8 ± 6.1 0.313 ± 0.036 50.53 ± 1.56 82.8 ± 6.5 unimodal_asymmetric PC 5 0.739 ± 0.035 0.951 ± 0.008 0.608 ± 0.044 0.554 ± 0.020 0.716 ± 0.021 0.455 ± 0.025 41.6 ± 4.3 0.350 ± 0.024 43663.92 ± 7887.31 45.4 ± 2.6 unimodal_asymmetric FCI 5 0.439 ± 0.043 1.000 0.285 ± 0.035 0.384 ± 0.039 0.877 ± 0.031 0.249 ± 0.031 54.0 ± 4.4 0.404 ± 0.027 46077.12 ± 8500.98 20.4 ± 2.6 unimodal_symmetric Spade-Hist 5 0.680 ± 0.037 0.954 ± 0.015 0.532 ± 0.045 0.661 ± 0.037 0.928 ± 0.015 0.518 ± 0.045 36.8 ± 4.9 0.204 ± 0.041 1.88 ± 0.08 39.6 ± 2.5 unimodal_symmetric Spade-Gauss 5 0.715 ± 0.023 0.901 ± 0.015 0.597 ± 0.035 0.646 ± 0.029 0.815 ± 0.037 0.539 ± 0.031 38.0 ± 3.5 0.226 ± 0.031 0.56 ± 0.04 47.4 ± 3.0 unimodal_symmetric Topic 5 0.668 ± 0.014 0.689 ± 0.014 0.653 ± 0.030 0.581 ± 0.019 0.601 ± 0.030 0.566 ± 0.020 52.2 ± 2.4 0.247 ± 0.026 173.44 ± 13.89 67.6 ± 3.1 unimodal_symmetric Cam 5 0.541 ± 0.013 0.386 ± 0.010 0.906 ± 0.021 0.449 ± 0.014 0.320 ± 0.010 0.754 ± 0.030 122.0 ± 10.0 0.190 ± 0.029 13791.18 ± 162.54 169.0 ± 11.3 unimodal_symmetric CauScale 5 0.760 ± 0.031 0.879 ± 0.019 0.672 ± 0.041 0.578 ± 0.050 0.665 ± 0.043 0.512 ± 0.056 42.0 ± 5.8 0.288 ± 0.039 5.94 ± 0.05 54.4 ± 2.4 unimodal_symmetric SDCD 5 0.644 ± 0.023 0.760 ± 0.041 0.575 ± 0.048 0.512 ± 0.020 0.603 ± 0.035 0.456 ± 0.038 53.6 ± 4.6 0.347 ± 0.013 138.24 ± 3.74 55.0 ± 6.0 unimodal_symmetric Globe 5 0.513 ± 0.046 0.890 ± 0.048 0.377 ± 0.056 0.464 ± 0.041 0.810 ± 0.062 0.341 ± 0.051 52.0 ± 4.3 0.431 ± 0.027 1138.47 ± 176.66 31.0 ± 5.7 unimodal_symmetric Das 5 0.556 ± 0.019 0.520 ± 0.021 0.599 ± 0.024 0.430 ± 0.014 0.403 ± 0.018 0.463 ± 0.014 78.6 ± 6.0 0.297 ± 0.035 246.92 ± 9.73 82.4 ± 4.0 unimodal_symmetric Score 5 0.564 ± 0.014 0.413 ± 0.015 0.888 ± 0.013 0.411 ± 0.013 0.301 ± 0.012 0.647 ± 0.013 116.4 ± 8.5 0.297 ± 0.035 236.19 ± 6.34 154.8 ± 10.3 unimodal_symmetric NoGAM 5 0.564 ± 0.016 0.413 ± 0.015 0.892 ± 0.018 0.401 ± 0.016 0.294 ± 0.014 0.634 ± 0.022 118.2 ± 10.1 0.307 ± 0.021 1307.26 ± 25.40 155.6 ± 10.8 unimodal_symmetric Avici 5 0.587 ± 0.039 0.942 ± 0.025 0.432 ± 0.040 0.535 ± 0.033 0.860 ± 0.031 0.393 ± 0.034 45.6 ± 3.6 0.317 ± 0.021 24.35 ± 0.44 33.0 ± 3.3 unimodal_symmetric NOTEARS 5 0.642 ± 0.013 0.650 ± 0.052 0.666 ± 0.055 0.508 ± 0.011 0.517 ± 0.048 0.525 ± 0.037 62.8 ± 5.0 0.315 ± 0.012 3844.85 ± 910.58 76.2 ± 10.3 unimodal_symmetric DAGMA 5 0.628 ± 0.011 0.701 ± 0.052 0.581 ± 0.022 0.417 ± 0.036 0.471 ± 0.065 0.381 ± 0.022 63.6 ± 5.7 0.460 ± 0.031 4538.36 ± 294.24 61.0 ± 6.1 unimodal_symmetric DiffAN 5 0.691 ± 0.019 0.613 ± 0.013 0.794 ± 0.037 0.489 ± 0.041 0.433 ± 0.034 0.563 ± 0.053 66.6 ± 3.0 0.350 ± 0.055 93.35 ± 1.07 92.6 ± 5.5 unimodal_symmetric SciNO 5 0.699 ± 0.016 0.612 ± 0.011 0.815 ± 0.025 0.500 ± 0.024 0.438 ± 0.022 0.583 ± 0.028 66.8 ± 4.0 0.312 ± 0.049 278.76 ± 0.64 95.4 ± 5.3 unimodal_symmetric Caps 5 0.741 ± 0.022 0.726 ± 0.023 0.756 ± 0.024 0.531 ± 0.031 0.520 ± 0.028 0.543 ± 0.035 53.8 ± 5.9 0.296 ± 0.039 44.94 ± 3.11 75.0 ± 5.4 unimodal_symmetric PC 5 0.766 ± 0.038 0.954 ± 0.013 0.646 ± 0.053 0.572 ± 0.028 0.714 ± 0.024 0.482 ± 0.038 39.8 ± 4.6 0.308 ± 0.031 53241.93 ± 8030.47 48.4 ± 4.1 unimodal_symmetric FCI 5 0.465 ± 0.060 0.993 ± 0.007 0.312 ± 0.052 0.417 ± 0.064 0.880 ± 0.022 0.281 ± 0.053 52.2 ± 6.0 0.339 ± 0.048 58771.18 ± 9388.03 22.2 ± 3.3 Table 39: Qualitative comparison without marginalization (n=2000n=2000, d=50d=50, degree 4, mechanism additive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.798 ± 0.029 0.961 ± 0.008 0.686 ± 0.039 0.783 ± 0.031 0.942 ± 0.017 0.673 ± 0.040 34.4 ± 5.0 0.099 ± 0.032 2.58 ± 0.10 68.2 ± 4.2 gaussian Spade-Gauss 5 0.808 ± 0.026 0.974 ± 0.007 0.693 ± 0.037 0.805 ± 0.025 0.971 ± 0.009 0.691 ± 0.036 32.0 ± 5.0 0.044 ± 0.025 0.84 ± 0.05 68.0 ± 3.7 gaussian Topic 5 0.760 ± 0.026 0.821 ± 0.012 0.709 ± 0.037 0.751 ± 0.028 0.811 ± 0.016 0.701 ± 0.039 44.2 ± 6.1 0.058 ± 0.035 207.03 ± 13.31 82.4 ± 3.7 gaussian Cam 5 0.676 ± 0.013 0.549 ± 0.014 0.882 ± 0.016 0.673 ± 0.011 0.546 ± 0.013 0.878 ± 0.014 81.0 ± 2.6 0.024 ± 0.011 13761.87 ± 161.24 154.0 ± 5.7 gaussian CauScale 5 0.773 ± 0.021 0.890 ± 0.010 0.686 ± 0.031 0.610 ± 0.022 0.703 ± 0.025 0.541 ± 0.024 52.4 ± 4.3 0.270 ± 0.028 5.95 ± 0.10 73.6 ± 3.4 gaussian SDCD 5 0.686 ± 0.018 0.808 ± 0.007 0.599 ± 0.029 0.546 ± 0.013 0.644 ± 0.018 0.476 ± 0.018 64.0 ± 3.5 0.302 ± 0.020 137.10 ± 2.35 71.0 ± 4.5 gaussian Globe 5 0.648 ± 0.054 0.931 ± 0.028 0.506 ± 0.059 0.642 ± 0.051 0.923 ± 0.024 0.501 ± 0.057 52.6 ± 8.0 0.175 ± 0.044 1918.53 ± 190.59 51.6 ± 5.1 gaussian Das 5 0.577 ± 0.020 0.551 ± 0.026 0.611 ± 0.026 0.439 ± 0.027 0.420 ± 0.032 0.463 ± 0.026 99.2 ± 4.3 0.304 ± 0.024 216.28 ± 1.89 106.2 ± 3.3 gaussian Score 5 0.586 ± 0.019 0.452 ± 0.023 0.847 ± 0.025 0.426 ± 0.029 0.329 ± 0.028 0.612 ± 0.027 137.0 ± 9.9 0.304 ± 0.024 203.54 ± 6.24 181.2 ± 11.3 gaussian NoGAM 5 0.582 ± 0.015 0.446 ± 0.016 0.843 ± 0.029 0.444 ± 0.026 0.341 ± 0.022 0.643 ± 0.038 135.2 ± 8.4 0.273 ± 0.031 1332.70 ± 14.93 181.8 ± 10.4 gaussian Avici 5 0.554 ± 0.030 0.939 ± 0.008 0.395 ± 0.029 0.507 ± 0.031 0.859 ± 0.017 0.362 ± 0.029 64.4 ± 6.7 0.282 ± 0.025 24.35 ± 0.43 39.8 ± 1.4 gaussian NOTEARS 5 0.669 ± 0.019 0.677 ± 0.025 0.663 ± 0.024 0.441 ± 0.031 0.446 ± 0.034 0.437 ± 0.031 85.0 ± 7.1 0.375 ± 0.036 3980.24 ± 617.06 94.0 ± 4.9 gaussian DAGMA 5 0.614 ± 0.015 0.682 ± 0.019 0.562 ± 0.024 0.274 ± 0.025 0.304 ± 0.029 0.250 ± 0.025 97.2 ± 4.7 0.558 ± 0.032 4583.65 ± 216.05 78.8 ± 3.4 gaussian DiffAN 5 0.738 ± 0.004 0.691 ± 0.025 0.802 ± 0.033 0.553 ± 0.020 0.520 ± 0.038 0.597 ± 0.011 73.8 ± 4.8 0.304 ± 0.013 96.70 ± 0.30 112.2 ± 9.2 gaussian SciNO 5 0.641 ± 0.017 0.548 ± 0.028 0.782 ± 0.025 0.342 ± 0.028 0.293 ± 0.029 0.415 ± 0.030 118.4 ± 4.9 0.506 ± 0.029 963.42 ± 91.13 137.8 ± 8.4 gaussian Caps 5 0.691 ± 0.022 0.699 ± 0.024 0.687 ± 0.032 0.512 ± 0.030 0.519 ± 0.034 0.508 ± 0.029 75.6 ± 4.7 0.324 ± 0.030 36.83 ± 1.27 94.4 ± 5.9 gaussian PC 4 0.757 ± 0.018 0.977 ± 0.010 0.619 ± 0.023 0.563 ± 0.017 0.728 ± 0.023 0.460 ± 0.017 55.2 ± 4.4 0.330 ± 0.023 63217.59 ± 9127.11 63.2 ± 5.7 gaussian FCI 3 0.489 ± 0.027 1.000 0.324 ± 0.023 0.441 ± 0.032 0.901 ± 0.020 0.293 ± 0.026 66.7 ± 5.8 0.325 ± 0.045 66011.57 ± 10650.40 30.3 ± 1.9 multimodal Spade-Hist 5 0.876 ± 0.008 0.983 ± 0.004 0.791 ± 0.012 0.876 ± 0.008 0.983 ± 0.004 0.791 ± 0.012 21.6 ± 2.3 0.017 ± 0.007 2.81 ± 0.12 77.4 ± 4.9 multimodal Spade-Gauss 5 0.796 ± 0.018 0.941 ± 0.016 0.691 ± 0.025 0.726 ± 0.025 0.858 ± 0.023 0.631 ± 0.029 40.2 ± 5.3 0.104 ± 0.010 0.91 ± 0.06 70.2 ± 2.9 multimodal Topic 5 0.780 ± 0.010 0.836 ± 0.020 0.734 ± 0.020 0.713 ± 0.017 0.764 ± 0.024 0.671 ± 0.022 45.6 ± 3.2 0.106 ± 0.009 226.09 ± 13.70 83.8 ± 1.9 multimodal Cam 5 0.713 ± 0.018 0.586 ± 0.021 0.914 ± 0.009 0.674 ± 0.019 0.553 ± 0.021 0.864 ± 0.015 75.0 ± 4.4 0.069 ± 0.007 13533.95 ± 145.28 150.0 ± 7.0 multimodal CauScale 5 0.798 ± 0.015 0.929 ± 0.022 0.700 ± 0.016 0.708 ± 0.022 0.824 ± 0.027 0.621 ± 0.022 41.6 ± 3.9 0.149 ± 0.020 5.95 ± 0.11 72.2 ± 3.1 multimodal SDCD 5 0.691 ± 0.011 0.806 ± 0.023 0.606 ± 0.014 0.549 ± 0.027 0.642 ± 0.041 0.481 ± 0.020 64.2 ± 5.3 0.269 ± 0.021 131.02 ± 4.49 72.0 ± 3.0 multimodal Globe 5 0.617 ± 0.029 0.896 ± 0.028 0.479 ± 0.039 0.584 ± 0.024 0.850 ± 0.038 0.452 ± 0.034 59.0 ± 5.4 0.247 ± 0.027 1972.64 ± 149.22 50.8 ± 3.0 multimodal Das 5 0.599 ± 0.021 0.562 ± 0.027 0.645 ± 0.022 0.460 ± 0.031 0.432 ± 0.034 0.494 ± 0.030 97.6 ± 8.6 0.291 ± 0.024 196.81 ± 9.03 111.2 ± 8.6 multimodal Score 5 0.608 ± 0.014 0.467 ± 0.012 0.870 ± 0.019 0.451 ± 0.014 0.347 ± 0.010 0.646 ± 0.025 129.0 ± 6.4 0.291 ± 0.024 211.81 ± 5.00 178.6 ± 9.2 multimodal NoGAM 5 0.632 ± 0.008 0.495 ± 0.009 0.875 ± 0.016 0.492 ± 0.009 0.385 ± 0.009 0.680 ± 0.016 116.0 ± 4.5 0.256 ± 0.007 1276.72 ± 33.07 169.6 ± 9.5 multimodal Avici 5 0.441 ± 0.043 0.942 ± 0.025 0.293 ± 0.038 0.347 ± 0.041 0.738 ± 0.040 0.231 ± 0.034 76.2 ± 7.0 0.482 ± 0.039 25.97 ± 1.73 29.6 ± 3.6 multimodal NOTEARS 5 0.653 ± 0.015 0.636 ± 0.019 0.673 ± 0.022 0.355 ± 0.040 0.347 ± 0.041 0.365 ± 0.040 99.4 ± 11.7 0.452 ± 0.050 4351.67 ± 381.47 101.8 ± 7.1 multimodal DAGMA 5 0.624 ± 0.011 0.667 ± 0.032 0.594 ± 0.029 0.274 ± 0.037 0.295 ± 0.044 0.260 ± 0.034 100.8 ± 8.2 0.566 ± 0.041 4209.50 ± 100.74 85.6 ± 5.1 multimodal DiffAN 5 0.742 ± 0.015 0.682 ± 0.027 0.816 ± 0.008 0.562 ± 0.017 0.517 ± 0.024 0.618 ± 0.012 73.0 ± 2.2 0.278 ± 0.017 94.21 ± 0.78 114.6 ± 2.7 multimodal SciNO 5 0.632 ± 0.023 0.530 ± 0.035 0.791 ± 0.008 0.374 ± 0.025 0.314 ± 0.030 0.465 ± 0.013 120.0 ± 9.8 0.459 ± 0.025 861.23 ± 34.22 145.0 ± 10.5 multimodal Caps 5 0.697 ± 0.010 0.705 ± 0.011 0.689 ± 0.012 0.530 ± 0.027 0.536 ± 0.026 0.525 ± 0.028 73.6 ± 6.2 0.318 ± 0.025 39.22 ± 2.02 94.2 ± 6.9 multimodal PC 4 0.742 ± 0.032 0.947 ± 0.011 0.612 ± 0.039 0.526 ± 0.027 0.672 ± 0.035 0.433 ± 0.025 54.5 ± 2.1 0.340 ± 0.025 69080.88 ± 9697.23 59.2 ± 5.8 multimodal FCI 4 0.514 ± 0.022 1.000 0.347 ± 0.021 0.446 ± 0.025 0.869 ± 0.042 0.301 ± 0.020 64.0 ± 4.9 0.290 ± 0.025 71612.79 ± 9222.01 31.5 ± 1.5 unimodal_asymmetric Spade-Hist 5 0.829 ± 0.014 0.972 ± 0.011 0.725 ± 0.024 0.826 ± 0.013 0.969 ± 0.011 0.722 ± 0.023 29.2 ± 3.5 0.054 ± 0.013 2.84 ± 0.10 71.2 ± 2.3 unimodal_asymmetric Spade-Gauss 5 0.777 ± 0.017 0.894 ± 0.025 0.690 ± 0.025 0.695 ± 0.029 0.802 ± 0.045 0.616 ± 0.025 45.0 ± 4.4 0.148 ± 0.019 0.91 ± 0.03 73.6 ± 1.4 unimodal_asymmetric Topic 5 0.744 ± 0.017 0.789 ± 0.020 0.706 ± 0.026 0.662 ± 0.021 0.704 ± 0.030 0.628 ± 0.021 54.0 ± 4.0 0.151 ± 0.009 226.76 ± 12.35 85.4 ± 1.6 unimodal_asymmetric Cam 5 0.623 ± 0.016 0.481 ± 0.022 0.893 ± 0.017 0.580 ± 0.022 0.448 ± 0.024 0.830 ± 0.024 109.0 ± 4.6 0.074 ± 0.014 13282.65 ± 62.75 178.0 ± 4.0 unimodal_asymmetric CauScale 5 0.776 ± 0.025 0.894 ± 0.020 0.688 ± 0.034 0.585 ± 0.018 0.677 ± 0.030 0.518 ± 0.016 54.6 ± 5.2 0.316 ± 0.022 5.86 ± 0.06 73.6 ± 3.7 unimodal_asymmetric SDCD 5 0.718 ± 0.022 0.796 ± 0.025 0.657 ± 0.029 0.551 ± 0.011 0.611 ± 0.011 0.504 ± 0.020 64.4 ± 5.9 0.292 ± 0.025 141.35 ± 2.71 79.0 ± 3.9 unimodal_asymmetric Globe 5 0.523 ± 0.038 0.897 ± 0.014 0.374 ± 0.040 0.464 ± 0.044 0.792 ± 0.027 0.333 ± 0.043 68.4 ± 5.6 0.338 ± 0.040 1237.48 ± 113.95 39.8 ± 4.0 unimodal_asymmetric Das 5 0.614 ± 0.014 0.569 ± 0.017 0.672 ± 0.027 0.456 ± 0.024 0.425 ± 0.030 0.496 ± 0.019 97.2 ± 5.5 0.306 ± 0.041 217.98 ± 2.54 113.0 ± 5.7 unimodal_asymmetric Score 5 0.577 ± 0.016 0.432 ± 0.018 0.875 ± 0.013 0.414 ± 0.031 0.311 ± 0.028 0.623 ± 0.029 146.4 ± 7.3 0.306 ± 0.041 212.08 ± 7.57 194.6 ± 7.2 unimodal_asymmetric NoGAM 5 0.588 ± 0.019 0.443 ± 0.023 0.879 ± 0.012 0.439 ± 0.035 0.333 ± 0.032 0.652 ± 0.029 139.0 ± 6.5 0.281 ± 0.040 1316.89 ± 5.53 190.4 ± 4.8 unimodal_asymmetric Avici 5 0.417 ± 0.016 0.958 ± 0.023 0.267 ± 0.012 0.300 ± 0.021 0.685 ± 0.028 0.192 ± 0.015 79.0 ± 5.5 0.475 ± 0.018 24.28 ± 0.36 26.8 ± 2.0 unimodal_asymmetric NOTEARS 5 0.642 ± 0.026 0.614 ± 0.017 0.675 ± 0.040 0.367 ± 0.020 0.351 ± 0.014 0.386 ± 0.027 99.8 ± 7.5 0.427 ± 0.034 4499.62 ± 570.26 104.6 ± 3.8 unimodal_asymmetric DAGMA 5 0.638 ± 0.010 0.669 ± 0.018 0.615 ± 0.026 0.278 ± 0.022 0.292 ± 0.023 0.268 ± 0.024 99.6 ± 4.1 0.541 ± 0.031 4426.80 ± 117.95 87.6 ± 1.6 unimodal_asymmetric DiffAN 5 0.672 ± 0.004 0.576 ± 0.007 0.807 ± 0.018 0.457 ± 0.019 0.393 ± 0.022 0.547 ± 0.013 100.0 ± 3.6 0.336 ± 0.027 96.21 ± 0.95 133.8 ± 4.9 unimodal_asymmetric SciNO 5 0.630 ± 0.012 0.523 ± 0.015 0.797 ± 0.020 0.336 ± 0.021 0.280 ± 0.021 0.424 ± 0.023 126.2 ± 10.5 0.486 ± 0.025 874.53 ± 37.91 146.8 ± 9.2 unimodal_asymmetric Caps 5 0.686 ± 0.012 0.666 ± 0.018 0.714 ± 0.032 0.504 ± 0.024 0.492 ± 0.036 0.521 ± 0.014 80.8 ± 6.4 0.317 ± 0.037 45.52 ± 1.37 102.6 ± 5.7 unimodal_asymmetric PC 4 0.778 ± 0.021 0.947 ± 0.015 0.662 ± 0.027 0.603 ± 0.022 0.734 ± 0.026 0.512 ± 0.023 47.8 ± 3.0 0.288 ± 0.008 68533.96 ± 8115.21 63.5 ± 2.2 unimodal_asymmetric FCI 3 0.411 ± 0.066 0.987 ± 0.013 0.264 ± 0.053 0.368 ± 0.056 0.888 ± 0.027 0.236 ± 0.045 71.7 ± 1.2 0.349 ± 0.049 70284.22 ± 4960.23 25.7 ± 6.4 unimodal_symmetric Spade-Hist 5 0.825 ± 0.014 0.966 ± 0.007 0.721 ± 0.024 0.825 ± 0.014 0.966 ± 0.007 0.721 ± 0.024 29.4 ± 2.9 0.038 ± 0.011 2.74 ± 0.14 71.6 ± 3.9 unimodal_symmetric Spade-Gauss 5 0.798 ± 0.019 0.931 ± 0.025 0.705 ± 0.037 0.767 ± 0.017 0.896 ± 0.036 0.675 ± 0.027 36.6 ± 3.3 0.063 ± 0.011 0.90 ± 0.09 72.6 ± 4.7 unimodal_symmetric Topic 5 0.781 ± 0.013 0.833 ± 0.031 0.741 ± 0.025 0.741 ± 0.021 0.790 ± 0.031 0.703 ± 0.032 43.0 ± 2.3 0.093 ± 0.028 225.83 ± 19.85 85.2 ± 3.2 unimodal_symmetric Cam 5 0.627 ± 0.012 0.484 ± 0.018 0.896 ± 0.026 0.600 ± 0.016 0.464 ± 0.021 0.856 ± 0.020 105.2 ± 2.1 0.055 ± 0.009 13373.40 ± 144.10 177.0 ± 4.6 unimodal_symmetric CauScale 5 0.786 ± 0.017 0.883 ± 0.013 0.711 ± 0.030 0.610 ± 0.030 0.683 ± 0.017 0.553 ± 0.040 52.6 ± 5.4 0.270 ± 0.023 5.96 ± 0.05 76.8 ± 3.1 unimodal_symmetric SDCD 5 0.664 ± 0.026 0.754 ± 0.023 0.595 ± 0.032 0.549 ± 0.026 0.623 ± 0.020 0.493 ± 0.033 68.0 ± 7.2 0.246 ± 0.022 136.83 ± 0.78 75.2 ± 2.9 unimodal_symmetric Globe 5 0.593 ± 0.016 0.899 ± 0.017 0.444 ± 0.020 0.578 ± 0.016 0.876 ± 0.021 0.433 ± 0.019 59.6 ± 4.4 0.277 ± 0.009 1683.70 ± 97.45 47.6 ± 3.6 unimodal_symmetric Das 5 0.588 ± 0.026 0.555 ± 0.026 0.629 ± 0.038 0.454 ± 0.029 0.428 ± 0.023 0.488 ± 0.042 97.4 ± 4.5 0.304 ± 0.049 219.18 ± 5.20 108.6 ± 6.2 unimodal_symmetric Score 5 0.556 ± 0.012 0.414 ± 0.014 0.848 ± 0.020 0.414 ± 0.031 0.309 ± 0.024 0.631 ± 0.046 150.0 ± 4.3 0.304 ± 0.049 198.95 ± 8.14 196.0 ± 5.2 unimodal_symmetric NoGAM 5 0.566 ± 0.015 0.425 ± 0.016 0.851 ± 0.019 0.433 ± 0.034 0.325 ± 0.027 0.649 ± 0.050 143.6 ± 6.9 0.290 ± 0.051 1285.58 ± 6.54 191.4 ± 4.8 unimodal_symmetric Avici 5 0.582 ± 0.032 0.927 ± 0.012 0.426 ± 0.032 0.544 ± 0.025 0.869 ± 0.012 0.398 ± 0.027 61.6 ± 6.4 0.270 ± 0.025 24.89 ± 0.50 43.6 ± 2.2 unimodal_symmetric NOTEARS 5 0.658 ± 0.018 0.637 ± 0.026 0.686 ± 0.028 0.473 ± 0.028 0.459 ± 0.036 0.490 ± 0.020 87.0 ± 7.2 0.316 ± 0.050 4708.45 ± 315.28 103.2 ± 4.6 unimodal_symmetric DAGMA 5 0.618 ± 0.028 0.698 ± 0.020 0.557 ± 0.034 0.366 ± 0.019 0.413 ± 0.017 0.329 ± 0.021 88.0 ± 7.8 0.416 ± 0.041 4738.06 ± 195.30 75.8 ± 2.3 unimodal_symmetric DiffAN 5 0.672 ± 0.019 0.581 ± 0.023 0.802 ± 0.035 0.504 ± 0.041 0.435 ± 0.034 0.603 ± 0.060 93.4 ± 4.9 0.326 ± 0.058 95.41 ± 1.41 131.8 ± 3.7 unimodal_symmetric SciNO 5 0.622 ± 0.019 0.520 ± 0.019 0.778 ± 0.025 0.360 ± 0.029 0.300 ± 0.024 0.451 ± 0.039 123.2 ± 12.6 0.458 ± 0.039 868.74 ± 25.80 144.0 ± 8.2 unimodal_symmetric Caps 5 0.686 ± 0.020 0.665 ± 0.021 0.715 ± 0.036 0.529 ± 0.035 0.511 ± 0.028 0.552 ± 0.049 77.4 ± 4.0 0.314 ± 0.042 40.82 ± 0.88 102.8 ± 5.0 unimodal_symmetric PC 3 0.727 ± 0.013 0.952 ± 0.015 0.587 ± 0.012 0.532 ± 0.033 0.697 ± 0.041 0.430 ± 0.027 56.0 ± 1.0 0.326 ± 0.038 61772.78 ± 11417.61 58.0 ± 3.5 unimodal_symmetric FCI 3 0.501 ± 0.017 0.990 ± 0.010 0.336 ± 0.016 0.464 ± 0.018 0.917 ± 0.003 0.311 ± 0.017 65.0 ± 3.1 0.249 ± 0.042 67949.90 ± 13056.85 32.0 ± 3.0 Table 40: Qualitative comparison without marginalization (n=2000n=2000, d=50d=50, degree 4, mechanism neveradditive). noise method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges gaussian Spade-Hist 5 0.624 ± 0.026 0.948 ± 0.017 0.469 ± 0.033 0.554 ± 0.025 0.842 ± 0.026 0.416 ± 0.030 58.8 ± 4.5 0.366 ± 0.023 1.95 ± 0.09 47.6 ± 4.3 gaussian Spade-Gauss 5 0.683 ± 0.034 0.927 ± 0.012 0.546 ± 0.045 0.665 ± 0.033 0.904 ± 0.019 0.532 ± 0.044 49.4 ± 5.5 0.162 ± 0.025 0.62 ± 0.04 56.6 ± 5.3 gaussian Topic 5 0.671 ± 0.024 0.761 ± 0.010 0.603 ± 0.037 0.597 ± 0.029 0.678 ± 0.027 0.537 ± 0.037 62.8 ± 5.3 0.208 ± 0.042 167.81 ± 15.06 75.6 ± 4.3 gaussian Cam 5 0.567 ± 0.004 0.417 ± 0.006 0.888 ± 0.014 0.507 ± 0.009 0.373 ± 0.008 0.794 ± 0.018 139.6 ± 9.1 0.128 ± 0.019 13762.52 ± 185.92 205.2 ± 13.5 gaussian CauScale 5 0.686 ± 0.032 0.859 ± 0.033 0.577 ± 0.042 0.523 ± 0.054 0.655 ± 0.066 0.439 ± 0.050 63.2 ± 6.2 0.287 ± 0.038 6.20 ± 0.14 64.2 ± 4.5 gaussian SDCD 5 0.622 ± 0.025 0.827 ± 0.022 0.504 ± 0.035 0.512 ± 0.029 0.678 ± 0.013 0.416 ± 0.036 66.6 ± 4.6 0.336 ± 0.027 132.93 ± 5.03 58.8 ± 5.3 gaussian Globe 5 0.421 ± 0.021 0.871 ± 0.041 0.281 ± 0.021 0.364 ± 0.025 0.750 ± 0.033 0.244 ± 0.023 77.4 ± 5.3 0.444 ± 0.024 1137.90 ± 88.59 30.8 ± 1.8 gaussian Das 5 0.560 ± 0.020 0.546 ± 0.017 0.578 ± 0.029 0.429 ± 0.012 0.418 ± 0.010 0.443 ± 0.020 100.2 ± 7.7 0.301 ± 0.017 231.31 ± 5.21 102.2 ± 8.3 gaussian Score 5 0.551 ± 0.016 0.408 ± 0.014 0.848 ± 0.017 0.406 ± 0.016 0.301 ± 0.013 0.625 ± 0.024 154.8 ± 11.6 0.301 ± 0.017 211.34 ± 7.14 200.4 ± 14.0 gaussian NoGAM 5 0.563 ± 0.017 0.420 ± 0.015 0.854 ± 0.017 0.404 ± 0.022 0.302 ± 0.018 0.614 ± 0.033 151.0 ± 11.3 0.333 ± 0.031 1309.17 ± 29.22 195.8 ± 11.6 gaussian Avici 5 0.449 ± 0.045 0.967 ± 0.014 0.297 ± 0.038 0.383 ± 0.054 0.810 ± 0.040 0.255 ± 0.042 73.6 ± 8.4 0.492 ± 0.044 23.99 ± 0.55 28.8 ± 2.6 gaussian NOTEARS 5 0.666 ± 0.025 0.684 ± 0.022 0.655 ± 0.039 0.485 ± 0.031 0.498 ± 0.031 0.476 ± 0.035 79.2 ± 5.1 0.338 ± 0.033 2723.67 ± 370.65 92.4 ± 8.2 gaussian DAGMA 5 0.607 ± 0.010 0.746 ± 0.035 0.519 ± 0.027 0.371 ± 0.019 0.459 ± 0.040 0.315 ± 0.015 83.6 ± 4.8 0.482 ± 0.026 4736.97 ± 291.11 67.0 ± 4.6 gaussian DiffAN 5 0.680 ± 0.035 0.621 ± 0.045 0.759 ± 0.026 0.461 ± 0.041 0.421 ± 0.043 0.514 ± 0.039 92.8 ± 10.2 0.370 ± 0.035 95.74 ± 0.85 119.0 ± 9.0 gaussian SciNO 5 0.655 ± 0.031 0.591 ± 0.049 0.747 ± 0.023 0.413 ± 0.032 0.374 ± 0.043 0.468 ± 0.018 104.8 ± 14.8 0.431 ± 0.018 877.42 ± 33.89 125.4 ± 14.4 gaussian Caps 5 0.688 ± 0.021 0.672 ± 0.031 0.710 ± 0.024 0.485 ± 0.018 0.473 ± 0.020 0.501 ± 0.024 82.4 ± 8.0 0.367 ± 0.016 44.43 ± 3.00 102.4 ± 8.2 gaussian PC 5 0.784 ± 0.022 0.950 ± 0.017 0.669 ± 0.030 0.516 ± 0.025 0.624 ± 0.019 0.441 ± 0.028 57.4 ± 5.3 0.421 ± 0.015 58241.53 ± 8347.31 67.4 ± 3.4 gaussian FCI 5 0.304 ± 0.031 0.991 ± 0.009 0.181 ± 0.022 0.243 ± 0.030 0.789 ± 0.044 0.145 ± 0.021 82.8 ± 6.7 0.481 ± 0.035 66500.16 ± 9240.37 17.4 ± 1.9 multimodal Spade-Hist 5 0.651 ± 0.023 0.949 ± 0.011 0.497 ± 0.025 0.641 ± 0.028 0.933 ± 0.018 0.489 ± 0.028 52.4 ± 6.4 0.173 ± 0.020 2.09 ± 0.09 49.8 ± 1.5 multimodal Spade-Gauss 5 0.667 ± 0.027 0.887 ± 0.018 0.537 ± 0.032 0.565 ± 0.019 0.754 ± 0.029 0.455 ± 0.021 59.4 ± 5.6 0.239 ± 0.010 0.65 ± 0.04 57.6 ± 2.4 multimodal Topic 5 0.650 ± 0.017 0.727 ± 0.013 0.591 ± 0.030 0.541 ± 0.015 0.608 ± 0.024 0.492 ± 0.020 70.4 ± 4.6 0.259 ± 0.013 185.76 ± 20.30 77.4 ± 2.7 multimodal Cam 5 0.585 ± 0.012 0.442 ± 0.016 0.871 ± 0.020 0.501 ± 0.006 0.378 ± 0.010 0.747 ± 0.021 130.8 ± 7.9 0.171 ± 0.025 14014.65 ± 109.09 189.6 ± 10.4 multimodal CauScale 5 0.738 ± 0.020 0.906 ± 0.004 0.624 ± 0.028 0.621 ± 0.027 0.762 ± 0.026 0.525 ± 0.030 52.4 ± 5.6 0.210 ± 0.021 6.08 ± 0.14 65.6 ± 1.7 multimodal SDCD 5 0.663 ± 0.015 0.856 ± 0.027 0.546 ± 0.026 0.535 ± 0.023 0.693 ± 0.045 0.439 ± 0.021 63.2 ± 4.3 0.346 ± 0.027 137.54 ± 0.54 61.0 ± 3.0 multimodal Globe 5 0.486 ± 0.032 0.849 ± 0.037 0.345 ± 0.031 0.452 ± 0.026 0.795 ± 0.046 0.321 ± 0.025 71.8 ± 5.5 0.394 ± 0.025 1691.39 ± 159.18 38.6 ± 2.3 multimodal Das 5 0.545 ± 0.012 0.526 ± 0.018 0.571 ± 0.028 0.429 ± 0.015 0.414 ± 0.007 0.451 ± 0.032 102.6 ± 5.2 0.240 ± 0.029 208.90 ± 6.73 104.0 ± 6.3 multimodal Score 5 0.567 ± 0.012 0.427 ± 0.016 0.851 ± 0.020 0.434 ± 0.016 0.326 ± 0.012 0.654 ± 0.038 143.8 ± 8.3 0.240 ± 0.029 223.21 ± 14.58 192.2 ± 12.4 multimodal NoGAM 5 0.572 ± 0.006 0.438 ± 0.011 0.831 ± 0.020 0.451 ± 0.017 0.344 ± 0.010 0.658 ± 0.041 136.0 ± 7.4 0.238 ± 0.038 1260.25 ± 15.65 182.6 ± 10.9 multimodal Avici 5 0.397 ± 0.035 0.881 ± 0.026 0.260 ± 0.029 0.285 ± 0.032 0.625 ± 0.014 0.187 ± 0.025 82.0 ± 6.5 0.486 ± 0.032 24.90 ± 0.92 28.0 ± 2.8 multimodal NOTEARS 5 0.677 ± 0.020 0.704 ± 0.036 0.662 ± 0.041 0.479 ± 0.041 0.501 ± 0.055 0.468 ± 0.044 78.6 ± 6.3 0.368 ± 0.030 4508.91 ± 1050.08 90.2 ± 5.2 multimodal DAGMA 5 0.638 ± 0.035 0.752 ± 0.019 0.566 ± 0.055 0.351 ± 0.040 0.409 ± 0.028 0.314 ± 0.047 84.8 ± 7.5 0.522 ± 0.050 4552.31 ± 148.16 71.0 ± 4.4 multimodal DiffAN 5 0.704 ± 0.022 0.671 ± 0.037 0.747 ± 0.014 0.524 ± 0.016 0.498 ± 0.024 0.556 ± 0.019 79.0 ± 6.2 0.311 ± 0.019 95.18 ± 1.00 107.6 ± 6.0 multimodal SciNO 5 0.619 ± 0.028 0.542 ± 0.040 0.732 ± 0.026 0.326 ± 0.018 0.286 ± 0.024 0.385 ± 0.017 121.0 ± 10.6 0.481 ± 0.025 997.24 ± 80.39 131.6 ± 10.1 multimodal Caps 5 0.689 ± 0.016 0.708 ± 0.026 0.674 ± 0.019 0.519 ± 0.019 0.533 ± 0.021 0.508 ± 0.023 74.8 ± 6.6 0.286 ± 0.013 44.10 ± 1.25 91.4 ± 4.5 multimodal PC 4 0.778 ± 0.015 0.961 ± 0.005 0.654 ± 0.021 0.493 ± 0.039 0.610 ± 0.047 0.415 ± 0.035 59.0 ± 5.2 0.419 ± 0.031 53831.09 ± 7653.20 65.8 ± 5.7 multimodal FCI 4 0.367 ± 0.058 0.965 ± 0.013 0.232 ± 0.048 0.304 ± 0.032 0.815 ± 0.035 0.191 ± 0.029 79.5 ± 8.3 0.432 ± 0.023 61389.40 ± 8807.89 22.5 ± 3.4 unimodal_asymmetric Spade-Hist 5 0.653 ± 0.047 0.968 ± 0.014 0.497 ± 0.049 0.625 ± 0.037 0.930 ± 0.009 0.475 ± 0.041 52.2 ± 5.9 0.205 ± 0.022 2.13 ± 0.07 48.8 ± 4.2 unimodal_asymmetric Spade-Gauss 5 0.684 ± 0.027 0.857 ± 0.017 0.570 ± 0.032 0.557 ± 0.027 0.698 ± 0.020 0.465 ± 0.030 61.2 ± 6.7 0.315 ± 0.027 0.75 ± 0.04 63.6 ± 3.9 unimodal_asymmetric Topic 5 0.665 ± 0.021 0.733 ± 0.017 0.611 ± 0.030 0.542 ± 0.030 0.598 ± 0.030 0.498 ± 0.034 70.0 ± 6.4 0.296 ± 0.036 185.60 ± 7.41 79.4 ± 2.8 unimodal_asymmetric Cam 5 0.535 ± 0.025 0.383 ± 0.023 0.894 ± 0.016 0.413 ± 0.028 0.296 ± 0.023 0.690 ± 0.034 171.0 ± 16.8 0.263 ± 0.033 13993.95 ± 132.63 227.0 ± 16.2 unimodal_asymmetric CauScale 5 0.718 ± 0.021 0.864 ± 0.014 0.618 ± 0.033 0.581 ± 0.031 0.700 ± 0.034 0.500 ± 0.035 57.6 ± 4.7 0.241 ± 0.032 6.16 ± 0.13 68.0 ± 1.7 unimodal_asymmetric SDCD 5 0.634 ± 0.052 0.756 ± 0.031 0.552 ± 0.060 0.502 ± 0.059 0.594 ± 0.050 0.438 ± 0.060 70.8 ± 8.0 0.356 ± 0.039 138.01 ± 1.29 69.4 ± 6.8 unimodal_asymmetric Globe 5 0.444 ± 0.041 0.804 ± 0.022 0.311 ± 0.038 0.360 ± 0.051 0.642 ± 0.042 0.254 ± 0.043 79.8 ± 8.3 0.490 ± 0.038 1311.30 ± 101.72 36.4 ± 3.0 unimodal_asymmetric Das 5 0.562 ± 0.027 0.525 ± 0.031 0.607 ± 0.028 0.458 ± 0.027 0.428 ± 0.028 0.495 ± 0.030 102.2 ± 9.5 0.260 ± 0.030 230.69 ± 7.25 111.6 ± 7.4 unimodal_asymmetric Score 5 0.538 ± 0.022 0.389 ± 0.022 0.884 ± 0.015 0.409 ± 0.029 0.296 ± 0.025 0.671 ± 0.037 167.2 ± 14.4 0.260 ± 0.030 239.19 ± 12.99 220.2 ± 14.2 unimodal_asymmetric NoGAM 5 0.547 ± 0.021 0.398 ± 0.022 0.885 ± 0.012 0.407 ± 0.032 0.297 ± 0.028 0.655 ± 0.031 164.0 ± 16.2 0.282 ± 0.028 1314.14 ± 13.50 216.0 ± 16.6 unimodal_asymmetric Avici 5 0.468 ± 0.056 0.903 ± 0.030 0.323 ± 0.052 0.362 ± 0.048 0.697 ± 0.044 0.250 ± 0.043 75.8 ± 7.5 0.441 ± 0.050 25.37 ± 0.69 33.2 ± 3.2 unimodal_asymmetric NOTEARS 5 0.636 ± 0.027 0.586 ± 0.028 0.695 ± 0.030 0.470 ± 0.018 0.434 ± 0.019 0.514 ± 0.018 94.6 ± 9.1 0.391 ± 0.020 2843.95 ± 238.83 114.6 ± 9.4 unimodal_asymmetric DAGMA 5 0.602 ± 0.030 0.672 ± 0.034 0.547 ± 0.032 0.374 ± 0.035 0.418 ± 0.040 0.340 ± 0.033 88.8 ± 5.4 0.500 ± 0.035 4445.28 ± 179.02 77.8 ± 2.7 unimodal_asymmetric DiffAN 5 0.657 ± 0.019 0.570 ± 0.030 0.784 ± 0.021 0.449 ± 0.022 0.391 ± 0.029 0.534 ± 0.014 103.2 ± 10.1 0.352 ± 0.027 95.50 ± 1.60 134.2 ± 13.2 unimodal_asymmetric SciNO 5 0.613 ± 0.018 0.507 ± 0.019 0.778 ± 0.020 0.327 ± 0.031 0.271 ± 0.027 0.414 ± 0.036 130.4 ± 13.3 0.482 ± 0.034 873.99 ± 37.31 148.2 ± 11.1 unimodal_asymmetric Caps 5 0.683 ± 0.014 0.635 ± 0.015 0.739 ± 0.015 0.522 ± 0.029 0.486 ± 0.028 0.565 ± 0.031 82.6 ± 6.2 0.267 ± 0.025 56.16 ± 0.86 112.0 ± 7.2 unimodal_asymmetric PC 3 0.768 ± 0.023 0.953 ± 0.009 0.646 ± 0.036 0.556 ± 0.018 0.690 ± 0.011 0.467 ± 0.027 53.3 ± 4.6 0.331 ± 0.013 53802.59 ± 10849.01 63.3 ± 0.7 unimodal_asymmetric FCI 3 0.412 ± 0.007 1.000 0.259 ± 0.006 0.349 ± 0.012 0.848 ± 0.039 0.220 ± 0.007 73.3 ± 4.1 0.393 ± 0.016 59385.60 ± 14714.03 24.3 ± 0.9 unimodal_symmetric Spade-Hist 5 0.658 ± 0.038 0.927 ± 0.023 0.517 ± 0.048 0.625 ± 0.033 0.883 ± 0.035 0.490 ± 0.040 53.2 ± 5.2 0.221 ± 0.019 2.18 ± 0.10 53.0 ± 3.7 unimodal_symmetric Spade-Gauss 5 0.697 ± 0.041 0.907 ± 0.024 0.573 ± 0.052 0.669 ± 0.049 0.868 ± 0.033 0.551 ± 0.057 49.4 ± 7.0 0.113 ± 0.028 0.69 ± 0.04 60.2 ± 4.6 unimodal_symmetric Topic 5 0.661 ± 0.034 0.743 ± 0.033 0.599 ± 0.044 0.594 ± 0.040 0.667 ± 0.036 0.540 ± 0.049 64.0 ± 5.5 0.192 ± 0.013 181.02 ± 13.79 76.8 ± 3.4 unimodal_symmetric Cam 5 0.562 ± 0.019 0.412 ± 0.020 0.889 ± 0.022 0.499 ± 0.029 0.366 ± 0.025 0.789 ± 0.041 141.8 ± 5.1 0.133 ± 0.031 13775.31 ± 162.19 208.0 ± 11.2 unimodal_symmetric CauScale 5 0.704 ± 0.025 0.847 ± 0.026 0.603 ± 0.028 0.527 ± 0.020 0.634 ± 0.015 0.452 ± 0.024 63.2 ± 4.7 0.293 ± 0.015 5.82 ± 0.11 68.0 ± 2.7 unimodal_symmetric SDCD 5 0.623 ± 0.031 0.803 ± 0.021 0.518 ± 0.044 0.496 ± 0.023 0.640 ± 0.021 0.411 ± 0.034 69.4 ± 5.0 0.381 ± 0.032 134.29 ± 3.66 61.8 ± 6.0 unimodal_symmetric Globe 5 0.489 ± 0.047 0.864 ± 0.030 0.347 ± 0.044 0.453 ± 0.054 0.796 ± 0.042 0.322 ± 0.048 71.2 ± 8.2 0.380 ± 0.043 1329.81 ± 101.42 37.8 ± 3.4 unimodal_symmetric Das 5 0.532 ± 0.017 0.507 ± 0.038 0.569 ± 0.011 0.394 ± 0.027 0.377 ± 0.038 0.420 ± 0.021 111.0 ± 8.3 0.334 ± 0.040 212.85 ± 4.65 110.0 ± 8.9 unimodal_symmetric Score 5 0.554 ± 0.018 0.412 ± 0.021 0.856 ± 0.020 0.391 ± 0.031 0.291 ± 0.027 0.603 ± 0.038 156.2 ± 9.3 0.334 ± 0.040 208.79 ± 8.74 201.4 ± 13.6 unimodal_symmetric NoGAM 5 0.556 ± 0.018 0.412 ± 0.020 0.863 ± 0.023 0.398 ± 0.024 0.295 ± 0.020 0.618 ± 0.036 155.4 ± 8.1 0.325 ± 0.037 1336.56 ± 5.71 202.8 ± 14.4 unimodal_symmetric Avici 5 0.517 ± 0.050 0.952 ± 0.014 0.363 ± 0.049 0.468 ± 0.046 0.860 ± 0.018 0.328 ± 0.044 66.8 ± 6.5 0.363 ± 0.043 25.44 ± 1.03 36.2 ± 4.2 unimodal_symmetric NOTEARS 5 0.624 ± 0.029 0.620 ± 0.029 0.640 ± 0.047 0.468 ± 0.039 0.465 ± 0.037 0.479 ± 0.048 89.2 ± 9.5 0.340 ± 0.035 3593.50 ± 545.52 99.4 ± 9.1 unimodal_symmetric DAGMA 5 0.585 ± 0.028 0.688 ± 0.022 0.519 ± 0.050 0.306 ± 0.031 0.360 ± 0.032 0.272 ± 0.035 92.8 ± 4.5 0.504 ± 0.038 4247.84 ± 181.24 71.8 ± 5.7 unimodal_symmetric DiffAN 5 0.680 ± 0.018 0.605 ± 0.032 0.786 ± 0.022 0.495 ± 0.019 0.440 ± 0.026 0.572 ± 0.020 91.6 ± 6.2 0.326 ± 0.020 94.22 ± 0.61 126.2 ± 9.1 unimodal_symmetric SciNO 5 0.607 ± 0.009 0.523 ± 0.021 0.734 ± 0.028 0.368 ± 0.016 0.318 ± 0.023 0.442 ± 0.008 118.4 ± 5.9 0.458 ± 0.013 865.99 ± 35.85 135.2 ± 8.7 unimodal_symmetric Caps 5 0.682 ± 0.021 0.667 ± 0.040 0.705 ± 0.023 0.475 ± 0.029 0.465 ± 0.038 0.491 ± 0.027 83.4 ± 5.1 0.349 ± 0.042 47.04 ± 1.49 102.8 ± 7.7 unimodal_symmetric PC 3 0.806 ± 0.041 0.968 ± 0.012 0.693 ± 0.059 0.540 ± 0.027 0.649 ± 0.006 0.464 ± 0.039 52.3 ± 4.3 0.419 ± 0.009 48241.10 ± 11727.02 67.3 ± 6.7 unimodal_symmetric FCI 3 0.355 ± 0.033 1.000 0.217 ± 0.025 0.314 ± 0.037 0.881 ± 0.022 0.192 ± 0.027 76.0 ± 5.5 0.347 ± 0.020 57868.23 ± 14074.27 20.3 ± 2.4 Table 41: Real-world benchmark table for the Causal Chamber setting used in the main paper. dataset n_samples n_features method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges causal_chamber 10000 23 Spade-Hist 5 0.514 ± 0.009 0.591 ± 0.013 0.456 ± 0.013 0.502 ± 0.014 0.577 ± 0.014 0.446 ± 0.017 34.0 ± 0.7 0.087 ± 0.006 0.97 ± 0.09 30.2 ± 1.2 causal_chamber 10000 23 Topic 5 0.471 ± 0.010 0.409 ± 0.010 0.554 ± 0.010 0.188 ± 0.016 0.164 ± 0.015 0.221 ± 0.017 61.6 ± 1.5 0.431 ± 0.032 285.54 ± 17.29 52.8 ± 0.7 causal_chamber 10000 23 Cam 5 0.334 ± 0.002 0.212 ± 0.001 0.795 ± 0.008 0.082 ± 0.010 0.052 ± 0.007 0.195 ± 0.025 146.8 ± 1.7 0.656 ± 0.031 1861.98 ± 23.31 146.4 ± 2.0 causal_chamber 10000 23 CauScale 5 0.365 ± 0.020 0.335 ± 0.024 0.405 ± 0.022 0.274 ± 0.014 0.253 ± 0.020 0.303 ± 0.005 59.2 ± 3.5 0.267 ± 0.022 10.00 ± 2.62 47.8 ± 3.5 causal_chamber 10000 23 SDCD 5 0.628 ± 0.005 0.785 ± 0.011 0.523 ± 0.006 0.493 ± 0.015 0.617 ± 0.023 0.410 ± 0.011 28.6 ± 0.7 0.123 ± 0.010 519.69 ± 15.35 26.0 ± 0.5 causal_chamber 10000 23 Globe 5 0.254 ± 0.025 0.483 ± 0.039 0.174 ± 0.022 0.124 ± 0.037 0.253 ± 0.077 0.082 ± 0.025 43.2 ± 1.8 0.221 ± 0.076 1535.25 ± 188.01 14.2 ± 1.6 causal_chamber 10000 23 Das 5 0.334 ± 0.015 0.228 ± 0.011 0.621 ± 0.029 0.107 ± 0.006 0.073 ± 0.004 0.200 ± 0.013 113.2 ± 2.9 0.769 4909.44 ± 36.80 106.2 ± 3.1 causal_chamber 10000 23 Score 5 0.362 ± 0.008 0.232 ± 0.007 0.831 ± 0.024 0.082 ± 0.002 0.053 ± 0.001 0.190 ± 0.010 139.6 ± 5.2 0.769 3043.07 ± 48.84 140.4 ± 6.3 causal_chamber 10000 23 NoGAM 5 0.367 ± 0.010 0.235 ± 0.007 0.846 ± 0.034 0.064 ± 0.004 0.041 ± 0.003 0.149 ± 0.013 141.2 ± 5.3 0.841 ± 0.005 16487.01 ± 206.37 141.0 ± 6.5 causal_chamber 10000 23 Avici 5 0.076 ± 0.018 0.533 ± 0.086 0.041 ± 0.010 0.000 0.000 0.000 40.4 ± 0.4 0.554 ± 0.041 233.68 ± 1.39 3.0 ± 0.4 causal_chamber 10000 23 NOTEARS 5 0.510 ± 0.020 0.543 ± 0.029 0.482 ± 0.015 0.178 ± 0.017 0.189 ± 0.016 0.169 ± 0.017 48.4 ± 1.2 0.426 ± 0.031 14160.34 ± 306.35 34.8 ± 1.0 causal_chamber 10000 23 DAGMA 5 0.177 ± 0.026 0.664 ± 0.078 0.103 ± 0.016 0.017 ± 0.017 0.050 ± 0.050 0.010 ± 0.010 40.6 ± 0.6 0.713 ± 0.047 11392.53 ± 333.34 6.0 ± 0.6 causal_chamber 10000 23 DiffAN 5 0.476 ± 0.015 0.367 ± 0.014 0.682 ± 0.019 0.100 ± 0.030 0.077 ± 0.023 0.144 ± 0.043 79.6 ± 3.1 0.779 ± 0.037 88.05 ± 0.52 72.8 ± 2.4 causal_chamber 10000 23 SciNO 5 0.469 ± 0.017 0.371 ± 0.019 0.641 ± 0.018 0.168 ± 0.019 0.133 ± 0.015 0.231 ± 0.029 73.0 ± 3.4 0.672 ± 0.030 3979.48 ± 25.22 68.0 ± 3.9 Table 42: Sachs observational-condition benchmark under the 17-edge bnlearn reference DAG. Rows compare preprocessing transformations for condition 1 of the Benchpress Sachs data; entries are means with standard errors over bootstrap seeds. sachs_transform n_samples n_features method n skel_f1 skel_prec skel_rec f1 prec rec shd order runtime pred_edges true_edges none 853 11 Spade-Hist 5 0.422 ± 0.038 1.000 0.271 ± 0.030 0.348 ± 0.047 0.810 ± 0.055 0.224 ± 0.034 13.2 ± 0.6 0.506 ± 0.040 0.13 ± 0.07 4.6 ± 0.5 17.0 none 853 11 Spade-Gauss 5 0.595 ± 0.011 1.000 0.424 ± 0.012 0.413 ± 0.027 0.696 ± 0.049 0.294 ± 0.019 12.0 ± 0.3 0.424 ± 0.051 0.03 ± 0.00 7.2 ± 0.2 17.0 none 853 11 Topic 5 0.655 ± 0.038 0.835 ± 0.038 0.541 ± 0.039 0.454 ± 0.050 0.575 ± 0.050 0.376 ± 0.048 12.4 ± 0.7 0.306 ± 0.029 2.89 ± 0.10 11.0 ± 0.5 17.0 none 853 11 Cam 5 0.605 ± 0.014 0.535 ± 0.036 0.718 ± 0.043 0.377 ± 0.036 0.333 ± 0.035 0.447 ± 0.051 20.6 ± 1.7 0.482 ± 0.051 27.39 ± 0.37 23.4 ± 2.4 17.0 none 853 11 CauScale 5 0.506 ± 0.045 0.925 ± 0.075 0.353 ± 0.037 0.201 ± 0.030 0.356 ± 0.040 0.141 ± 0.024 15.2 ± 0.6 0.624 ± 0.063 7.39 ± 1.83 6.6 ± 0.7 17.0 none 853 11 SDCD 5 0.527 ± 0.021 0.828 ± 0.023 0.388 ± 0.024 0.222 ± 0.035 0.344 ± 0.046 0.165 ± 0.029 15.6 ± 0.2 0.682 ± 0.030 47.37 ± 5.26 8.0 ± 0.5 17.0 none 853 11 Globe 5 0.396 ± 0.047 0.867 ± 0.057 0.259 ± 0.035 0.233 ± 0.086 0.497 ± 0.173 0.153 ± 0.058 15.0 ± 1.1 0.541 ± 0.086 29.68 ± 1.19 5.0 ± 0.4 17.0 none 853 11 Das 5 0.529 ± 0.039 0.588 ± 0.063 0.494 ± 0.035 0.152 ± 0.028 0.169 ± 0.034 0.141 ± 0.024 21.2 ± 2.1 0.718 ± 0.012 3.48 ± 0.06 15.0 ± 2.2 17.0 none 853 11 Score 5 0.591 ± 0.022 0.524 ± 0.021 0.694 ± 0.060 0.171 ± 0.018 0.151 ± 0.016 0.200 ± 0.024 24.6 ± 1.6 0.718 ± 0.012 2.96 ± 0.06 22.8 ± 2.7 17.0 none 853 11 NoGAM 5 0.596 ± 0.022 0.533 ± 0.021 0.694 ± 0.060 0.172 ± 0.018 0.154 ± 0.016 0.200 ± 0.024 24.2 ± 1.5 0.694 ± 0.022 9.32 ± 0.06 22.4 ± 2.5 17.0 none 853 11 Avici 5 0.533 ± 0.036 0.865 ± 0.011 0.388 ± 0.035 0.275 ± 0.029 0.444 ± 0.038 0.200 ± 0.024 14.6 ± 0.4 0.624 ± 0.030 6.32 ± 0.77 7.6 ± 0.6 17.0 none 853 11 NOTEARS 5 0.543 ± 0.015 0.536 ± 0.019 0.553 ± 0.024 0.358 ± 0.016 0.353 ± 0.013 0.365 ± 0.022 19.0 ± 0.4 0.376 ± 0.066 107.34 ± 21.84 17.6 ± 0.9 17.0 none 853 11 DiffAN 5 0.624 ± 0.012 0.684 ± 0.020 0.576 ± 0.022 0.230 ± 0.028 0.254 ± 0.035 0.212 ± 0.024 18.0 ± 0.7 0.588 ± 0.049 9.17 ± 0.52 14.4 ± 0.8 17.0 none 853 11 SciNO 5 0.679 ± 0.024 0.774 ± 0.036 0.612 ± 0.035 0.104 ± 0.025 0.117 ± 0.026 0.094 ± 0.024 18.6 ± 0.9 0.788 ± 0.014 329.29 ± 7.81 13.6 ± 1.1 17.0 none 853 11 Caps 5 0.653 ± 0.011 0.740 ± 0.026 0.588 ± 0.019 0.170 ± 0.026 0.193 ± 0.029 0.153 ± 0.024 18.0 ± 0.6 0.718 ± 0.022 3.43 ± 0.24 13.6 ± 0.7 17.0 none 853 11 PC 5 0.243 ± 0.063 0.468 ± 0.120 0.165 ± 0.043 0.175 ± 0.062 0.344 ± 0.127 0.118 ± 0.042 17.0 ± 0.7 0.729 ± 0.055 147.20 ± 12.83 4.8 ± 1.2 17.0 none 853 11 FCI 5 0.063 ± 0.026 0.300 ± 0.122 0.035 ± 0.014 0.042 ± 0.026 0.200 ± 0.122 0.024 ± 0.014 17.4 ± 0.2 0.812 ± 0.034 153.16 ± 12.29 1.4 ± 0.4 17.0 log1p 853 11 Spade-Hist 5 0.569 ± 0.006 0.925 ± 0.031 0.412 0.326 ± 0.028 0.532 ± 0.055 0.235 ± 0.019 13.6 ± 0.5 0.341 ± 0.057 0.06 ± 0.00 7.6 ± 0.2 17.0 log1p 853 11 Spade-Gauss 5 0.606 ± 0.014 1.000 0.435 ± 0.014 0.361 ± 0.020 0.596 ± 0.036 0.259 ± 0.014 12.6 ± 0.2 0.482 ± 0.047 0.03 ± 0.00 7.4 ± 0.2 17.0 log1p 853 11 Topic 5 0.586 ± 0.014 0.745 ± 0.018 0.482 ± 0.012 0.371 ± 0.027 0.473 ± 0.034 0.306 ± 0.022 14.6 ± 0.5 0.435 ± 0.040 3.19 ± 0.06 11.0 17.0 log1p 853 11 Cam 5 0.554 ± 0.020 0.413 ± 0.017 0.847 ± 0.040 0.331 ± 0.035 0.247 ± 0.027 0.506 ± 0.055 29.0 ± 1.6 0.424 ± 0.057 26.62 ± 0.33 35.0 ± 1.6 17.0 log1p 853 11 CauScale 5 0.453 ± 0.022 1.000 0.294 ± 0.019 0.254 ± 0.032 0.563 ± 0.071 0.165 ± 0.022 14.2 ± 0.4 0.506 ± 0.055 5.43 ± 0.02 5.0 ± 0.3 17.0 log1p 853 11 SDCD 5 0.554 ± 0.007 0.735 ± 0.021 0.447 ± 0.014 0.263 ± 0.019 0.351 ± 0.031 0.212 ± 0.014 16.2 ± 0.5 0.518 ± 0.022 51.55 ± 4.02 10.4 ± 0.6 17.0 log1p 853 11 Globe 5 0.456 ± 0.012 0.905 ± 0.039 0.306 ± 0.012 0.245 ± 0.016 0.486 ± 0.035 0.165 ± 0.012 14.8 ± 0.2 0.506 ± 0.024 27.99 ± 1.18 5.8 ± 0.4 17.0 log1p 853 11 Das 5 0.534 ± 0.019 0.509 ± 0.026 0.565 ± 0.014 0.290 ± 0.040 0.277 ± 0.042 0.306 ± 0.039 21.2 ± 1.2 0.459 ± 0.022 3.35 ± 0.05 19.0 ± 0.8 17.0 log1p 853 11 Score 5 0.546 ± 0.024 0.439 ± 0.014 0.729 ± 0.055 0.282 ± 0.026 0.227 ± 0.021 0.376 ± 0.040 26.4 ± 1.0 0.459 ± 0.022 2.86 ± 0.00 28.2 ± 1.6 17.0 log1p 853 11 NoGAM 5 0.535 ± 0.023 0.420 ± 0.016 0.741 ± 0.048 0.222 ± 0.036 0.175 ± 0.029 0.306 ± 0.051 29.2 ± 1.4 0.529 ± 0.037 9.13 ± 0.02 30.0 ± 1.4 17.0 log1p 853 11 Avici 5 0.603 ± 0.020 0.886 ± 0.005 0.459 ± 0.022 0.340 ± 0.036 0.498 ± 0.049 0.259 ± 0.030 13.6 ± 0.5 0.635 ± 0.060 5.46 ± 0.09 8.8 ± 0.4 17.0 log1p 853 11 NOTEARS 5 0.593 ± 0.040 0.590 ± 0.049 0.600 ± 0.034 0.430 ± 0.023 0.429 ± 0.033 0.435 ± 0.014 17.0 ± 1.6 0.388 ± 0.024 55.38 ± 5.52 17.6 ± 1.2 17.0 log1p 853 11 DiffAN 5 0.606 ± 0.014 0.627 ± 0.019 0.588 ± 0.019 0.328 ± 0.056 0.340 ± 0.058 0.318 ± 0.055 17.6 ± 1.3 0.482 ± 0.094 9.43 ± 0.47 16.0 ± 0.7 17.0 log1p 853 11 SciNO 5 0.621 ± 0.033 0.677 ± 0.042 0.576 ± 0.034 0.278 ± 0.035 0.302 ± 0.040 0.259 ± 0.035 17.4 ± 0.8 0.447 ± 0.055 329.02 ± 6.64 14.6 ± 0.9 17.0 log1p 853 11 Caps 5 0.610 ± 0.014 0.705 ± 0.037 0.541 ± 0.012 0.321 ± 0.039 0.376 ± 0.056 0.282 ± 0.029 16.2 ± 1.1 0.388 ± 0.030 3.12 ± 0.06 13.2 ± 0.8 17.0 log1p 853 11 PC 5 0.423 ± 0.021 0.597 ± 0.022 0.329 ± 0.024 0.270 ± 0.048 0.378 ± 0.062 0.212 ± 0.040 17.2 ± 0.7 0.576 ± 0.054 304.85 ± 22.86 9.4 ± 0.7 17.0 log1p 853 11 FCI 5 0.269 ± 0.051 0.800 ± 0.122 0.165 ± 0.034 0.154 ± 0.021 0.447 ± 0.034 0.094 ± 0.014 16.2 ± 0.4 0.753 ± 0.022 308.93 ± 22.41 3.6 ± 0.5 17.0 arcsinh 853 11 Spade-Hist 5 0.574 ± 0.006 0.950 ± 0.031 0.412 0.393 ± 0.031 0.650 ± 0.053 0.282 ± 0.022 12.6 ± 0.4 0.306 ± 0.034 0.06 ± 0.00 7.4 ± 0.2 17.0 arcsinh 853 11 Spade-Gauss 5 0.601 ± 0.012 0.978 ± 0.022 0.435 ± 0.014 0.358 ± 0.021 0.585 ± 0.044 0.259 ± 0.014 12.8 ± 0.4 0.459 ± 0.068 0.03 ± 0.00 7.6 ± 0.4 17.0 arcsinh 853 11 Topic 5 0.600 ± 0.017 0.764 ± 0.022 0.494 ± 0.014 0.371 ± 0.027 0.473 ± 0.034 0.306 ± 0.022 14.4 ± 0.4 0.459 ± 0.054 3.18 ± 0.11 11.0 17.0 arcsinh 853 11 Cam 5 0.561 ± 0.014 0.417 ± 0.009 0.859 ± 0.040 0.325 ± 0.038 0.243 ± 0.030 0.494 ± 0.051 29.0 ± 1.5 0.435 ± 0.058 26.73 ± 0.38 35.0 ± 1.4 17.0 arcsinh 853 11 CauScale 5 0.422 ± 0.038 1.000 0.271 ± 0.030 0.277 ± 0.039 0.667 ± 0.097 0.176 ± 0.026 14.0 ± 0.4 0.529 ± 0.059 5.45 ± 0.01 4.6 ± 0.5 17.0 arcsinh 853 11 SDCD 5 0.509 ± 0.031 0.671 ± 0.060 0.412 ± 0.019 0.229 ± 0.045 0.293 ± 0.052 0.188 ± 0.039 17.4 ± 0.4 0.612 ± 0.080 52.34 ± 2.56 10.6 ± 0.5 17.0 arcsinh 853 11 Globe 5 0.462 ± 0.024 0.971 ± 0.029 0.306 ± 0.022 0.249 ± 0.015 0.526 ± 0.033 0.165 ± 0.012 14.4 ± 0.2 0.529 ± 0.042 26.97 ± 1.49 5.4 ± 0.5 17.0 arcsinh 853 11 Das 5 0.522 ± 0.049 0.508 ± 0.040 0.541 ± 0.063 0.270 ± 0.043 0.260 ± 0.037 0.282 ± 0.051 21.0 ± 1.0 0.459 ± 0.022 3.38 ± 0.03 18.0 ± 1.0 17.0 arcsinh 853 11 Score 5 0.546 ± 0.042 0.441 ± 0.032 0.718 ± 0.060 0.278 ± 0.036 0.225 ± 0.029 0.365 ± 0.047 26.2 ± 1.7 0.459 ± 0.022 2.87 ± 0.01 27.6 ± 0.8 17.0 arcsinh 853 11 NoGAM 5 0.536 ± 0.039 0.429 ± 0.030 0.718 ± 0.060 0.267 ± 0.040 0.215 ± 0.033 0.353 ± 0.049 27.2 ± 1.8 0.471 ± 0.053 9.10 ± 0.03 28.4 ± 1.2 17.0 arcsinh 853 11 Avici 5 0.593 ± 0.014 0.883 ± 0.003 0.447 ± 0.014 0.341 ± 0.018 0.500 ± 0.025 0.259 ± 0.014 13.8 ± 0.2 0.624 ± 0.048 5.40 ± 0.04 8.8 ± 0.2 17.0 arcsinh 853 11 NOTEARS 5 0.570 ± 0.035 0.572 ± 0.040 0.576 ± 0.043 0.408 ± 0.017 0.411 ± 0.029 0.412 ± 0.019 17.6 ± 1.3 0.400 ± 0.022 50.97 ± 4.76 17.4 ± 1.5 17.0 arcsinh 853 11 DiffAN 5 0.625 ± 0.026 0.618 ± 0.035 0.635 ± 0.022 0.416 ± 0.031 0.411 ± 0.031 0.424 ± 0.034 16.6 ± 1.0 0.318 ± 0.055 9.37 ± 0.35 17.6 ± 0.8 17.0 arcsinh 853 11 SciNO 5 0.581 ± 0.013 0.614 ± 0.023 0.553 ± 0.014 0.259 ± 0.022 0.274 ± 0.024 0.247 ± 0.022 18.8 ± 0.7 0.459 ± 0.047 338.00 ± 3.97 15.4 ± 0.7 17.0 arcsinh 853 11 Caps 5 0.617 ± 0.004 0.722 ± 0.018 0.541 ± 0.012 0.377 ± 0.021 0.442 ± 0.031 0.329 ± 0.014 15.0 ± 0.5 0.341 ± 0.022 3.04 ± 0.02 12.8 ± 0.6 17.0 arcsinh 853 11 PC 5 0.442 ± 0.026 0.602 ± 0.023 0.353 ± 0.032 0.279 ± 0.039 0.377 ± 0.050 0.224 ± 0.034 17.2 ± 0.6 0.576 ± 0.047 282.17 ± 25.22 10.0 ± 0.9 17.0 arcsinh 853 11 FCI 5 0.275 ± 0.070 0.831 ± 0.092 0.176 ± 0.053 0.126 ± 0.054 0.277 ± 0.124 0.082 ± 0.035 16.4 ± 0.4 0.776 ± 0.043 295.84 ± 29.21 3.8 ± 1.1 17.0 A.7 Implementation Details We used the default or author-provided hyperparameters of the corresponding implementations wherever possible. This choice is deliberate: in causal discovery, tuning method-specific hyperparameters without access to the ground-truth graph is not straightforward, and using fixed defaults avoids giving methods different amounts of oracle information. We did not tune hyperparameters against the benchmark ground truth; computational batch sizes and subsampling parameters were adjusted only when needed to make the methods fit GPU memory. For methods with stochastic optimization, we ran one predetermined seed per method–dataset pair. We did not use random restarts or select the best run across seeds. The seed policy and exact seeds are provided in the released code. Spade. To ensure a fair comparison, we fixed sensible defaults for all hyperparameters before evaluation, and did not perform any hyperparameter tuning. Both variants of Spade use the same additive univariate spline regression for local parent-set scores. The spline basis is the SplineTransformer basis from scikit-learn: cubic B-splines (degree=3), no bias column (include_bias=False), and linear extrapolation outside the fitted knot range. The implementation uses an internal n_splines=20 setting, which is converted to n_knots=18 for scikit-learn; with cubic splines and include_bias=False, this gives 1919 output basis columns per univariate parent block. The resulting basis columns are centered before scoring. The local regression uses ridge regularization 10−610^-6 and a second-difference smoothing penalty with weight 0.50.5. Scores are evaluated on the deviance scale with a BIC penalty. For a candidate parent set S, Spade-Gauss uses the local score jGauss(S)=nlog(RSSj,S/n)+kSlogn,S^Gauss_j(S)=n \! (RSS_j,S/n )+k_S n, up to the additive Gaussian constant independent of S. Here RSSj,SRSS_j,S is the residual sum of squares of the fitted spline regression and kSk_S is the number of active spline-design columns for the selected parents. Spade-Hist uses the same spline basis, search settings, and kSlognk_S n penalty but replaces the Gaussian residual term by a histogram residual likelihood. For residuals N^i N_i, the histogram range is split into B equal-width bins using the Freedman–Diaconis rule, clipped to 4≤B≤644≤ B≤ 64. If b(i)b(i) is the bin of residual i, cb(i)c_b(i) is the corresponding bin count, w is the bin width, and α=0.5α=0.5, the leave-one-out density estimate is p^−i(N^i)=cb(i)−1+αn−1+αB1w. p_-i( N_i)= c_b(i)-1+αn-1+α B\, 1w. The histogram variant therefore uses jHist(S)=2∑i=1n−logp^−i(N^i)+kSlogn.S^Hist_j(S)=2 _i=1^n- p_-i( N_i)+k_S n. Baselines. For CAM, Score, NoGAM, and DAS we used the dodiscover implementations with their default spline settings (1010 splines, degree 33) and the default significance level 0.050.05. Topic was run with published implementation and its default continuous-data GAM score. Globe was run through its published default implementation with maximum interaction order 33. NoTears uses the nonlinear MLP variant with hidden layers (10,1)(10,1), λ1=λ2=0.01 _1= _2=0.01, 100100 outer iterations, and edge threshold 0.30.3. Dagma uses the nonlinear MLP variant with hidden width 1010, λ1=0.02 _1=0.02, λ2=0.005 _2=0.005, L2 loss, and edge threshold 0.30.3. DiffAN was run with masking enabled, batch size 256256, ordering batch size 128128, learning rate 10−310^-3, and 30003000 epochs. We used the residue-free DiffAN variant for the scalability experiments and the standard DiffAN variant for the qualitative quality experiment. SciNO uses the SciNO model with batch size 6464, ordering batch size 3232, evaluation batch size 88, 33 votes, learning rate 10−310^-3, 20002000 epochs, Fourier depth 1010, batch normalization, and CAM pruning at cutoff 10−310^-3. SDCD uses the observational-only setup with parameterized variance, no Gumbel relaxation, stage-1 and stage-2 training for 20002000 epochs each, batch size 256256, one hidden layer of width 1010, 1515 power-iteration steps, validation fraction 0.20.2, and threshold 0.10.1. Avici uses the pretrained scm-v0 checkpoint and thresholds returned edge probabilities at 0.50.5. CauScale uses the OpenCausaLab/causcale-model checkpoint synthetic/auprc=0.905_migrated.ckpt, evaluates one graph per batch, uses at most 500500 samples for feature subsampling, thresholds edge probabilities at 0.50.5, and removes cycles by deleting the lowest-probability edge on a detected cycle. Caps uses ηG=ηH=10−3 _G= _H=10^-3, λ1=λ2=50 _1= _2=50, pre-pruning, edge addition, median-bandwidth score estimation using at most 20482048 samples, and Stein batches of size 6464. PC and FCI use the causal-learn implementations with the kernel conditional independence test.