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Sequential Probabilistic Uncertainty Estimation for Parallel Multi-Agent Reasoning Systems
Tunyu Zhang, Zihao Zhao, Yusong Zhao, Haizhou Shi, Zhuohang Li, Haoxian Chen, Hao Wang, Dimitris N. Metaxas
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
The paper introduces SAUCE (Sequential Agent Uncertainty through Consensus Evolution), a lightweight, training-free uncertainty estimator for parallel multi-agent reasoning systems (MAS). SAUCE models MAS uncertainty as a sequential inference problem over a latent system-level belief, aggregating round-level agreement and generation-uncertainty signals through a filtering-style update. It outperforms baselines in misclassification detection, selective prediction, and calibration across five backbones, five benchmarks, and two MAS protocols.
Entities (19)
Relation Signals (18)
SAUCE → evaluatedon → Debate
confidence 95% · Empirically, SAUCE outperforms log-likelihood baselines and MAS-specific uncertainty estimators across Debate Du et al. (2024) and DyLAN Liu et al. (2023) protocols.
SAUCE → evaluatedon → DyLAN
confidence 95% · Empirically, SAUCE outperforms log-likelihood baselines and MAS-specific uncertainty estimators across Debate Du et al. (2024) and DyLAN Liu et al. (2023) protocols.
SAUCE → improves → Selective Prediction
confidence 95% · SAUCE improves misclassification detection, selective prediction, and calibration over a broad set of uncertainty estimation baselines
SAUCE → improves → calibration
confidence 95% · SAUCE improves misclassification detection, selective prediction, and calibration over a broad set of uncertainty estimation baselines
SAUCE → improves → Misclassification Detection
confidence 95% · SAUCE improves misclassification detection, selective prediction, and calibration over a broad set of uncertainty estimation baselines
SAUCE → isbasedon → filtering-style update
confidence 95% · SAUCE aggregates round-level agreement and generation-uncertainty signals through a filtering-style update.
SAUCE → isbasedon → sequential inference
confidence 95% · SAUCE formulates MAS uncertainty as sequential inference over a latent system-level belief.
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Abstract
Abstract:LLM-based multi-agent systems (MAS) have attracted growing attention for improving reasoning through interaction among multiple agents. In this work, we focus on parallel multi-agent reasoning systems, where several agents solve the same problem over multiple rounds and aggregate their outputs into a final answer. Despite their strong reasoning performance, uncertainty estimation for such systems remains underexplored: the reliability of a MAS depends not only on individual generations, but also on how agents interact and evolve across rounds. We propose SAUCE (Sequential Agent Uncertainty through Consensus Evolution), a lightweight, training-free uncertainty estimator that formulates MAS uncertainty as sequential inference over a latent system-level belief. SAUCE aggregates round-level agreement and generation-uncertainty signals through a filtering-style update. Across five backbones, five benchmarks, and two MAS protocols, SAUCE improves misclassification detection, selective prediction, and calibration over a broad set of uncertainty estimation baselines, including standard log-likelihood-based methods and MAS-specific estimators.
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- Source: https://arxiv.org/abs/2610.08901v1
- Canonical: https://arxiv.org/abs/2610.08901v1
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Sequential Probabilistic Uncertainty Estimation for Parallel Multi-Agent Reasoning Systems Tunyu Zhang Zihao Zhao Yusong Zhao Haizhou Shi Zhuohang Li Haoxian Chen Hao Wang Dimitris N. Metaxas Abstract LLM-based multi-agent systems (MAS) have attracted growing attention for improving reasoning through interaction among multiple agents. In this work, we focus on parallel multi-agent reasoning systems, where several agents solve the same problem over multiple rounds and aggregate their outputs into a final answer. Despite their strong reasoning performance, uncertainty estimation for such systems remains underexplored: the reliability of a MAS depends not only on individual generations, but also on how agents interact and evolve across rounds. We propose SAUCE (Sequential Agent Uncertainty through Consensus Evolution), a lightweight, training-free uncertainty estimator that formulates MAS uncertainty as sequential inference over a latent system-level belief. SAUCE aggregates round-level agreement and generation-uncertainty signals through a filtering-style update. Across five backbones, five benchmarks, and two MAS protocols, SAUCE improves misclassification detection, selective prediction, and calibration over a broad set of uncertainty estimation baselines, including standard log-likelihood-based methods and MAS-specific estimators. Code: github.com/Tyrion58/SAUCE Data: Tyrion279/SAUCE-Qwen3-4B-Rollouts 00footnotetext: 1Rutgers University. 2University of Illinois Urbana-Champaign. 3Vanderbilt University. 4Columbia University. †Correspondence to: Tunyu Zhang (ty.zhang@rutgers.edu), Hao Wang (hwangml@illinois.edu), Dimitris N. Metaxas (dmn@cs.rutgers.edu). ‡Equal advising. 1 Introduction Multi-agent systems (MAS) built with large language models (LLMs) have demonstrated strong capabilities in complex reasoning tasks Li et al. (2024); Talebirad and Nadiri (2023); He et al. (2025); Yu et al. (2024); Venkataramani et al. (2026). A common design is parallel multi-agent reasoning Cui et al. (2025); Du et al. (2024); Hu et al. (2025); Liu et al. (2023), such as debate-style systems Chan et al. (2023); Du et al. (2024), where multiple agents solve the same problem and exchange responses over several rounds. In these systems, agents act in parallel within each round, while interactions evolve sequentially across rounds. As such systems are explored for complex reasoning, planning, and decision-making tasks Xu et al. (2025); Coeckelbergh (2025); Guo et al. (2024); Lee et al. (2025), this motivates the need for reliable system-level uncertainty estimation for multi-agent interactions. Uncertainty estimation has been extensively studied for single LLMs, including sampling-based Farquhar et al. (2024); Manakul et al. (2023); Kossen et al. (2024); Malinin and Gales (2020), Bayesian Yang et al. (2024); Wang et al. (2024a); Shi et al. (2025b), and internal signal-based approaches Xia et al. (2025); Kossen et al. (2024); Ren et al. (2023); Fu et al. (2025). Extending these methods to MAS, however, is non-trivial. Sampling-based and Bayesian approaches become costly when repeated across agents and rounds. Signal-based methods are more computationally practical because they rely on quantities already available during interaction, but they introduce a different challenge: these signals are heterogeneous across agents and fluctuate over rounds. For parallel reasoning MAS, uncertainty estimation must aggregate heterogeneous agent-level signals into a coherent system-level estimate while tracking how these signals change across rounds. Crucially, multi-agent interaction is a sequential process Liang et al. (2024); Du et al. (2024). As agents exchange information over rounds, reliability may change, so uncertainty should be updated progressively rather than inferred from a single-step signal or a static summary (Fig. 1 (A)). Thus, MAS uncertainty is not merely a property of the final output, but of the interaction process itself. This raises a key question: Do the interaction dynamics of a MAS inform its internal uncertainty? We address this question by modeling MAS uncertainty as a sequential inference problem over interaction rounds. We propose SAUCE (Sequential Agent Uncertainty through Consensus Evolution), a lightweight, training-free uncertainty estimator for parallel reasoning MAS. As shown in Fig. 1 (B), inspired by hierarchical Bayesian deep learning Wang and Yeung (2016); Wang and Yeung (2020), SAUCE treats each round as noisy evidence about an evolving system-level belief, rather than as an independent snapshot. Round-level agreement provides the evidence, while generation uncertainty determines how strongly the evidence should update the belief. This yields a filtering-style update that propagates the previous belief state and revises uncertainty according to the current round’s generation uncertainty. As a result, uncertainty is inferred from the full interaction trajectory rather than from a single final answer. SAUCE uses only signals extracted from the MAS inference trace, requiring no additional sampling or access to model weights or hidden states. Empirically, SAUCE outperforms log-likelihood baselines and MAS-specific uncertainty estimators across Debate Du et al. (2024) and DyLAN Liu et al. (2023) protocols. Contributions. Our contributions are summarized as follows: 1. A system-dynamics perspective for MAS uncertainty estimation. We argue that uncertainty in parallel reasoning MAS should be estimated from how consensus and reliability evolve across rounds, rather than from a static final-output signal. 2. A sequential inference framework for consensus aggregation. We formulate MAS uncertainty estimation as a filtering-style update over a latent system-level belief, where round-level evidence is weighted by its estimated generation uncertainty. 3. A lightweight estimator for practical MAS. We instantiate SAUCE with majority-vote consensus and token-level generation entropy, yielding a training-free method that requires minimal model access and applies to both open- and closed-source models with top-k log-probabilities. Figure 1: Overview. (A) Two MAS trajectories can reach the same final answer but differ greatly in reliability: a stable run whose outputs converge across rounds (top) is more trustworthy than an unstable run whose outputs oscillate (bottom). A static view that only considers the final output cannot distinguish them, motivating uncertainty estimation from interaction dynamics. (B) SAUCE models multi-round interaction as a sequence of round-level observations (Mt,Rt)(M_t,R_t), where agreement is weighted by generation uncertainty, and performs a filtering-style update over a latent system-level belief. The resulting trajectory summary defines the SAUCE uncertainty score u(,τ)u( x,τ) (Eqn. 11). 2 Preliminaries In this section, we formalize the multi-agent reasoning process and define the problem of uncertainty estimation for MAS. 2.1 Multi-Agent Reasoning Trajectory Given an input problem x, a MAS consists of N agents =A1,…,ANA=\A_1,…,A_N\ that solve the problem through T rounds of interaction. At round t, agent AiA_i produces an output ti∼πi(⋅∣,ti), z_t^i _i(· x, h_t^i), (1) where ti h_t^i denotes the information available to agent AiA_i before round t, including its own previous outputs and the messages observed from other agents. The full multi-agent interaction trajectory is τ=(11:N,21:N,…,T1:N), τ= ( z_1^1:N, z_2^1:N,…, z_T^1:N ), (2) which is sampled from the MAS rollout distribution induced by the agents and the interaction protocol. The final prediction is obtained by applying an aggregation function g to the input and the trajectory: ^=g(,τ). y=g( x,τ). (3) Scope: Parallel Reasoning MAS. This paper focuses on parallel multi-agent reasoning Du et al. (2024); Liang et al. (2024); Liu et al. (2023), where all N agents generate responses in parallel at each round, conditioned on the same input x and the prior interaction history. In the protocols considered in this paper, the final prediction is typically obtained by aggregating the round-T parsed answers, e.g., by majority vote Wang et al. (2022); Kaesberg et al. (2025); Choi et al. (2025). Debate Du et al. (2024) and DyLAN Liu et al. (2023) are representative protocols of this form. 2.2 Sequential Uncertainty Estimation for MAS Given an input x, a reasoning trajectory τ, and the final prediction ^=g(,τ) y=g( x,τ), our goal is to estimate a trajectory-level uncertainty score u(,τ)∈ℝ, u( x,τ) , (4) where a larger value indicates lower confidence in y. Dynamics in MAS. Unlike static confidence measures that summarize only the final output or aggregate signals without temporal structure, we model MAS uncertainty as a quantity induced by the interaction process. At each round, agent outputs induce round-level evidence about the system state, and the uncertainty is updated sequentially as new evidence arrives: ut=Φ(ut−1,et),u(,τ)=(u1:T), u_t= (u_t-1,e_t), u( x,τ)=A(u_1:T), (5) where ete_t denotes the evidence extracted from round t, utu_t denotes the system uncertainty state after round t, Φ is a sequential update rule, and A maps the uncertainty trajectory to the final score. Challenges. This dynamic view is essential because the final prediction arises from multi-round communication, where agents may reinforce, revise, or contradict one another. A static summary, such as the log-likelihood of the final answer or an order-invariant average of per-round signals, is blind to the temporal order of evidence: two trajectories with identical average signals but opposite dynamics may receive the same score despite carrying very different reliability. MAS uncertainty should therefore be updated progressively as new evidence emerges across rounds, rather than inferred from a single-step signal or a static aggregation. Motivated by this view, we formulate MAS uncertainty estimation as a sequential inference problem. SAUCE instantiates this principle by integrating round-level signals into the uncertainty state utu_t, leading to the method in Sec. 3 and the probabilistic interpretation in Sec. 4. 3 Method We present SAUCE, a lightweight uncertainty estimator for parallel reasoning MAS. SAUCE extracts round-level evidence from generations of agents, and updates uncertainty across rounds through a filtering-style recursion. This recursion can be interpreted as posterior uncertainty inference over a latent system-level belief, with the full probabilistic view given in Sec. 4. 3.1 Round-Level Signals from Agent Outputs At each interaction round t, the N agents generate outputs in parallel, tii=1N\ z_t^i\_i=1^N. This parallel structure naturally exposes two round-level signals at no extra cost: a cross-agent agreement signal and a per-agent generation-uncertainty signal. Agreement Signal MtM_t. For each output ti z_t^i, we extract the parsed answer atia_t^i. Let mtm_t denote the modal answer at round t. We define MtM_t as the modal-answer share, i.e., the fraction of agents whose answer matches mtm_t: Mt=1N∑i=1N[ati=mt]. M_t\;=\; 1NΣ _i=1^N1[\,a_t^i=m_t\,]. (6) A higher MtM_t reflects stronger cross-agent consensus. Generation-Uncertainty Signal RtR_t. Cross-agent agreement should be trusted more when the underlying generations are confident. We measure round-level generation uncertainty by averaging token-level predictive entropy across agents: Rt=1N∑i=1NHtok(ti), R_t\;=\; 1NΣ _i=1^NH_tok( z_t^i), (7) where Htok(ti)H_tok( z_t^i) averages predictive entropy Malinin and Gales (2020) over the tokens of ti z_t^i, computed from the top-k log-probabilities. Larger RtR_t indicates higher generation uncertainty, so the round-level agreement should be trusted less. Together, (Mt,Rt)(M_t,R_t) captures two sources of MAS uncertainty: cross-agent disagreement, reflected by low MtM_t, and generation uncertainty, reflected by high RtR_t. 3.2 Sequential Uncertainty Update SAUCE maintains a round-level confidence estimate μt _t throughout the multi-agent interaction, together with a stability weight vtv_t that controls how strongly new evidence updates this estimate. Both quantities are initialized by hyperparameters, μ0 _0 and v0v_0. The final uncertainty score is obtained by negating the temporal average of μt _t. This sequential estimation process alternates between two steps: confidence propagation and reliability-weighted confidence update. Figure 2: Round-level μt _t on Llama-3.1-8B / MATH-500. Both classes grow more confident with each round, but correct trajectories converge to a consistently higher consensus. Shading shows ±1± 1 standard deviation across trajectories. Propagation. At round t, SAUCE first propagates the previous state: μ~t=λμt−1,v~t=λ2vt−1+q,t∈[1,T], μ_t=λ _t-1, v_t=λ^2v_t-1+q, t∈[1,T], (8) where λ∈(0,1)λ∈(0,1) controls temporal coupling across rounds, and q captures additional uncertainty introduced during interaction. The two limits motivate this open interval: as λ→0λ→ 0 the state resets each round (no temporal memory), and as λ→1λ→ 1 the previous belief is retained without decay; an interior λ balances these two extremes. SAUCE then combines the propagated state with the current round signals through a reliability-dependent weight: αt=v~t/(v~t+Rt),t∈[1,T]. _t\;=\; v_t/( v_t+R_t), t∈[1,T]. (9) Update. The weight αt _t determines how much SAUCE trusts the current round. When the round is reliable, RtR_t is small and αt _t is large, so the estimate moves quickly toward the current consensus signal MtM_t. When the round is noisy, αt _t becomes small, and SAUCE retains more of the propagated state. The update is μt=μ~t+αt(Mt−μ~t),vt=(1−αt)v~t. _t\;=\; μ_t+ _t(M_t- μ_t), v_t\;=\;(1- _t)\, v_t. (10) This recursion has the same form as the posterior mean and variance update of a linear-Gaussian filter with effective noise RtR_t, as formally derived in Sec. 4. We show the evolution of round-level confidence μt _t in Fig. 2. As the debate proceeds, μt _t gradually increases, indicating that the system becomes more confident over rounds. Moreover, correct samples consistently exhibit higher average μt _t than incorrect samples, suggesting that the sequential confidence estimate remains informative throughout the interaction. (a) Baseline M¯t M_t. (b) SAUCE μt _t. Figure 3: Distribution of (a) the baseline agreement signal M¯t M_t and (b) SAUCE’s posterior-mean estimate μt _t (each averaged over rounds) for correct vs. incorrect MAS outputs (MMLU-Pro, Gemini-2.5-FL, DyLAN). SAUCE sharpens the separation between classes (AUROC →0.8050.579\!→\!0.805). Densities are normalized independently in each panel. Finally, SAUCE aggregates the consensus estimate across the trajectory into a per-record uncertainty score: u(,τ)=−1T∑t=1Tμt. u( x,τ)\;=\;- 1TΣ _t=1^T _t. (11) We use the temporal average rather than the terminal value μT _T so that the score reflects the full deliberation process. Remark. As shown in Fig. 3, directly averaging MtM_t yields mixed and highly overlapping distributions for correct and incorrect trajectories, whereas SAUCE produces a much cleaner separation. This highlights the importance of sequentially modeling how confidence evolves across rounds. 4 Theoretical Analysis We provide a probabilistic interpretation of SAUCE by viewing multi-agent interaction as a sequential inference process over a latent system belief. This perspective is inspired by the hierarchical Bayesian deep learning framework Wang and Yeung (2016); Wang and Yeung (2020), where observable neural representations are connected to latent probabilistic variables through a hierarchical probabilistic graphical model Jordan et al. (1998). As illustrated in Fig. 4, we treat agent outputs at each round as noisy observations of a latent belief and estimate uncertainty through posterior inference. Figure 4: Probabilistic view of SAUCE. A latent system belief sts_t evolves across rounds and emits a noisy observation oto_t formed from the agents’ outputs. SAUCE’s per-round estimate is the posterior expectation [st∣o1:t]=μtE[s_t o_1:t]\!=\! _t. 4.1 Sequential Inference over Latent System Beliefs Let st∈ℝs_t denote the latent belief of the multi-agent system at round t, representing the system’s underlying belief about the correctness of its current solution. Transition Model. We assume that the latent belief evolves over rounds according to a first-order Markov process Rabiner (1989); Murphy (2012); Särkkä and Svensson (2023), p(st∣st−1), p(s_t s_t-1), (12) which captures the temporal evolution of the system belief during interaction. Observation Model. At round t, we extract a round-level signal oto_t from the collection of agent outputs and treat it as a noisy observation of the latent belief state sts_t. Formally, the observation process is described by p(ot∣st), p(o_t s_t), (13) which specifies the conditional distribution of the observed signal given the latent belief. Sequential Inference. Given the transition and observation models, posterior inference over the latent belief follows the Bayesian filtering Särkkä and Svensson (2023); Doucet et al. (2001); Kalman (1960) recursion p(st∣o1:t)∝p(ot∣st)∫p(st∣st−1)p(st−1∣o1:t−1)dst−1. p(s_t o_1:t) p(o_t s_t) p(s_t s_t-1)\,p(s_t-1 o_1:t-1)\,ds_t-1. (14) 4.2 Linear-Gaussian Instantiation and Posterior Variance Update To obtain a tractable update rule, we adopt a linear-Gaussian instantiation of the framework above. Specifically, we assume the latent belief evolves as st=λst−1+ϵt,ϵt∼(0,q), s_t=λ\,s_t-1+ _t, _t (0,q), (15) where λ∈(0,1)λ∈(0,1) controls the temporal dependence across rounds and q is the process noise that captures additional uncertainty introduced during interaction. The latent belief therefore evolves smoothly while being continually shaped by the accumulated interaction history. Treating λ as a constant is a deliberate simplification, which we justify empirically through ablations in Appendix C.1. Proposition 4.1. Under the linear-Gaussian instantiation above, let the observation be ot|st∼(st,rt2)o_t s_t (s_t,r_t^2). If the posterior belief remains Gaussian, st∣o1:t∼(μt,σt2)s_t o_1:t ( _t, _t^2), then the posterior mean and variance satisfy μt _t =μ~t+αt(ot−μ~t), = μ_t+ _t (o_t- μ_t ), (16) σt2 _t^2 =(1σ~t2+1rt2)−1=(1−αt)σ~t2, = ( 1 σ_t^2+ 1r_t^2 )^-1=(1- _t)\, σ_t^2, (17) where the predictive moments and Kalman gain Kalman (1960) are μ~t=λμt−1 μ_t=λ\, _t-1, σ~t2=λ2σt−12+q σ_t^2=λ^2 _t-1^2+q, and αt=σ~t2/(σ~t2+rt2) _t= σ_t^2/( σ_t^2+r_t^2). See Appendix D.1 for the proof. Proposition 4.1 shows that the SAUCE recursion (Eqn. 8–Eqn. 10) is the exact posterior of this linear-Gaussian model under the identification ot↔Mto_t\! \!M_t and rt2↔Rtr_t^2\! \!R_t. Substituting these into the proposition recovers the SAUCE update verbatim, with the agreement signal MtM_t acting as a noisy observation of the latent belief and the generation-uncertainty signal RtR_t as its observation-noise variance. The running estimate μt _t is therefore the posterior mean [st∣o1:t]E[s_t o_1:t], and the stability weight vtv_t its posterior variance Var[st∣o1:t]Var[s_t o_1:t]. Takeaway. SAUCE carries two quantities with a clean probabilistic reading. The round-level estimate μt _t is the posterior mean of the latent system belief sts_t, so the per-record score u(,τ)=−1T∑tμtu( x,τ)=- 1T _t _t summarizes how the system’s belief evolves across the entire trajectory rather than relying on the terminal round alone. 5 Experiments We evaluate SAUCE on two standard uncertainty-estimation tasks for large language models: (i) misclassification detection and (i) selective prediction. We further analyze calibration via the Brier score after Platt scaling, and include two diagnostic studies on long-debate robustness and the contribution of the sequential update. 5.1 Experimental Setup Models. We evaluate five backbones spanning both open- and closed-source models. The open-source models include Qwen3-4B Yang et al. (2025), Llama-3.1-8B-Instruct Grattafiori et al. (2024), and Gemma-4-E4B-it Google DeepMind (2026); the closed-source models include Gemini-2.5-Flash-Lite Comanici et al. (2025) and GPT-5.4-mini OpenAI (2026). Open-source models are served locally with vLLM Kwon et al. (2023), which provides per-token log-probabilities Murray and Chiang (2018); Fu et al. (2025); for closed-source models, we use the top-k log-probabilities returned by the API. MAS Architectures. We run two representative MAS protocols on every backbone. (i) Debate Du et al. (2024) uses N=3N\!=\!3 agents that first solve each problem independently and then exchange their full responses for T=3T\!=\!3 rounds. (i) DyLAN Liu et al. (2023) places the same agents in a dynamic agent network with a listwise ranker that retains the top k=2k\!=\!2 replies each round and an early-stopping consensus threshold of 0.6670.667. For the main experiments, both protocols use up to T=3T\!=\!3 rounds; the consensus answer is the majority vote of the surviving agents in the final round. We probe longer debates separately in Sec. 5.4. Benchmarks. We evaluate on five widely used reasoning suites covering mathematical problem solving, broad knowledge, and hard instruction-following. For open-source models we use MATH-500 Lightman et al. (2024), MMLU-Pro Wang et al. (2024b), and BBH Suzgun et al. (2022). For closed-source models we use OlympiadBench He et al. (2024), MMLU-Pro (subset), and BBEH-mini Kazemi et al. (2025). Benchmark sizes, formats, and answer verifiers are summarized in Appendix B.2. Baselines. We compare SAUCE with four training-free uncertainty estimators: PE, the predictive entropy of the final-round output Malinin and Gales (2020); L, the log-likelihood of the final answer Murray and Chiang (2018); UDPO Tang et al. (2026), a system-level variance score for debate; and MATU Chen et al. (2026), an uncertainty estimator designed for multi-agent systems. The original MATU samples multiple MAS runs at substantially higher inference cost, so for budget parity we use its single-run variant (N=1N\!=\!1). Metrics. Following standard practice in uncertainty estimation Farquhar et al. (2024); Chen et al. (2026); Zhang et al. (2026), we measure ranking quality with AUROC ↑ , which captures how well an estimator separates correct from incorrect MAS outputs, and AUARC ↑ Hanley and McNeil (1982); Boyd et al. (2013), which summarizes the accuracy–coverage curve obtained by rejecting increasingly uncertain inputs. For calibration, we report the Brier score after Platt scaling Glenn (1950); Platt and others (1999); Guo et al. (2017) with 5-fold cross-validation (lower is better). All three metrics are reported as percentages (×100× 100). Formal definitions are given in Appendix B.1. Implementation Details. All experiments use 3 agents and 3 interaction rounds, decoding with temperature 0.70.7, top-p=0.9p\!=\!0.9, and an 8192-token budget per agent. SAUCE uses transferred hyperparameters: for each MAS architecture, we tune on a single calibration cell and reuse the same values across every other model–dataset combination. The transferred values are listed in Appendix B.3, and per-axis sensitivity is reported in Appendix C.2. Table 1: Misclassification detection (AUROC ↑ ) and selective prediction (AUARC ↑ ) on open-source models. Boldface and underlining denote the best and second-best performance. Method MATH-500 MMLU-Pro BBH Avg Debate DyLAN Debate DyLAN Debate DyLAN ROC ARC ROC ARC ROC ARC ROC ARC ROC ARC ROC ARC ROC ARC Qwen3-4B PE 64.89 83.45 64.58 82.70 59.54 73.63 63.53 74.92 68.90 87.13 59.13 83.79 63.43 80.94 L 65.93 83.14 66.36 84.08 68.39 78.49 69.88 79.27 69.56 87.62 60.61 83.82 66.79 82.74 UDPO 57.29 79.14 72.50 84.10 58.22 66.43 67.95 73.43 59.18 84.38 64.97 88.00 63.35 79.25 MATU (N=1) 62.11 82.90 63.30 84.05 58.55 73.11 57.81 73.46 58.59 84.56 63.62 85.76 60.66 80.64 SAUCE (Ours) 68.81 84.12 73.11 85.60 77.05 82.72 78.14 83.37 73.68 89.09 70.18 87.93 73.50 85.47 Llama-3.1-8B-Instruct PE 35.69 34.10 46.96 43.83 45.90 46.51 58.68 57.33 53.92 58.99 63.45 71.06 50.77 51.97 L 39.20 35.68 47.62 46.44 48.69 47.12 57.69 56.99 57.78 61.44 63.66 71.53 52.44 53.20 UDPO 70.94 63.13 70.21 66.81 66.96 61.75 68.71 66.58 66.71 71.44 62.82 72.22 67.72 66.99 MATU (N=1) 55.55 49.77 73.85 70.71 52.85 53.23 63.26 65.32 61.31 64.20 66.24 72.51 62.18 62.62 SAUCE (Ours) 83.63 72.12 80.69 74.46 75.83 71.00 74.44 72.30 74.07 74.81 72.35 76.94 76.84 73.61 Gemma-4-E4B-it PE 58.20 87.47 66.08 88.45 69.33 83.64 75.50 87.58 53.67 84.69 54.47 79.97 62.88 85.30 L 62.43 87.55 65.12 88.58 74.04 86.63 76.19 87.89 66.28 89.59 56.53 80.70 66.77 86.82 UDPO 55.83 85.47 55.53 84.45 62.01 75.64 72.28 81.36 64.01 93.83 80.36 92.30 65.00 85.51 MATU (N=1) 54.01 86.99 60.64 87.72 55.37 77.41 65.76 82.36 57.73 90.82 74.98 88.82 61.41 85.69 SAUCE (Ours) 62.29 87.95 64.05 87.89 82.24 89.75 82.57 90.08 83.62 96.33 83.53 91.94 76.38 90.66 Table 2: Misclassification detection and selective prediction on closed-source models. Metrics and notation follow Table 1. Boldface and underlining denote the best and second-best performance. Method Olympiad MMLU-Pro BBEH-mini Avg Debate DyLAN Debate DyLAN Debate DyLAN ROC ARC ROC ARC ROC ARC ROC ARC ROC ARC ROC ARC ROC ARC Gemini-2.5-Flash-Lite PE 65.51 66.86 66.35 65.23 48.71 73.35 70.22 89.09 49.98 19.13 41.73 16.06 57.08 54.95 L 68.67 68.13 67.42 66.67 64.90 82.82 72.18 90.05 46.71 18.50 39.41 15.46 59.88 56.94 UDPO 63.08 48.89 81.58 71.05 54.59 77.52 57.83 82.71 59.23 25.12 75.41 27.29 65.29 55.43 MATU (N=1) 68.96 70.73 64.31 70.23 54.37 78.22 61.06 85.46 56.50 23.83 66.08 25.37 61.88 58.97 SAUCE (Ours) 81.27 76.82 81.85 77.13 79.60 89.84 78.72 91.95 65.68 28.07 67.48 24.44 75.77 64.71 GPT-5.4-mini PE 55.56 54.90 72.58 63.59 54.61 80.26 63.26 84.77 46.36 16.34 39.92 13.00 55.38 52.14 L 70.25 61.87 75.03 65.69 61.57 83.56 65.51 85.97 41.67 14.18 45.03 13.76 59.84 54.17 UDPO 65.99 46.61 77.73 59.55 56.29 81.71 67.70 86.20 58.58 19.59 54.73 16.08 63.50 51.62 MATU (N=1) 53.45 54.91 65.23 63.74 56.44 81.52 60.31 85.04 48.64 16.96 55.78 18.93 56.64 53.51 SAUCE (Ours) 85.15 74.46 83.46 72.57 70.63 86.44 73.83 88.29 50.08 17.72 47.36 14.85 68.42 59.06 5.2 Uncertainty Estimation for Multi-Agent Systems Tables 1 and 2 report AUROC and AUARC on the full grid of backbones, MAS protocols, and benchmarks of Sec. 5.1. SAUCE leads the large majority of cells and every per-backbone average; the gap to the strongest baseline holds under both ranking (AUROC) and accuracy–coverage (AUARC) views and on both Debate and DyLAN. Among the baselines, single-LLM scores (PE, L) frequently land near or below chance, and UDPO and MATU each lead only on isolated (model, protocol) cells. Results. Single-round token statistics break down under MAS: interaction reshapes the very entropy and log-likelihood that PE and L rely on (Sec. 5.4). Specialized MAS estimators tie their gains to a single protocol and do not transfer across backbones or datasets. Only SAUCE’s filtering-style aggregation across rounds wins consistently across backbones, datasets, and protocols, and the same property lets it carry over to the closed-source setting where only top-k token log-probabilities are exposed. 5.3 Uncertainty Estimation for Calibration Figure 5: Brier score after Platt scaling (5-fold cross-validation) on MMLU-Pro under the Debate protocol, across all five backbones. SAUCE attains the lowest Brier-Platt score on four of the five backbones. Figure 6: Per-round behavior on a 15-round Qwen3-4B Debate run on MMLU-Pro, grouped by final-answer correctness. Left: SAUCE posterior mean μt _t; the correct/incorrect gap persists through round 15. Middle: Average per-agent log-likelihood at round t; both classes become increasingly token-confident and the gap vanishes. Right: AUROC against final-consensus correctness as a function of debate length T; SAUCE plateaus while L degrades toward chance (dashed line). Fig. 5 reports the Brier score after Platt scaling on MMLU-Pro under the Debate protocol across all five backbones. SAUCE attains the lowest Brier-Platt score on four of the five backbones; on Gemini-FL it is within 0.50.5 p of the strongest baseline (L). Calibration results on additional benchmarks and settings are deferred to Appendix C.3. Results. Brier penalizes both miscalibrated magnitudes and unresolved predictions, so this view complements AUROC/AUARC: a method can rank well yet pay a Brier penalty when its confidences cluster in a narrow band—trivially calibrated, but unable to separate reliable from unreliable trajectories. SAUCE’s calibrated probabilities, by contrast, remain well spread across the probability range while still landing close to empirical accuracy. The score therefore has the right shape as well as the right order, and a two-parameter Platt transform suffices to turn it into a per-trajectory confidence usable as a downstream gating signal. 5.4 Robustness in Long Debates Our main experiments use T=3T\!=\!3. To test whether SAUCE remains reliable in longer interactions, we run Qwen3-4B Debate on MMLU-Pro up to T=15T\!=\!15 and compare SAUCE with a last-round log-likelihood (L) baseline at each truncation. As the debate proceeds, agents become increasingly token-confident on both correct and incorrect trajectories, causing the L scores to concentrate and the correct/incorrect gap to vanish (Fig. 6, middle). In contrast, SAUCE’s posterior mean μt _t also rises as agreement accumulates, but preserves a clear class separation throughout the sweep (left); correspondingly, SAUCE’s AUROC plateaus after T=3T\!=\!3, whereas L steadily degrades toward chance (right). These results suggest that retaining earlier-round evidence makes SAUCE more robust than last-round confidence scores when debate length increases. 5.5 Contribution of the Sequential Update Table 3: Component-wise ablation on Qwen3-4B / MMLU-Pro / Debate. Row (b) is the static joint of MtM_t and RtR_t; row (c) is the full sequential update with μ¯ μ readout. AUROC ↑ and Brier-Platt ↓ . Method AUROC Brier-Platt Last round, 1−Mt1-M_t 58.4 21.3 (a) Round-mean of MtM_t 68.6 20.0 (b) Static joint with RtR_t 68.4 20.9 (c) SAUCE (Ours) 77.1 19.2 A natural concern is that SAUCE’s improvement may come simply from combining agreement MtM_t with generation uncertainty RtR_t, rather than from the sequential update itself. Row (b) of Table 3 tests this possibility with a static precision-weighted aggregation ∑tMt/Rt∑t1/Rt _tM_t/R_t _t1/R_t. Results. Table 3 shows that the static joint score performs closely to the MtM_t-only round mean, while the full SAUCE update yields a clear gain. This indicates that SAUCE’s improvement comes from the sequential update rather than a static combination of MtM_t and RtR_t. To further probe the contribution of interaction history, we compare simpler trajectory summaries and alternative round orders (Appendix C.4; statistical analyses in Appendix C.5) and examine whether answer-switch histories remain informative when final agreement is held fixed (Appendix C.6). 6 Related Work Uncertainty Estimation for LLMs. Uncertainty estimation Wang et al. (2016); Wang and Wang (2023) has been widely studied for hallucination detection and calibration in LLMs Farquhar et al. (2024); Liu et al. (2024); Bakman et al. (2025); Liu et al. (2025); Vashurin et al. (2025). Existing methods include sampling-based approaches, such as Semantic Entropy Farquhar et al. (2024) and SAR Duan et al. (2024), which estimate uncertainty from multiple generations and their semantic variation; Bayesian approaches, which approximate posterior uncertainty through weight perturbations or low-rank adaptation Zhang et al. (2026); Shi et al. (2025b); Wang et al. (2024a); Yang et al. (2024); and internal-signal methods, which use token entropy, log-likelihood, or related generation statistics with little additional cost Fu et al. (2025); Van der Poel et al. (2022); Ren et al. (2022); Darrin et al. (2023); Duan et al. (2024). While effective for single LLMs, these methods do not directly address the structure of multi-agent reasoning, where uncertainty depends not only on individual generations but also on how agents interact and evolve across rounds. Our work addresses this gap by formulating uncertainty estimation in parallel multi-agent reasoning systems as a sequential inference problem over the interaction trajectory. Parallel Multi-Agent Reasoning Systems. LLM-based multi-agent systems (MAS) have emerged as a promising paradigm for complex reasoning, where multiple agents collaborate, debate, or critique one another Talebirad and Nadiri (2023); Li et al. (2024); Shi et al. (2025a); Zhang et al. (2024). We focus on parallel multi-agent reasoning systems, such as debate-style methods Choi et al. (2025); Du et al. (2024); Cui et al. (2025) and DyLAN Liu et al. (2023), where agents solve the same input at each round, exchange information, and produce a verifiable final prediction. This setting naturally exposes system-level signals Oh et al. (2026); Tang et al. (2026), such as cross-agent agreement, disagreement, and their evolution over rounds. Recent work has begun to study uncertainty in MAS, including general frameworks for uncertainty quantification in LLM agents Oh et al. (2026), uncertainty-driven optimization for mitigating debate collapse Tang et al. (2026), and MATU Chen et al. (2026), which estimates trajectory-level uncertainty from repeated MAS runs. In contrast, SAUCE is a lightweight, training-free estimator that updates uncertainty online from standard rollout signals within a single parallel multi-agent reasoning trajectory. 7 Conclusion We studied uncertainty estimation in parallel multi-agent reasoning systems. We formulated MAS uncertainty as a sequential inference problem over the multi-round interaction trajectory, and proposed SAUCE: a filtering-style update that propagates a latent consensus state across rounds. Across Debate and DyLAN protocols on multiple reasoning benchmarks, SAUCE consistently outperforms log-likelihood baselines and recent MAS-specific uncertainty estimators on both ranking and calibration, and remains stable as debates extend over more rounds where last-round confidence scores collapse. More broadly, our results suggest that interaction dynamics are a useful primitive for reasoning about the reliability of multi-agent systems, opening the way toward uncertainty-aware orchestration and adaptive control of MAS. Limitations. SAUCE focuses on parallel multi-agent reasoning systems, where agents produce comparable per-round answers. This assumption limits its direct applicability to more general agentic systems with heterogeneous roles, tool use, or open-ended planning, where round-level consensus is not naturally defined. Our current instantiation also uses simple agreement and uncertainty signals, namely modal-answer share and token-level predictive entropy; developing principled alternatives remains an important direction for future work. Acknowledgement We thank all reviewers, AC, and SAC for their valuable comments. 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Appendix In Appendix A, we present the full algorithmic description of SAUCE as a single-pass procedure over a multi-agent reasoning trajectory. In Appendix B, we provide our implementation details, including: • evaluation metrics (Appendix B.1), • benchmark formats and subset construction (Appendix B.2), • and hyperparameter configuration (Appendix B.3). In Appendix C, we present additional empirical studies, including: • a comparison with a learned, context-dependent transition coefficient (Appendix C.1), • per-axis hyperparameter sensitivity (Appendix C.2), • the full Brier-Platt calibration grid (Appendix C.3), • trajectory-summary comparisons and temporal-order ablations (Appendix C.4), • bootstrap comparisons and repeated-rollout variability (Appendix C.5), • and trajectory analysis under unanimous final agreement (Appendix C.6). Finally, in Appendix D, we provide proofs and derivations supporting our theoretical claims, including: • the proof of Proposition 4.1 (Appendix D.1), • the innovations form of the marginal likelihood (Appendix D.2), • and the equivalence with the evidence lower bound (Appendix D.3). Appendix A Algorithm Algorithm 1 summarizes SAUCE as a single-pass procedure over a parallel-MAS rollout. Given τ=(t1:N)t=1Tτ=( z_t^1:N)_t=1^T, SAUCE traverses the rounds once, extracts the round-level signals (Mt,Rt)(M_t,R_t) defined in Sec. 3.1, and applies the propagation–update recursion of Sec. 3.2. The procedure is training-free, requires no additional forward passes, and operates entirely on quantities available in a standard rollout (parsed answers and top-k token log-probabilities). All hyperparameters are scalars and are shared across (model, dataset) cells within each MAS protocol (Appendix C.2). Algorithm 1 SAUCE: Sequential Agent Uncertainty through Consensus Evolution. 1: Rollout τ=(t1:N)t=1Tτ=( z_t^1:N)_t=1^T for input x; hyperparameters λ∈(0,1)λ∈(0,1), q>0q>0, μ0 _0, v0>0v_0>0. 2: Per-record uncertainty score u(,τ)u( x,τ). 3: (μ,v)←(μ0,v0)(μ,v)←( _0,v_0) 4: for t=1,…,Tt=1,…,T do 5: Extract parsed answers atii=1N\a_t^i\_i=1^N and per-token log-probabilities from tii=1N\ z_t^i\_i=1^N 6: mt←mode(atii=1N)m_t (\a_t^i\_i=1^N) ⊳ round modal answer 7: Mt←1N∑i=1N[ati=mt]M_t← 1N _i=1^N1[a_t^i=m_t] ⊳ agreement signal (Eqn. 6) 8: Rt←1N∑i=1NHtok(ti)R_t← 1N _i=1^NH_tok( z_t^i) ⊳ generation uncertainty (Eqn. 7) 9: μ~←λμ μ←λ\,μ, v~←λ2v+q v←λ^2v+q ⊳ predict (Eqn. 8) 10: α←v~/(v~+Rt)α← v/( v+R_t) ⊳ Kalman gain (Eqn. 9) 11: μ←μ~+α(Mt−μ~)μ← μ+α\,(M_t- μ), v←(1−α)v~v←(1-α)\, v ⊳ update (Eqn. 10) 12: μt←μ _t←μ 13: end for 14: return u(,τ)=−1T∑t=1Tμtu( x,τ)=- 1T _t=1^T _t ⊳ (Eqn. 11) Appendix B Implementation Details This appendix collects the metric definitions, benchmark formats, and hyperparameter values used in the main results. B.1 Metrics Each estimator outputs an uncertainty score si=u(i,τi)s_i=u( x_i, _i) per record (higher = less reliable), evaluated against a binary correctness label yi∈0,1y_i∈\0,1\ on a dataset of size N. Let =i:yi=1P=\i:y_i=1\ and =i:yi=0N=\i:y_i=0\. AUROC. The probability that a uniformly drawn correct trajectory has a lower uncertainty score than a uniformly drawn incorrect one, with ties contributing one half: AUROC=1||||∑i∈∑j∈([si<sj]+12 1[si=sj]), \;=\; 1|P||N| _i _j (1[s_i<s_j]+ 12\,1[s_i=s_j] ), (18) equivalent to the Mann–Whitney U statistic, with tied pairs contributing 12 12. AUARC. The area under the accuracy–coverage curve obtained by rejecting the most uncertain trajectories first. Let π be the permutation that orders records by ascending uncertainty, sπ(1)≤sπ(2)≤…≤sπ(N)s_π(1)≤ s_π(2)≤…≤ s_π(N). Then AUARC=1N∑k=1N1k∑i=1kyπ(i), \;=\; 1N _k=1^N 1k _i=1^ky_π(i), (19) i.e., the average accuracy on the top-k retained subset over all coverage levels k/Nk/N. Brier after Platt scaling. The Brier score evaluated on calibrated probabilities, Brier=1N∑i=1N(p^i−yi)2,p^i=σ(a⋅(−si)+b), \;=\; 1N _i=1^N ( p_i-y_i )^2, p_i\;=\;σ\! (a·(-s_i)+b ), (20) where Platt parameters (a,b)(a,b) are fit by minimising the binary cross-entropy on training data. We use 5-fold cross-validation: (a,b)(a,b) is refit on each training fold, the Brier score is computed on the corresponding held-out fold, and we report the average across folds. All three metrics are reported as percentages (×100× 100). B.2 Benchmarks We follow each benchmark’s standard test split or, where listed in Table 4, a fixed random subset shared across all backbones. The same prompt template and decoding configuration are used for every (model, dataset, MAS) cell. The consensus answer of a MAS rollout is judged by an automatic verifier matched to the answer format: Benchmark Choice. Open-source models in our suite (Qwen3-4B, Llama-3.1-8B-Instruct, Gemma-4-E4B-it) are at the smaller-scale end of LLMs and already exhibit a wide accuracy spread on MATH-500, MMLU-Pro, and BBH, yielding a well-balanced split of correct and incorrect MAS outputs that uncertainty estimation can be meaningfully evaluated against. Closed-source models (Gemini-2.5-Flash-Lite, GPT-5.4-mini) largely saturate the original MATH-500 and BBH, leaving too little correct/incorrect variance for the same purpose. We therefore replace those with the harder variants, e.g., OlympiadBench for mathematics and BBEH-mini for hard instruction-following, to restore an informative spread at this stronger model class. MMLU-Pro spans a wide enough difficulty range to remain useful for both classes; we use the full split for open-source models and the 5,0005,000 subset above for closed-source models to control API cost. Multiple-Choice (MMLU-Pro, BBH, BBEH-mini): canonicalised exact match on the parsed option after format-aware normalisation (whitespace, parenthesisation, casing). Mathematical Reasoning (MATH-500, OlympiadBench): symbolic equivalence via the Math-Verify framework, which handles algebraic simplification (e.g. fraction/decimal forms, sign). Subset Construction. For closed-source models we use a 5,0005,000-question random subset of the MMLU-Pro test split, sampled uniformly from its 12,03212,032 items with a fixed seed and shared across all closed-source backbones and MAS protocols. Drawing a subset (rather than the full split) keeps API inference cost manageable while preserving the broad subject coverage of MMLU-Pro. BBEH-mini is used as released by [27], a compact ≈460≈\!460-question collection spanning 2323 Big-Bench Extra Hard tasks; we use it without further filtering. Table 4: Benchmark formats and evaluation sizes used in the main results. “–” denotes a benchmark not used for that model class. Dataset Format Open-source size Closed-source size MATH-500 Free-form math 500500 – MMLU-Pro Multiple-choice 12,03212,032 5,0005,000 (subset) BBH MCQ / short string 1,2501,250 – OlympiadBench Free-form math – 674674 BBEH-mini MCQ / short string – ≈460≈\!460 B.3 Hyperparameter Configuration SAUCE uses a single tuple (λ,q,σ02,μ0)(λ,q, _0^2, _0) per MAS protocol, transferred across every (model, dataset) cell of Tables 1–2. We tune on a single calibration cell (Qwen3-4B / MMLU-Pro) by joint grid search; per-axis sensitivity around these defaults is reported in Appendix C.2. Table 5: SAUCE hyperparameter defaults transferred across all cells. MAS protocol λ q σ02 _0^2 μ0 _0 Debate 0.950.95 0.100.10 0.250.25 0.00.0 DyLAN 0.700.70 0.0010.001 0.250.25 0.00.0 Appendix C Additional Experiments This appendix collects additional empirical studies supporting the main results: a comparison between constant and context-dependent transition coefficients (Appendix C.1), hyperparameter sensitivity (Appendix C.2), the full calibration grid (Appendix C.3), comparisons with simpler trajectory summaries and temporal-order ablations (Appendix C.4), statistical analyses (Appendix C.5), and a diagnostic study of trajectories with identical final-round agreement (Appendix C.6). C.1 Constant vs. Context-Dependent Transition Coefficient The main text instantiates the latent belief as st=λst−1+ϵt,ϵt∼(0,q), s_t=λ\,s_t-1+ _t, _t (0,q), (21) with a constant transition coefficient λ. A natural generalisation is to let λ depend on the interaction history t h_t, since how strongly an agent should carry its prior belief into the next round may plausibly vary with the conversation so far. This appendix shows the extra capacity is not exercised: under a tight variational objective, a learnable λθ(t)=σ(MLP(t−1)) _θ( h_t)=σ\! (MLP( h_t-1) ) collapses to a near-degenerate constant whose downstream AUROC is close to that of a fixed scalar, so we adopt the constant λ∗λ^* as the paper’s main configuration. Training Objective: An ELBO. We learn θ by maximising the evidence lower bound (ELBO) on the marginal likelihood of the observation sequence, an objective that does not use correctness labels. For any variational posterior q(s1:T)q(s_1:T), logpθ(o1:T) p_θ(o_1:T) =ℒ(θ,q)+KL(q(s1:T)∥pθ(s1:T∣o1:T)), =L(θ,q)+KL\! (q(s_1:T)\,\|\,p_θ(s_1:T o_1:T) ), (22) ℒ(θ,q) (θ,q) =q[logpθ(o1:T,s1:T)−logq(s1:T)]. =E_q\! [ p_θ(o_1:T,s_1:T)- q(s_1:T) ]. (23) Under the linear-Gaussian model the Kalman filter induced by λθ _θ yields the exact posterior, so the bound is tight (KL=0KL=0) and reduces to the closed-form innovations decomposition ℒ(θ)=−T2log(2π)−12∑t=1T[logSt+et2St],et=ot−λθ(t)μt−1,St=σ~t2+Rt. (θ)=- T2 (2π)- 12 _t=1^T [ S_t+ e_t^2S_t ], e_t=o_t- _θ( h_t)\, _t-1, S_t= σ_t^2+R_t. (24) The full derivation, including the equivalence between the ELBO and the innovations form, is given in Appendix D.3. Maximising Eqn. 24 is therefore variational learning of the transition coefficient, with no looser surrogate to blame if the procedure fails to extract context-dependent structure. Downstream Comparison. Table 6: Learned vs. constant transition coefficient on Qwen3-4B / Debate (AUROC↑ , ×100× 100). The two filter variants achieve similar AUROC on both datasets; both improve substantially over the no-filter baseline. Method MATH-500 MMLU-Pro mean(Rt)(R_t) – no filter 77.00 64.93 Learned λθ(t) _θ( h_t) 82.85 75.18 Const. λ∗λ^* (ours) 82.94 74.76 We optimise Eqn. 24 end-to-end with respect to θ on Qwen3-4B Debate trajectories from MATH-500 and MMLU-Pro and compare against the constant λ∗λ^* on the same trajectories. Table 6 reports downstream AUROC: the learned context-dependent λθ(t) _θ( h_t) and constant λ∗λ^* differ by only 0.090.09 and 0.420.42 AUROC points on MATH-500 and MMLU-Pro, respectively, and both substantially outperform the no-filter baseline. Because the bound is tight, the absence of an improvement cannot be attributed to a loose variational surrogate, and the optimiser nonetheless fails to extract any context-dependent signal that downstream uncertainty estimation can use. Inspection of the trained network confirms this picture: the learned λθ(t) _θ( h_t) concentrates on a near-degenerate spike with no measurable conditioning on correctness. We therefore adopt the constant transition coefficient as the paper’s main configuration: it achieves similar AUROC to the learned variant while remaining fully training-free, and it side-steps the additional engineering overhead (hidden-state access, optimiser hyperparameters, validation splits) that the learnable alternative drags along. Implementation Details. The MLP has a 64-dimensional bottleneck and a 32-dimensional ReLU hidden layer; q, μ0 _0 and σ02 _0^2 are fixed buffers shared with the constant baseline. We optimise the innovations NLL (the negative of Eqn. 24) with Adam at learning rate 10−310^-3 and early-stop on validation AUROC, which uses correctness labels. Trajectories are split 70/15/1570/15/15 into train, validation and test; sample sizes are n=500n\!=\!500 for MATH-500 and n=12,032n\!=\!12,032 for MMLU-Pro. C.2 Hyperparameter Sensitivity Figure 7: Per-axis sensitivity of SAUCE’s four hyperparameters at the calibration cell (Qwen3-4B / MMLU-Pro). Top row: Debate, selected values (λ,q,σ02,μ0)=(0.95,0.10,0.25,0.0)(λ,q, _0^2, _0)=(0.95,0.10,0.25,0.0). Bottom row: DyLAN, selected values (0.70,0.001,0.25,0.0)(0.70,0.001,0.25,0.0). In each panel, one hyperparameter is swept while the other three are held at their per-protocol selected value (marked by ★ and a dotted vertical line). Setup. SAUCE’s four scalar hyperparameters (λ,q,σ02,μ0)(λ,q, _0^2, _0) are tuned by joint grid search on the Qwen3-4B / MMLU-Pro calibration cell, separately for each MAS protocol. The resulting tuples, which are (0.95,0.10,0.25,0.0)(0.95,0.10,0.25,0.0) for Debate and (0.70,0.001,0.25,0.0)(0.70,0.001,0.25,0.0) for DyLAN, are then transferred verbatim to every other (model, dataset) cell in Tables 1–2. Fig. 7 sweeps each axis independently around the selected tuple. In Debate, AUROC drops by roughly three points as λ decreases from 0.950.95 to 0.10.1. Decreasing λ erodes the temporal coupling that aggregates evidence across rounds, consistent with the role of sequential dynamics highlighted in Table 3. The curves also show sensitivity to the initial mean and variance over the full swept range; these axes are not uniformly flat. We transfer the selected tuple to every other (model, dataset) cell instead of tuning each cell separately. The sensitivity study characterises the calibration cell; it does not bound the potential benefit of per-cell tuning on other models or datasets. C.3 Calibration on All Benchmarks Table 7 and Table 8 report the full Brier-Platt grid across all backbones, MAS protocols, and benchmarks. SAUCE attains the best per-backbone average on every model and is the leading or near-leading method on the large majority of cells, in line with the headline result of Fig. 5. Table 7: Calibration quality on open-source models, measured by Brier score after Platt scaling with 5-fold cross-validation (↓ , ×100× 100). Bold denotes the lowest score per (model, protocol, dataset) cell, and per-backbone Avg. Method MATH-500 MMLU-Pro BBH Avg Debate DyLAN Debate DyLAN Debate DyLAN Qwen3-4B PE 16.92 17.06 22.44 21.64 15.23 16.48 18.30 L 17.12 16.97 22.46 21.65 15.41 16.55 18.36 UDPO 17.16 15.13 21.40 19.84 15.54 15.85 17.49 MATU 17.86 17.94 22.38 22.34 16.32 16.16 18.83 SAUCE (Ours) 15.98 15.06 19.19 18.52 14.77 15.62 16.52 Llama-3.1-8B-Instruct PE 24.02 25.26 24.84 24.45 24.59 22.83 24.33 L 24.71 22.51 24.74 23.65 24.56 22.49 23.78 UDPO 20.70 22.38 22.03 21.67 21.94 22.51 21.87 MATU 25.02 20.76 24.94 23.47 23.68 22.32 23.36 SAUCE (Ours) 16.57 17.71 19.76 20.18 20.67 20.73 19.27 Gemma-4-E4B-it PE 13.43 12.79 18.12 16.79 11.68 17.23 15.01 L 13.41 12.78 18.14 16.78 11.69 17.21 15.00 UDPO 13.45 14.10 17.72 16.20 10.91 13.05 14.24 MATU 13.52 13.79 19.79 18.77 11.71 14.95 15.42 SAUCE (Ours) 13.30 13.79 15.53 15.29 10.06 12.40 13.40 Table 8: Calibration quality on closed-source models, measured by Brier score after Platt scaling with 5-fold cross-validation (↓ , ×100× 100). Notation follows Table 7. Method OlympiadBench MMLU-Pro BBEH-mini Avg Debate DyLAN Debate DyLAN Debate DyLAN Gemini-2.5-Flash-Lite PE 22.35 21.87 18.70 14.84 17.21 15.05 18.34 L 22.47 22.18 14.83 13.53 17.18 15.56 17.63 UDPO 21.60 16.63 18.47 15.93 16.80 14.21 17.27 MATU 22.02 23.45 18.62 15.93 17.06 14.56 18.61 SAUCE (Ours) 17.10 16.71 15.28 15.36 16.30 14.55 15.88 GPT-5.4-mini PE 24.87 20.50 16.41 16.12 15.67 13.80 17.90 L 24.86 20.67 16.48 16.03 15.59 13.83 17.91 UDPO 20.92 17.92 15.69 14.57 15.30 13.97 16.40 MATU 25.06 22.96 16.48 16.47 15.63 13.83 18.41 SAUCE (Ours) 16.38 17.04 14.58 14.18 15.54 13.98 15.28 C.4 Comparison with Simpler Trajectory Summary Methods In Sec. 5.5, we compared the full update with static agreement and reliability-weighted aggregation. Here, we extend the analysis to simpler trajectory summaries on Qwen3-4B and GPT-5.4-mini across six Debate settings. The experiments in this section follow the same protocol-specific hyperparameter settings as the main experiments, as specified in Appendix B.3. Evaluation Records. For the single-rollout comparisons, MMLU-Pro uses 12,03212,032 Qwen examples and 4,9994,999 evaluable GPT examples from the 5,0005,000-question subset; the remaining evaluation sizes are listed in Table 9. Alternative Summaries. All scores are oriented so that higher values indicate greater uncertainty: • EWMA: a fixed-weight moving average of agreement, initialized with h1=M1h_1=M_1 and updated as ht=(1−β)ht−1+βMth_t=(1-β)h_t-1+β M_t with β=0.3β=0.3. Its score is −T−1∑tht-T^-1 _th_t, matching SAUCE’s trajectory-averaged readout. • Slope: the negative least-squares slope of MtM_t over rounds. • Switches: the number of adjacent rounds in which the plurality answer changes. • TTC (time to consensus): the first round with unanimous agreement, with trajectories that never reach unanimity assigned T+1T+1. • Mean disagreement: T−1∑t(1−Mt)T^-1 _t(1-M_t). • Mean answer entropy: T−1∑tHtT^-1 _tH_t, where HtH_t is the entropy of the empirical distribution of parsed answers at round t. This measures cross-agent answer diversity and is distinct from token-level generation entropy RtR_t. Table 9: Simpler trajectory summaries under Debate (AUROC ↑ , ×100× 100). Boldface and underlining denote the best and second-best performance. Evaluation sizes and score definitions follow Appendix C.4. Method Qwen3-4B GPT-5.4-mini MATH-500 MMLU-Pro BBH OlympiadBench MMLU-Pro BBEH-mini EWMA 66.80 68.43 69.56 81.63 66.95 53.46 Slope 37.79 38.76 40.75 32.36 39.56 57.25 Switches 61.76 60.59 61.47 71.08 57.58 52.43 TTC 66.47 68.29 68.29 79.97 66.31 56.94 Mean 1−Mt1-M_t 66.60 68.56 69.72 82.33 67.08 55.95 Mean HtH_t 66.78 68.35 69.33 82.37 66.80 56.63 SAUCE (Ours) 68.59 76.42 72.86 83.83 69.53 49.96 n 500 12,032 1,250 674 4,999 460 Results. SAUCE achieves the highest AUROC among the listed summaries in five of the six settings. It also outperforms EWMA in those five settings, indicating that its gains generally extend beyond fixed-weight temporal smoothing. The exception is GPT-5.4-mini on BBEH-mini, where SAUCE is near chance and the slope summary performs best. The paired-bootstrap comparison with EWMA is reported in Appendix C.5. Temporal-Order Ablation. We keep each trajectory and its final prediction fixed, but process the paired signals (Mt,Rt)(M_t,R_t) in their original order, in reverse order, or after a random permutation. Shuffled AUROC is averaged over three permutation seeds. The evaluation records and filter parameters are unchanged across conditions. Table 10: Temporal-order ablation under Debate (AUROC ↑ , ×100× 100). Shuffled averages three seeds; Δ is original minus the better alternative. Best and second-best scores are bold and underlined. Method Qwen3-4B GPT-5.4-mini MATH-500 MMLU-Pro BBH OlympiadBench MMLU-Pro BBEH-mini Original 68.59 76.42 72.86 83.83 69.53 49.96 Reversed 68.30 71.68 74.26 82.29 65.21 49.72 Shuffled 68.41 72.45 73.84 82.99 66.22 49.12 Δ +0.19+0.19 +3.97+3.97 −1.41-1.41 +0.84+0.84 +3.31+3.31 +0.24+0.24 Table 10 shows that the original order outperforms both alternatives in five of the six settings. The clearest differences occur on MMLU-Pro: original processing exceeds the better alternative by 3.973.97 AUROC points for Qwen3-4B and 3.313.31 for GPT-5.4-mini. The effect is smaller elsewhere, and reversing the order improves Qwen3-4B on BBH. Thus, chronological order provides useful information in several settings, but its benefit is not uniform across tasks. C.5 Statistical Significance The experiments in this section follow the same protocol-specific hyperparameter settings as the main experiments, as specified in Appendix B.3. Prompt-Level Bootstrap. We perform 1,0001,000 paired bootstrap resamples of the evaluation records in each setting of Table 9. Each resample selects records with replacement and applies the same indices to all estimators. The 2.52.5th and 97.597.5th percentiles give the reported 95%95\% AUROC intervals. For each baseline B, we also report the empirical tail fraction pBp_B, the proportion of paired resamples in which AUROC(SAUCE )−AUROC(B)≤0AUROC(SAUCE )-AUROC(B)≤ 0. These comparisons are one-sided and unadjusted for multiple comparisons. A reported value below 0.0010.001 means no nonpositive difference was observed in 1,0001,000 resamples; it does not establish a zero population probability. Table 11: Paired-bootstrap comparisons under Debate. The first two rows report SAUCE’s AUROC (×100× 100) and 95% interval; subsequent rows give its tail fraction against each baseline. Bold indicates p<0.05p<0.05; <0.001<0.001 denotes zero tail counts in 1,000 resamples. Statistic Qwen3-4B GPT-5.4-mini MATH-500 MMLU-Pro BBH OlympiadBench MMLU-Pro BBEH-mini SAUCE 68.59 76.42 72.86 83.83 69.53 49.96 95% CI 62.6–74.7 75.6–77.3 69.4–76.3 80.7–87.1 67.4–71.5 43.8–56.3 vs. PE 0.011 <0.001<0.001 0.019 <0.001<0.001 <0.001<0.001 0.121 vs. L 0.020 <0.001<0.001 0.034 <0.001<0.001 <0.001<0.001 0.002 vs. UDPO <0.001<0.001 <0.001<0.001 <0.001<0.001 <0.001<0.001 <0.001<0.001 1.000 vs. MATU 0.016 <0.001<0.001 <0.001<0.001 <0.001<0.001 <0.001<0.001 0.378 vs. EWMA 0.156 <0.001<0.001 <0.001<0.001 0.002 <0.001<0.001 0.998 In these comparisons, L averages token log-likelihood over agents and rounds, and MATU uses a single rollout (N=1N=1). In these supplementary comparisons, the tail fraction is below 0.050.05 against PE, averaged L, UDPO, and single-run MATU in five of the six settings. Against EWMA, it is below 0.050.05 in four settings. Qwen3-4B / MATH-500 has a positive point-estimate difference from EWMA but a tail fraction of 0.1560.156, while GPT-5.4-mini / BBEH-mini does not show an advantage over EWMA. These intervals quantify prompt-sampling uncertainty for a fixed rollout; they do not measure variability across independently generated rollouts. Repeated-Rollout Variability. We separately summarize independently generated Debate rollouts at temperature 0.70.7 on four settings, evaluating SAUCE, averaged L, and final-round PE on each run. Table 12 reports the arithmetic mean and sample standard deviation of AUROC, together with the actual run counts and evaluation sizes. MATH-500, BBH, and BBEH-mini each have five runs; OlympiadBench has six, with 673673–674674 evaluable records per run. We restrict this summary to settings with comparable evaluation sizes across runs. Table 12: Repeated Debate rollouts (AUROC ↑ , mean ± sample standard deviation, ×100× 100). Boldface and underlining denote the best and second-best means. Method Qwen3-4B GPT-5.4-mini MATH-500 BBH OlympiadBench BBEH-mini PE 67.04 ± 1.64 69.83 ± 0.99 53.65 ± 1.92 46.11 ± 1.31 L 67.67 ± 1.22 71.57 ± 1.59 70.17 ± 0.68 43.34 ± 2.54 SAUCE (Ours) 72.26 ± 2.29 72.58 ± 1.76 82.79 ± 0.87 49.01 ± 1.51 Runs 5 5 6 5 n per run 500 1,250 673–674 460 SAUCE has the highest mean AUROC among the three estimators in all four settings, with sample standard deviations ranging from 0.870.87 to 2.292.29 AUROC points. On BBEH-mini its mean remains near chance, so the relative advantage over L and PE should not be interpreted as reliable discrimination on that task. The repeated-run summaries and prompt-level bootstrap address different sources of variability and are reported separately. C.6 Trajectory Analysis under Unanimous Final Agreement The experiment in this section follows the same protocol-specific hyperparameter settings as the main experiments, as specified in Appendix B.3. The motivating example in Fig. 1 suggests that identical final agreement can conceal different interaction histories. We examine this directly on Qwen3-4B / Debate / MMLU-Pro by restricting attention to the 10,69610,696 of 12,03212,032 trajectories whose agents give the same parsed answer in the final round. We group these trajectories by the total number of individual agents’ answer changes between adjacent rounds, rather than the plurality-answer switches used as a baseline in Appendix C.4. Table 13: Answer-switch analysis among final-unanimous trajectories on Qwen3-4B / Debate / MMLU-Pro (n=10,696n=10,696). Switches count changes in individual agents’ answers. Switches n Accuracy (%) Mean confidence 0 8,004 76.9 0.915 1 1,595 49.5 0.758 2 486 44.4 0.676 3+ 611 38.8 0.622 As the number of agent-level switches increases from zero to three or more, accuracy decreases from 76.9%76.9\% to 38.8%38.8\%, while mean SAUCE confidence μ¯=−u μ=-u decreases from 0.9150.915 to 0.6220.622. Final-round agreement is constant across these groups, but SAUCE’s score tracks their different empirical reliability. This is a descriptive association between interaction history, confidence, and correctness; the analysis does not establish a causal effect of switching, and μ¯ μ is not a calibrated correctness probability. Appendix D Theoretical Analysis and Proofs This appendix collects the formal derivations and proofs supporting the linear-Gaussian instantiation of SAUCE in Sec. 4. Appendix D.1 proves Proposition 4.1 (the closed-form posterior update that defines the SAUCE recursion). Appendix D.2 derives the innovations form of the marginal likelihood used as our training objective in Appendix C.1. Appendix D.3 shows that, under our linear-Gaussian model, this objective is also the tight evidence lower bound, which underwrites the variational interpretation of the empirical analysis. D.1 Proof of Proposition 4.1 Proof. We proceed by induction on t. The base case at t=0t=0 holds by the prior s0∼(μ0,σ02)s_0 ( _0, _0^2). Assume the inductive hypothesis st−1∣o1:t−1∼(μt−1,σt−12)s_t-1 o_1:t-1 ( _t-1, _t-1^2); we show that the same Gaussian form propagates to round t with the stated mean and variance. Step 1: Predictive Distribution. The transition st=λst−1+ϵts_t=λ s_t-1+ _t with ϵt∼(0,q) _t (0,q) independent of (st−1,o1:t−1)(s_t-1,o_1:t-1) is an affine transformation of a Gaussian plus an independent Gaussian, so st∣o1:t−1s_t o_1:t-1 is Gaussian with [st∣o1:t−1] [s_t o_1:t-1] =[λst−1+ϵt∣o1:t−1]=λμt−1=:μ~t, \;=\;E[λ s_t-1+ _t o_1:t-1]\;=\;λ _t-1\;=:\; μ_t, Var[st∣o1:t−1] [s_t o_1:t-1] =λ2Var[st−1∣o1:t−1]+Var[ϵt]=λ2σt−12+q=:σ~t2. \;=\;λ^2\,Var[s_t-1 o_1:t-1]+Var[ _t]\;=\;λ^2 _t-1^2+q\;=:\; σ_t^2. Hence st∣o1:t−1∼(μ~t,σ~t2)s_t o_1:t-1 ( μ_t, σ_t^2). Step 2: Joint Distribution of State and Observation. The observation model ot|st∼(st,rt2)o_t s_t (s_t,r_t^2) is equivalent to ot=st+ηto_t=s_t+ _t with ηt∼(0,rt2) _t (0,r_t^2) independent of sts_t given o1:t−1o_1:t-1. Conditional on o1:t−1o_1:t-1, both sts_t and oto_t are linear functions of jointly Gaussian variables, so the conditional joint (st,ot)∣o1:t−1(s_t,o_t) o_1:t-1 is itself Gaussian. Its first two moments are [ot∣o1:t−1] [o_t o_1:t-1] =[st+ηt∣o1:t−1]=μ~t, \;=\;E[s_t+ _t o_1:t-1]\;=\; μ_t, Var[ot∣o1:t−1] [o_t o_1:t-1] =Var[st∣o1:t−1]+Var[ηt]=σ~t2+rt2, \;=\;Var[s_t o_1:t-1]+Var[ _t]\;=\; σ_t^2+r_t^2, Cov(st,ot∣o1:t−1) (s_t,\,o_t o_1:t-1) =Cov(st,st+ηt∣o1:t−1)=Var[st∣o1:t−1]=σ~t2, \;=\;Cov(s_t,\,s_t+ _t o_1:t-1)\;=\;Var[s_t o_1:t-1]\;=\; σ_t^2, where the last line uses the conditional independence Cov(st,ηt∣o1:t−1)=0Cov(s_t, _t o_1:t-1)=0. Collecting these moments, (stot)|o1:t−1∼((μ~tμ~t),(σ~t2σ~t2σ~t2σ~t2+rt2)). pmatrixs_t\\ o_t pmatrix\, |\,o_1:t-1\; \;N\! (\, pmatrix μ_t\\ μ_t pmatrix,\; pmatrix σ_t^2& σ_t^2\\ σ_t^2& σ_t^2+r_t^2 pmatrix ). (25) Step 3: Conditioning on the New Observation. For any jointly Gaussian pair (X,Y)(X,Y) with means (μX,μY)( _X, _Y), variances (ΣXX,ΣYY)( _X, _Y), and cross-covariance ΣXY _XY, the standard conditioning identity gives X|Y∼(μX+ΣXYΣYY−1(Y−μY),ΣXX−ΣXYΣYY−1ΣYX). X Y\; \;N\! ( _X+ _XY _Y^-1(Y- _Y),\;\; _X- _XY _Y^-1 _YX ). (26) Applying Eqn. 26 to the joint in Eqn. 25 with X=stX=s_t and Y=otY=o_t yields st∣o1:t∼(μ~t+σ~t2σ~t2+rt2(ot−μ~t),σ~t2−(σ~t2)2σ~t2+rt2). s_t o_1:t\; \;N\! (\, μ_t+ σ_t^2 σ_t^2+r_t^2 (o_t- μ_t ),\;\; σ_t^2- ( σ_t^2)^2 σ_t^2+r_t^2\, ). Define the Kalman gain αt:=σ~t2σ~t2+rt2∈[0,1]. _t\;:=\; σ_t^2 σ_t^2+r_t^2\;∈\;[0,1]. Substituting back, the posterior mean and variance simplify to μt=μ~t+αt(ot−μ~t),σt2=σ~t2⋅rt2σ~t2+rt2=(1−αt)σ~t2. _t\;=\; μ_t+ _t (o_t- μ_t ), _t^2\;=\; σ_t^2· r_t^2 σ_t^2+r_t^2\;=\;(1- _t)\, σ_t^2. The variance further admits a harmonic-of-precisions form: starting from σt2=σ~t2rt2/(σ~t2+rt2) _t^2= σ_t^2r_t^2/( σ_t^2+r_t^2), 1σt2=σ~t2+rt2σ~t2rt2=1σ~t2+1rt2, 1 _t^2\;=\; σ_t^2+r_t^2 σ_t^2\,r_t^2\;=\; 1 σ_t^2+ 1r_t^2, which is exactly the precision-addition statement in the proposition. Since st∣o1:ts_t o_1:t is Gaussian with the stated mean and variance, the inductive hypothesis carries to round t, completing the induction. ∎ D.2 Innovations Form of the Marginal Likelihood By the chain rule of probability, the log marginal likelihood factorises as logp(o1:T)=∑t=1Tlogp(ot∣o1:t−1) p(o_1:T)= _t=1^T p(o_t o_1:t-1). Under Proposition 4.1 the posterior from the previous round is p(st−1∣o1:t−1)=(μt−1,σt−12)p(s_t-1 o_1:t-1)=N( _t-1, _t-1^2); marginalising over the linear-Gaussian transition gives the predictive distribution p(st∣o1:t−1)=(μ^t,σ~t2),μ^t=λt(t)μt−1,σ~t2=λt(t)2σt−12+q, p(s_t o_1:t-1)=N\! ( μ_t,\; σ_t^2 ), μ_t= _t( h_t)\, _t-1, σ_t^2= _t( h_t)^2\, _t-1^2+q, (27) by the convolution of two Gaussians. Marginalising once more over sts_t under the linear-Gaussian observation model yields p(ot∣o1:t−1)=(μ^t,St),St=σ~t2+Rt, p(o_t o_1:t-1)=N\! ( μ_t,\;S_t ), S_t= σ_t^2+R_t, (28) and substituting the Gaussian log-density and summing over t produces the innovations form logp(o1:T)=−T2log(2π)−12∑t=1T[logSt+et2St],et=ot−μ^t. p(o_1:T)=- T2 (2π)- 12 _t=1^T [ S_t+ e_t^2S_t ], e_t=o_t- μ_t. (29) The two terms balance prediction error against predicted uncertainty: et2/Ste_t^2/S_t penalises errors relative to the predictive variance, while logSt S_t penalises overconfidence. D.3 Equivalence with the Evidence Lower Bound For the following decomposition, we assume a Markov-structured variational posterior, q(s1:T)=q(s1)∏t=2Tq(st∣st−1),q(s_1:T)=q(s_1) _t=2^Tq(s_t s_t-1), where dependence on the observed sequence o1:To_1:T is left implicit. For any variational posterior q(s1:T)q(s_1:T) over the latent trajectory, the standard decomposition gives logp(o1:T) p(o_1:T) =ℒ(q)+KL(q(s1:T)∥p(s1:T∣o1:T)), =L(q)+KL\! (q(s_1:T)\,\|\,p(s_1:T o_1:T) ), (30) ℒ(q) (q) =q[logp(o1:T,s1:T)−logq(s1:T)]. =E_q\! [ p(o_1:T,s_1:T)- q(s_1:T) ]. (31) Using the Markov structure p(o1:T,s1:T)=p(s1)∏t=2Tp(st∣st−1)∏t=1Tp(ot∣st)p(o_1:T,s_1:T)=p(s_1) _t=2^Tp(s_t s_t-1) _t=1^Tp(o_t s_t), the ELBO splits into three interpretable terms, ℒ(q) (q) =∑t=1Tq(st)[logp(ot∣st)]⏟(I) observation fit−KL(q(s1)∥p(s1))⏟(I) prior regularisation \;=\; _t=1^TE_q(s_t)\! [ p(o_t s_t) ]_(I) observation fit\;-\; KL\! (q(s_1)\,\|\,p(s_1) )_(I) prior regularisation −∑t=2Tq(st−1)[KL(q(st∣st−1)∥p(st∣st−1))]⏟(I) transition regularisation, \;-\; _t=2^TE_q(s_t-1)\! [KL\! (q(s_t s_t-1)\,\|\,p(s_t s_t-1) ) ]_(I) transition regularisation, (32) each of which has a closed form under Gaussian q (omitted for brevity). Tightness then follows from a standard property of linear-Gaussian state-space models: the Kalman smoother yields the exact posterior p(s1:T∣o1:T)p(s_1:T o_1:T), so setting q(s1:T)=p(s1:T∣o1:T)q(s_1:T)=p(s_1:T o_1:T) makes KL(q∥p)=0KL(q\|p)=0 and the ELBO coincides with the exact log marginal likelihood. Evaluating the three terms above at the Kalman posterior and simplifying recovers the innovations form of Appendix D.2. The training objective in Appendix C.1 is therefore simultaneously the exact marginal likelihood and the tight ELBO, a property we use implicitly when arguing that the failure of λθ(t) _θ( h_t) to outperform a constant cannot be attributed to a loose variational bound.