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Multi-Level Causal Embeddings
Willem Schooltink, Fabio Massimo Zennaro
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This paper introduces a framework for 'causal embeddings,' a generalization of causal abstraction that allows multiple detailed low-level models to be mapped into sub-systems of a coarser high-level causal model. The authors define causal embeddings, present a generalized notion of consistency (both functional and graphical), and demonstrate their relevance to the multi-resolution marginal problem. The work highlights the practical utility of causal embeddings for merging datasets derived from models with different representations and levels of detail.
Entities (9)
Relation Signals (7)
Fabio Massimo Zennaro → authored → Multi-Level Causal Embeddings
confidence 99% · Fabio Massimo Zennaro
Willem Schooltink → authored → Multi-Level Causal Embeddings
confidence 99% · Multi-Level Causal Embeddings Willem Schooltink
Causal Embeddings → generalizes → Causal Abstraction
confidence 95% · We define causal embeddings as a generalization of abstraction
Causal Embeddings → usedfor → Merging Datasets
confidence 92% · illustrate its practical use in merging datasets coming from models with different representations
Causal Embeddings → appliesto → Multi-Resolution Marginal Problem
confidence 90% · By defining a multi-resolution marginal problem, we showcase the relevance of causal embeddings
Structural Causal Models → underpins → Causal Embeddings
confidence 90% · The formalism of Structural Causal Models (SCMs) ... rigorously captures reasoning ... We define an SCM as follows
Cluster DAG → usedin → Graphical Consistency
confidence 85% · On the graphical level abstractions are defined as relations between the graphs of causal models. We will consider the framework of the Cluster DAG
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Abstract
Abstract:Abstractions of causal models allow for the coarsening of models such that relations of cause and effect are preserved. Whereas abstractions focus on the relation between two models, in this paper we study a framework for causal embeddings which enable multiple detailed models to be mapped into sub-systems of a coarser causal model. We define causal embeddings as a generalization of abstraction, and present a generalized notion of consistency. By defining a multi-resolution marginal problem, we showcase the relevance of causal embeddings for both the statistical marginal problem and the causal marginal problem; furthermore, we illustrate its practical use in merging datasets coming from models with different representations.
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- Source: https://arxiv.org/abs/2602.22287v2
- Canonical: https://arxiv.org/abs/2602.22287v2
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Multi-Level Causal Embeddings Willem Schooltink willem.schooltink@uib.no Fabio Massimo Zennaro fabio.zennaro@uib.no (Department of Informatics, University of Bergen ) Abstract Abstractions of causal models allow for the coarsening of models such that relations of cause and effect are preserved. Whereas abstractions focus on the relation between two models, in this paper we study a framework for causal embeddings which enable multiple detailed models to be mapped into sub-systems of a coarser causal model. We define causal embeddings as a generalization of abstraction, and present a generalized notion of consistency. By defining a multi-resolution marginal problem, we showcase the relevance of causal embeddings for both the statistical marginal problem and the causal marginal problem; furthermore, we illustrate its practical use in merging datasets coming from models with different representations. INTRODUCTION Causality enables us to reason about real-world systems on a level beyond statistics, allowing us to answer questions on the effects of interventions and hypotheticals. Such queries naturally show in many fields, such as medicine, biology or economics, where studying correlation is not enough. The formalism of Structural Causal Models (SCMs) [Pearl, 2009] rigorously captures reasoning about observations, interventions and counterfactuals (hypotheticals). However, causal models of real-world systems often grow very large to a point where reasoning becomes impractical, as SCMs, based on Directed Acyclic Graphs (DAGs), do not scale very well. One solution to the issue is to work with models at a coarser resolution. For example, when modeling the wildlife populations in a forest, we may have data on all subspecies of deer and rodents present, but for our purposes we may only care about the combined population of all deer subspecies. In such cases we can use causal abstraction to describe how a detailed (low-level) model maps to a coarser (high-level) model, whilst preserving causal relations when merging variables and the values they take. Such frameworks of causal abstraction [Rischel, 2020, Beckers and Halpern, 2019a, Beckers et al., 2020] have rigorous mathematical foundations, and provide measures to evaluate consistency among the models. Another solution to deal with large SCMs is to reason about sub-systems. In such cases we wish to map, or embed, the detailed models into sub-parts of a coarse model. Fig.1 illustrates how embeddings compare against abstractions. While abstraction deals with one-to-one mappings of models, embeddings deal with the common challenge in the sciences of having a high-level model of a system, for example an overarching climate model, and a combination of low-level sub-models, each describing only a part of the global model. Low-level:AbstractionEmbeddingX1X_1X2X_2X3X_3YYHigh-level:XXYYX1X_1X2X_2X3X_3YYXXWWYYZZ Figure 1: A visual comparison between abstractions (left) and embeddings (right). Note that abstractions (blue) have mappings to all variables in the high-level model, whereas the embeddings (orange) provides a fine-grained description only of the sub-system X→YX→ Y. Contribution. In this paper, we extend the idea of abstractions: whereas abstractions describe how an entire high-level SCM can be described by a more detailed low-level SCM, we study causal embeddings, describing how sub-systems of a high-level SCM can be described by detailed low-level models. We will (i) illustrate how this new point of view allows us to define a high-level causal model as the combination of multiple low-level sub-system models, (i) discuss the graphical and functional consistency of causal embeddings, (i) discuss theoretical applications of causal embeddings in a multi-level version of the marginal problem, and (iv) show the use of causal embeddings as a tool to merge overlapping datasets with differing levels of detail. RELATED WORKS Causal abstractions can be described as SCM-to-SCM mappings; two frameworks have been proposed: the category theoretical α framework [Rischel, 2020, Rischel and Weichwald, 2021] and the τ-ω framework [Rubenstein et al., 2017, Beckers and Halpern, 2019b, Beckers et al., 2020]. Alternatively, abstractions can be described as DAG-to-DAG mappings from one causal graph to another [Anand et al., 2023]. The relations between these abstraction frameworks have been studied by Schooltink and Zennaro [2025]. The closest work to our proposal is Otsuka and Saigo [2022], who adopts a non-surjective definition of abstraction in their proposed ϕφ framework. However, our definition is more flexible and extends to the case of embedding multiple low-level models into a single high-level model. Embeddings have relation to both the statistical [Kellerer, 1964] and causal marginal problem [Gresele et al., 2022]. Given multiple marginal SCMs Gresele et al. [2022] proposes a method to generate a family of compatible joint SCMs, and through falsification find those that are counterfactually consistent. We show that embeddings can be used to tackle a multi-resolution version of the causal marginal problem. Relatedly, Mejia et al. [2022] discusses how SCMs can be learned for overlapping statistical datasets under certain assumptions, while the Integration of Overlapping Datasets algorithm [Tillman and Spirtes, 2011] proposes a sound method to merge causal datasets and learn an equivalence class of graphs representing the data generation. This work was extended to a more special case by Bang and Didelez [2025] exploiting potential knowledge of variable ordering. Finally, embeddings can be used to merge causal datasets. Causal abstractions has already been used before to merge datasets in order to improve statistical power [Zennaro et al., 2023, Felekis et al., 2024], although limited to transporting data from one model to another one. In the causal literature, Janzing [2018] shows how causal models can help in merging overlapping statistical datasets, while exploiting data from different models has been studied in the context of data fusion and transportability [Bareinboim and Pearl, 2016, Pearl and Bareinboim, 2022], including the case of multiple environments [Bareinboim and Pearl, 2013]; these approaches, however, do not establish an explicit relation between SCMs, but rely on graphical calculus to take best advantage of observational and interventional data from a source and target model. PRELIMINARIES We first introduce the necessary background, specifically we present SCMs, Pearl’s Causal Hierarchy, causal abstraction, and the causal marginal problem. Notation. Throughout we will have a set of variables represented using bold uppercase V, and a specific variable using regular uppercase with index subscript when necessary Vi∈V_i . The value of a set of variables is indicated by bold lowercase v, and the value of a single variable using regular lowercase viv_i, again with index subscript when necessary. Additionally, a distribution over a variable V or set of variables V is denoted as P(V)P(V) and P()P(V), respectively. CAUSALITY Structural Causal Models. SCMs are formal descriptions of causal models, specifying causal variables and effects among them [Pearl, 2009]. We define an SCM as follows: Definition 1 (Structural Causal Model). An SCM is a 4-tuple ℳ:⟨,,ℱ,P()⟩M: ,V,F,P(U) , with: • U: a set of unobservable (exogenous) variables, which can take values in the range ℛ()R(U), • V: a set of observable (endogenous) variables, which can take values in the range ℛ()R(V), • ℱF: a collection of functions determining the value of the endogenous variables V, such that for each V∈V there exists a function fV(PaV,V)f_V(Pa_V,U_V) with PaV⊆∖V and V⊆,Pa_V V and U_V , • P()P(U): a probability distribution over the exogenous variables U. An SCM ℳM entails a unique directed graph G:⟨,⟩G: ,E with the vertices given by the endogenous variables V and an edge Vi→Vj∈V_i→ V_j if the value of VjV_j depends on ViV_i. Specifically, Vi→Vj∈V_i→ V_j if Vi∈PaVjV_i∈ Pa_V_j. Thus the set PaVjPa_V_j is the set of parents of VjV_j in a graph theoretical sense. Additionally, whenever two variables Vi,VjV_i,V_j share an unobservable parent (confounder) U∈U we add a dashed bidirected arrow ViVjV_i to25.35pt to5.8pt -2.59998pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-6.94427pt1.7pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke \;\;\;\;\; @invoke @endscope @invoke @endscope @beginscope @invoke @setdash @temp @x @invoke @moveto-8.67729pt0.0pt @lineto8.67729pt0.0pt @stroke @invoke @beginscope @invoke @transformcm-1.00.00.0-1.0-8.87729pt0.0pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm1.00.00.01.08.87729pt0.0pt @invoke @invoke @invoke @endscope @invoke @endscope @invoke @endscope @discardpath @invoke @endscope V_j. We assume the SCMs to have no cyclic relations: the graph entailed by the SCM is acyclic (DAG). Notice that the bidirected arrow is a shorthand for Vi←U→VjV_i← U→ V_j; hence the graph G remains acyclic. Causal models allow for reasoning about causality through interventions. In this paper we consider hard interventions: Definition 2 (Intervention). Given a causal model ℳ:=⟨,,ℱ,P()⟩M:= ,U,F,P(U) an intervention on a variable X∈X , denoted as do(X=x)do(X=x), is the replacement of the function fX∈ℱf_X with a constant function fX′=xf _X=x. As is common, we will apply the dodo-operator over sets of variables ⊆X , implying the replacement of the functions fX∈ℱf_X for all X∈X . In essence, when intervening on a variable X its value is no longer dependent on its parents but forcibly set to some value x. Consequently, as the DAG implied by an SCM has edges determined by the functions ℱF, interventions have graphical implications: specifically, an intervention do(X)do(X) removes all edges coming into X in the DAG. PEARL’S CAUSAL HIERARCHY. In causal reasoning there are three distinct types of questions one may wish to answer: (i) given the observation X what can be said about Y, (i) if X is set to X=xX=x what can be said about Y, (i) given observations Y=yY=y and X=xX=x what can be said about Y if X had been set to X=x^X= x? These questions can be categorized as (i) observing ℒ1L_1, (i) acting ℒ2L_2, and (i) imagining ℒ3L_3. This defines the Pearl’s Causal Hierarchy (PCH) [Pearl, 2009, Bareinboim et al., 2022], where each layer subsumes the previous layer, but cannot be reduced to the previous one: a query at layer ℒiL_i can be answered at layer ℒi+1L_i+1, but a query in layer ℒi+1L_i+1 cannot in general be reduced to layer ℒiL_i. Fully specified SCMs allow for reasoning about counterfactuals ℒ3L_3. Notation. For generalization, we will use the ℒiL_i operator in distributions: instead of P(Y|X)P(Y|X) we write P(Y|ℒ1(X))P(Y|L_1(X)), and instead of P(Y|do(X))P(Y|do(X)) we write P(Y|ℒ2(X))P(Y|L_2(X)). We will not consider ℒ3L_3 quantities in this work. CAUSAL ABSTRACTIONS Causal abstractions provide tools that allow us to map a detailed low-level causal model to a coarser high-level model. Similar to how causal models have a functional and graphical side (SCMs and DAGs, respectively), abstractions can be defined on both the functional and graphical level. Functional. We first consider the functional side through the framework of the α-abstraction [Rischel, 2020]. Definition 3 (α-abstraction). Let ℳ:⟨ℳ,ℳ,ℱℳ,M: _M,U_M,F_M, P(ℳ)⟩P(U_M) and ℳ′:⟨ℳ′,ℳ′,ℱℳ′,P(ℳ′)⟩M : _M ,U_M ,F_M ,P(U_M ) be two SCMs, then an α-abstraction :ℳ→ℳ′ α:M is given by a 3-tuple ⟨,φ,αV′⟩ , , _V with: 1. ⊆ℳR _M is a subset of relevant variables in ℳM. 2. φ:→ℳ′ :R _M is a surjective map from the relevant variables to the variables of ℳ′M . 3. αV′:ℛ(φ−1(V′))→ℛ(V′) _V :R( ^-1(V )) (V ), for each V′∈ℳ′V _M , is a surjective function from the range of the variables in the pre-image φ−1(V′)⊆ℳ ^-1(V ) _M in ℳM to the range of the variable V′V in ℳ′M . Importantly, the application of α _V to a distribution is given by the pushforward α[P()]=α#(P)() _V [P(V) ]= _V_\#(P)(V). α-abstractions do not enforce any form of causal consistency between a base model and an abstracted model; any mapping, as long as surjective, is permissible. For this reason, an error measure is used to define how much an abstracted model agrees with a base model. For α-abstractions the error measure is defined as follows: Definition 4 (ℒiL_i-Abstraction error). Let :ℳ→ℳ′ α:M be an α-abstraction and ′,′⊆ℳ′X ,Y _M , its ℒiL_i-error is given by the distance or divergence D between the distribution obtained by first abstracting and then evaluating: Pℳ′(′|α′[ℒi(φ−1(′))]), [rgb]0,0,1 [named]pgfstrokecolorrgb0,0,1P_M (Y \>|\> _X [L_i( ^-1(X ))]), and that obtained by first evaluating and then abstracting: α′[Pℳ(φ−1(′)|ℒi(φ−1(′)))], [rgb]1,.5,0 [named]pgfstrokecolorrgb1,.5,0 _Y [P_M( ^-1(Y )\>|\>L_i( ^-1(X ))) ], and taking the maximum over all ′,′⊆ℳ′X ,Y _M as the error. Visually this corresponds to the maximum distance between the paths in blue and orange over all diagrams of the following form: φ−1(′) ^-1 (Y )φ−1(′)|ℒi(φ−1(′)) ^-1 (Y )\>|\>L_i ( ^-1(X ) )′Y ′|ℒi(′)Y \>|\>L_i(X )ℒi(φ−1(′))L_i( ^-1(X ))ℒi(X′)L_i(X )α′ _Y α′ _X or formally by the following equation: eℒi()=max′,′⊆ℳ′D(Pℳ′(′|α′[ℒi(φ−1(′))]),α′[Pℳ(φ−1(′)|ℒi(φ−1(′)))]) -4.0pte_L_i( α)= _X ,Y _M D ( [rgb]0,0,1 [named]pgfstrokecolorrgb0,0,1P_M (Y | _X [L_i( ^-1(X ))]), .\\ . [rgb]1,.5,0 [named]pgfstrokecolorrgb1,.5,0 _Y [P_M( ^-1(Y )\>|\>L_i( ^-1(X ))) ] ) (1) An abstraction is ℒiL_i-consistent if its ℒiL_i-error is zero. Graphical. On the graphical level abstractions are defined as relations between the graphs of causal models. We will consider the framework of the Cluster DAG (CDAG), as introduced by Anand et al. [2023]. Definition 5 (Cluster DAG). Let ℳM and ℳ′M be two causal models admitting DAGs Gℳ:⟨ℳ,ℳ⟩G_M: _M,E_M and Gℳ′:⟨ℳ′,ℳ′⟩G_M : _M ,E_M , respectively; let φ:ℳ→ℳ′ :V_M _M be a surjective map from the variables of ℳM to the variables of ℳ′M . Gℳ′G_M is a Cluster DAG of GℳG_M if: 1. A directed edge Vi′→Vj′V_i → V_j is in Eℳ′E_M iff there exists an edge Vn→Vm∈EℳV_n→ V_m∈ E_M such that φ(Vn)=Vi′ (V_n)=V_i and φ(Vm)=Vj′ (V_m)=V_j . 2. A bidirected edge Vi′Vj′V_i to25.35pt to5.8pt -2.59998pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-6.94427pt1.7pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke \;\;\;\;\; @invoke @endscope @invoke @endscope @beginscope @invoke @setdash @temp @x @invoke @moveto-8.67729pt0.0pt @lineto8.67729pt0.0pt @stroke @invoke @beginscope @invoke @transformcm-1.00.00.0-1.0-8.87729pt0.0pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm1.00.00.01.08.87729pt0.0pt @invoke @invoke @invoke @endscope @invoke @endscope @invoke @endscope @discardpath @invoke @endscope V_j is in Eℳ′E_M iff there exists an edge VnVm∈EℳV_n to25.35pt to5.8pt -2.59998pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-6.94427pt1.7pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke \;\;\;\;\; @invoke @endscope @invoke @endscope @beginscope @invoke @setdash @temp @x @invoke @moveto-8.67729pt0.0pt @lineto8.67729pt0.0pt @stroke @invoke @beginscope @invoke @transformcm-1.00.00.0-1.0-8.87729pt0.0pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm1.00.00.01.08.87729pt0.0pt @invoke @invoke @invoke @endscope @invoke @endscope @invoke @endscope @discardpath @invoke @endscope V_m∈ E_M such that φ(Vn)=Vi′ (V_n)=V_i and φ(Vm)=Vj′ (V_m)=V_j . While graphical models do not specify distributions and functions explicitly they do entail constraints on distributional (in)equalities. For example an edge X→YX→ Y implies P(Y|X)≠P(Y)P(Y|X)≠ P(Y). We denote ℒi(G)G^L_i(G) the set of algebraic constraints on ℒiL_i distributions implied by the causal graph G. We can then define a notion of consistency for graphical abstractions similar to that in functional abstractions. Definition 6 (Graphical ℒiL_i-Consistency). Let GℳG_M be the causal graph induced by SCM ℳM, and Gℳ′G_M an abstraction of GℳG_M with the variable map φ:ℳ→ℳ′ :V_M _M . Let us define ℒi(Gℳ′−1)G^L_i(G_M ^-1) the set of all the constraints obtained from ℒi(Gℳ′)G^L_i(G_M ) substituting each variable V′∈ℳ′V _M with the respective pre-image φ−1(V′) ^-1(V ). Then Gℳ′G_M is graphically ℒiL_i-consistent with GℳG_M iff ℒi(Gℳ′−1)⊆ℒi(Gℳ).G^L_i (G_M ^-1 ) ^L_i (G_M ). (2) Graphical consistency and functional consistency align as follows: a graphically ℒiL_i-consistent abstraction from GℳG_M to Gℳ′G_M implies the existence of an SCM ℳ′M that is functionally ℒiL_i-consistent with a given ℳM; a functionally ℒiL_i-consistent abstraction implies graphical ℒiL_i-consistency only if all causal dependencies are preserved (see for more details Schooltink and Zennaro [2025]). (CAUSAL) MARGINAL PROBLEM We will show applications of our work in the marginal problem: the challenge of finding a joint probability given two or more separate but overlapping datasets [Kellerer, 1964], for example estimating the joint distribution P^(X,Y,Z) P(X,Y,Z) from the two distributions P^(X,Y) P(X,Y) and P^(Y,Z) P(Y,Z). Definition 7 (Marginal Problem). Given marginal datasets 1,…,nX_V_1,…,X_V_n, defined respectively over variables iV_i, with possibly non-empty intersections of variables between datasets: i∩j≠∅V_i _j≠ , find the joint distribution over the union ⋃i=1ni _i=1^nV_i. Overlap between the variables of the different datasets is a necessary condition in order to find a meaningful solution to the marginal problem. To illustrate this, consider the counter example where we have two datasets that do not overlap at all; this gives no information on dependencies and as such on the joint distribution of their variables. The marginal problem per Def.7 is a statistical problem: a question of combining observational distributions. We will focus on the causal extension of the marginal problem [Gresele et al., 2022]: finding a joint SCM from multiple overlapping marginal SCMs. Whereas in the statistical setting the object of interest is distributions, in the causal setting it is SCMs. This distinction is required as observational distributions alone cannot fully describe causality. Definition 8 (Causal Marginal Problem). Given SCMs ℳ1,…,ℳnM_1,…,M_n, with possibly non-empty intersections of endogenous variables between models ℳi∩ℳj≠∅V_M_i _M_j≠ , find the space of joint causal models ℳ∗M^* over the union ⋃i=1nℳi _i=1^nV_M_i consistent with the models ℳ1,…,ℳnM_1,…,M_n. Our proposed causal embeddings will tackle the more complicated setting of the causal marginal problem where the overlapping variables do not share the same level of detail. SCM PROJECTIONS In this section we will formalize a notion of projections for SCMs which will be instrumental in defining embeddings. First, recall that SCMs consist of a set of observable variables V and a set of unobservable variables U. One may want to move some variables V⊂V to the set of exogenous variables U, perhaps to simplify the model or since some variables have become unobservable. Formally we can describe a projection of SCMs as follows: Definition 9 (SCM Projection). Let GℳG_M be the graph induced by an SCM ℳM, ⊆ℳR _M a set of relevant variables to be preserved, ℳ′M an SCM over the relevant variables s.t. ℳ′=V_M =R, and Gℳ′G_M the graph induced by ℳ′M . Gℳ′G_M is a projection of GℳG_M iff Gℳ′G_M is graphically ℒ2L_2-consistent with GℳG_M with φ the identity mapping. Importantly, SCM projections enforce only graphical constraints. The SCM projection ℳ′M of an SCM ℳM implies the compatibility of causal (in)dependencies; that is, if there is a causal arrow X→YX→ Y in ℳM and both X and Y are preserved, then there must exist an arrow X→YX→ Y in the graph induced by ℳ′M . This definition follows as an extension of latent structure projections [Pearl and Verma, 1995] (see App.A.1) to the domain of SCMs. EMBEDDINGS We will now define embeddings as a generalization of abstractions and similarly provide the notions of functional and graphical consistency. First, recall that an α-abstraction describes the relation of a causal model to a coarser causal model. α-abstractions enforce surjectivity of both the mapping of variables φ and the mapping α _V of their ranges. The surjectivity requirement makes sure that all states of the abstracted model are represented in the base model. So a base model cannot represent only a sub-system of the coarse model. Dropping the requirement of surjectivity of φ allows for defining refinements of sub-systems of a larger system. To give more intuition to this idea, consider the following motivating example: Example 1 (Simplified Ecosystem Modeling). Imagine we model the causal dynamics of some ecosystem of deer and squirrels, and the effects of predators and human hunting on their population, as in the causal model shown in Fig.2: ℳ′:M :SquirrelsDeerHumansPredators Figure 2: A high-level causal model of a simplified ecosystem. However, for the area we are interested in, such causal models have not been defined. Instead other researchers have modeled two related systems as shown in Fig.3: model ℳ1M_1, describing causal interactions between human hunting, squirrels, deer, and berry bush availability; and model ℳ2M_2, describing the causal interactions between wolves, eagles, red deer, fallow deer, and squirrels. ℳ2:M_2:Red DeerFallow DeerSquirrelsWolvesEaglesℳ1:M_1:SquirrelsDeerHumansBerries Figure 3: Two low-level causal models, each modeling a sub-system of a simplified ecosystem. Notice that ℳ1M_1 and ℳ2M_2 together cover all variables we are interested in, albeit both ℳ1M_1 and ℳ2M_2 contain more detail. To construct a single model for the whole system, both models should be mapped to the same level of detail. Following Ex.1, we formalize causal embeddings. Formal Definition. We define an embedding using a generalization of the α-abstraction framework by first expanding the α-abstraction definition to allow for non-surjective maps φ wrt ℳ′V_M : Definition 10 (Non-surjective α-abstraction). A non-surjective α-abstraction is an α-abstraction with: 1. ⊆ℳR _M is a subset of relevant variables in ℳM. 2. ⊆ℳ′S _M is a subset of relevant variables in ℳ′M . 3. φ:→ :R is a surjective map between relevant variables. 4. αV′:ℛ(φ−1(V′))→ℛ(V′) _V :R( ^-1(V )) (V ), for each V′∈V , is a surjective function from the range of the pre-image φ−1(V′)⊆ ^-1(V ) in ℳM to the range of V′V in ℳ′M . Similar to α-abstraction, Def.10 does not enforce consistency or make any graphical guarantees. However, since embeddings are to encode a detailed description of a sub-system into a high-level model, it is important to ensure a compatible embedding. For this purpose we define embeddings as follows, by including graphical constraints: Definition 11 ( α-embedding). Given SCMs ℳM and ℳ′M and a non-surjective α-abstraction α with φ:→ :R , α is an α-embedding iff the projection of the graph Gℳ′G_M over S is a CDAG of the projection of the graph GℳG_M over R. Let us illustrate α-embeddings on our guiding example. Example 2. (Simplified Ecosystem Modeling) Let us consider again the ecosystem models in Ex.1 and suppose we define an α-embedding α1:ℳ1→ℳ′ _1:M_1 with: φ1:ℳ1→ℳ′:=Humans↦HumansSquirrels↦SquirrelsDeer↦Deer _1:R_M_1 _M := casesHumans& \\ Squirrels& \\ Deer& \\ cases α1 _1 satisfies the α-embedding definition as the projection of Gℳ′G_M over ℳ′=Humans,Squirrels,DeerS_M =\Humans,Squirrels,Deer\ is a CDAG of the projection of Gℳ1G_M_1 over ℳ1=Humans,Squirrels,DeerR_M_1=\Humans,Squirrels,Deer\. An alternative definition makes a more explicit reference to the graphical constraints allowing for easier construction and verification of embeddings. For this, we need the definition of mediated adjacencies and confounders [Massidda et al., 2024, Schooltink and Zennaro, 2025]. Definition 12 (Mediated adjacencies). Given a set of relevant variables ⊆R and X,Y∈X,Y , a mediated adjacency X↝YX Y wrt R is a directed path from X to Y such that all intermediate variables are not in R. Definition 13 (Mediated confounders). Given a set of relevant variables ⊆R and X,Y∈X,Y , a mediated confounder X YX to13.44pt to7.11pt -4.95274pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-4.16656pt0.0pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke $\;\;\;$ @invoke @endscope @invoke @endscope @moveto6.11934pt-2.35277pt @lineto2.51933pt-2.35277pt @lineto1.66576pt-3.05276pt @lineto-0.04141pt-1.65277pt @lineto-0.895pt-2.35277pt @lineto-6.11934pt-2.35277pt @stroke @invoke @beginscope @invoke @transformcm1.00.00.01.06.31934pt-2.35277pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm-1.00.00.0-1.0-6.31934pt-2.35277pt @invoke @invoke @invoke @endscope @invoke @endscope @discardpath @invoke @endscope Y wrt R is a hidden or observed confounder Z∉Z , with Z↝XZ X and Z↝YZ Y. We can now give a constructive definition of α-embedding. Definition 14 ( α-embedding (Alternative)). Given causal models ℳM and ℳ′M and a non-surjective α-abstraction with φ:→ :R , α is an α-embedding iff the following two conditions hold: 1. a mediated adjacency X′↝Y′X Y wrt S is in ℳ′M iff there exists X∈φ−1(X′),Y∈φ−1(Y′)X∈ ^-1(X ),Y∈ ^-1(Y ) such that there is a mediated adjacency X↝YX Y wrt R. 2. a mediated confounder X′ Y′X to13.44pt to7.11pt -4.95274pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-4.16656pt0.0pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke $\;\;\;$ @invoke @endscope @invoke @endscope @moveto6.11934pt-2.35277pt @lineto2.51933pt-2.35277pt @lineto1.66576pt-3.05276pt @lineto-0.04141pt-1.65277pt @lineto-0.895pt-2.35277pt @lineto-6.11934pt-2.35277pt @stroke @invoke @beginscope @invoke @transformcm1.00.00.01.06.31934pt-2.35277pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm-1.00.00.0-1.0-6.31934pt-2.35277pt @invoke @invoke @invoke @endscope @invoke @endscope @discardpath @invoke @endscope Y wrt S is in ℳ′M iff there exists X∈φ−1(X′),Y∈φ−1(Y′)X∈ ^-1(X ),Y∈ ^-1(Y ) such that there is a mediated confounder X YX to13.44pt to7.11pt -4.95274pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-4.16656pt0.0pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke $\;\;\;$ @invoke @endscope @invoke @endscope @moveto6.11934pt-2.35277pt @lineto2.51933pt-2.35277pt @lineto1.66576pt-3.05276pt @lineto-0.04141pt-1.65277pt @lineto-0.895pt-2.35277pt @lineto-6.11934pt-2.35277pt @stroke @invoke @beginscope @invoke @transformcm1.00.00.01.06.31934pt-2.35277pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm-1.00.00.0-1.0-6.31934pt-2.35277pt @invoke @invoke @invoke @endscope @invoke @endscope @discardpath @invoke @endscope Y wrt R. We can prove the equivalence of the two definitions (see proof in App.B.1): Lemma 1 (Equivalence Def.11 and Def.14). Def.11 in terms of projections is equivalent to Def.14 in terms of explicit graphical constraints. Note that the definition of α-embedding only restricts the edges that are part of mediated adjacencies and confounders, any edge in the graph of ℳ′M that is not part of either of these may exist (see following example). Example. (Example of an Embedding) In this example we illustrate an example of a permissible embedding, and highlight some graphical properties. The following diagram illustrates an α-embedding, with non-relevant variables denoted as ∙ : X1X_1X2X_2YYWW∙ ′X Y′Y W′W ∙ ∙ ∙ ∙ ∙ ∙ ∙ X1,X2↦X′\X_1,X_2\ X Y↦Y′\Y\ Y W↦W′\W\ W The diagram shows: (i) How paths in the low-level model (left) can be mapped to one or multiple paths in the high-level model. The path from the cluster X1,X2\X_1,X_2\ to Y\Y\ is represented in the high-level model with two paths from X′X to Y′Y , one containing two mediating variables. Embeddings preserve causal effects, but do not enforce causal pathways beyond consistency. (i) How confounders are mapped to confounders. The unobservable confounder in the low-level model between Y\Y\ and W\W\ is represented in the high-level model with an observable confounder between Y′Y and W′W . Embeddings require confounding effects to be consistent, but unobservable confounders may become observable or vice-versa. (i) How additional structures may exist in the high-level model. The high-level graph has an additional parent for X′X , child of Y′Y and collider structure with parents X′,W′,∙X ,W , [rgb].5,.5,.5 [named]pgfstrokecolorrgb.5,.5,.5 @color@gray@stroke.5 @color@gray@fill.5 . Any number of variables can exist that do not create new causal pathways between the relevant variables. (iv) How some embedding might be unintuitive. The high-level model may introduce an observable separation X′→∙→W′X → [rgb].5,.5,.5 [named]pgfstrokecolorrgb.5,.5,.5 @color@gray@stroke.5 @color@gray@fill.5 → W which is not represented in the low-level model X2→WX_2→ W. As illustrated, α-embedding are versatile. If patterns as (iv) in the example are deemed undesirable, one can adopt additional restrictions; for example, requiring a graph homomorphism from the graph of ℳM to a subgraph of ℳ′M respecting the variable mapping, as is in ϕφ-abstractions [Otsuka and Saigo, 2022]. For the sake of generality, we do not impose such a restriction. Reconsidering the motivating example, α-embeddings provide a formal tool to specify all the needed embeddings: Example 3. (Simplified Ecosystem Modeling) Let us consider again Ex.2 and let us define also the last α-embedding for model ℳ2M_2 onto ℳ′M . φ2:ℳ2→ℳ′:=Eagles↦PredatorsWolves↦PredatorsRed Deer↦DeerFallow Deer↦DeerSquirrels↦Squirrels _2:V_M_2 _M := casesEagles& \\ Wolves& \\ Red Deer& \\ Fallow Deer& \\ Squirrels& \\ cases This completes the example with a graphical specification of the α-embeddings necessary to map ℳ1,ℳ2M_1,M_2 onto ℳ′M . CAUSAL CONSISTENCY OF EMBEDDINGS Similar to abstractions, it is important to assess the consistency of causal embeddings. Therefore, we will define a notion of consistency for embeddings, both in terms of a functional consistency, wrt the functional descriptions of SCMs, and graphical consistency, wrt to the causal graphs. Functional Consistency. First let us consider the functional side of the α-embedding: the non-surjective α-abstraction. We adapt the ℒiL_i-abstraction error from Def.4: Definition 15 (ℒiL_i-Embedding error). Given an α-embedding :ℳ→ℳ′ α:M and ′,′⊆X ,Y , the ℒiL_i-error is given by the maximum distance or divergence between the distribution obtained by first embedding and then evaluating and the distribution obtained by first evaluating and then embedding, as given by the formula: eℒi()=max′,′⊆D(Pℳ′(′|α′[ℒi(φ−1(′))]),α′[Pℳ(φ−1(′)|ℒi(φ−1(′)))]) e_L_i( α)= _X ,Y D (P_M (Y \>|\> _X [L_i( ^-1(X ))]), .\\ . _Y [P_M( ^-1(Y )\>|\>L_i( ^-1(X ))) ] ) (3) An embedding with ℒiL_i-error zero is ℒiL_i-consistent. Functional consistency does not follow by construction, as this may be too limiting and some error may be tolerable, similar to abstractions. Notice that the embedding error subsumes the abstraction error and can be considered a generalization, mirroring the definition of the non-surjective α-abstraction. Remark 1. Let :=ℳ1,…,ℳnM:=\M_1,…,M_n\ be a set of n causal models and :=1,…,nA:=\ α_1,…, α_n\ be a set of n α-embeddings such that i:ℳi→ℳ′ α_i:M_i embeds into the same ℳ′M for all i. ℒiL_i-consistency of all ∈ α solely does not imply uniqueness of the ℒiL_i-distributions over the set of relevant variables S for any permissible ℳ′M , see App.C.1 for a concrete example. Instead, to guarantee uniqueness of all distributions stronger assumptions on the set of embeddings A are needed. We conjecture a sufficient condition would be that all causal and confounding edges are constrained by at least one of the embeddings ∈ α . Graphical Consistency. Additional to the functional side, embeddings have a graphical restriction, so we will also consider graphical consistency by adjusting for unmapped variables in the high-level model. For this we introduce the following notation: Definition 16 (Graphical ℒiL_i-Embedding Consistency). Given a variable map φ:→ :R , let G[]G_[R] be a projection of the DAG of ℳM over R and G[]G_[S] a projection of the DAG of ℳ′M over S. The two models ℳM and ℳ′M are graphically consistent iff ℒi(G[]−1)⊆ℒi(G[]).G^L_i (G_ [S ]^-1 ) ^L_i (G_ [R ] ). (4) As the α-embedding is defined in terms of projections and CDAGs it inherits graphical ℒ2L_2-consistency (see proof from Lem.1: App.B.1): Theorem 2 ( α-embedding is graphically ℒ2L_2-consistent). By definition an α-embedding is graphically ℒ2L_2-consistent. A relation between functional and graphical consistency for embeddings can be proved in analogy to the same relation proved for abstractions (proofs in App.B.2 and App.B.3): Theorem 3 (Graphical ℒ2L_2-consistent map ⇒ Functional ℒ2L_2-consistent embedding). Given a graphically ℒ2L_2-consistent φ:→ :R with ⊆ℳR _M and ⊆ℳ′S _M there exists a specification of ℳ′M and an α-embedding :ℳ→ℳ′ α:M with variable map φ s.t. α is functionally ℒ2L_2-consistent. Theorem 4 (Functional ℒ2L_2-consistent non-surjective abstraction ⇏ embedding). A non-surjective ℒ2L_2-consistent α-abstraction does not necessarily imply an α-embedding. Notice that the asymmetry between Thm.3 and Thm.4 follows from Thm.2, as Def.11 requires graphical ℒ2L_2-consistency but not functional ℒ2L_2-consistency. MULTI-RESOLUTION MARGINAL PROBLEM We now show how α-embeddings can be used to expand upon the causal marginal problem. Recall the causal marginal problem (Def.8) where marginal models have sets of overlapping variables. We consider the case when this assumption does not hold: that is, marginal models have different representations of the overlapping variables, either by (i) having differing resolutions, or (i) having the overlapping variables represented using multiple variables. We refer the reader again to Ex.1-3, where the detailed models have overlapping variables through the embedding. We define the Multi-Resolution Causal Marginal Problem as an extension of the causal marginal problem where the resolution of the marginal models can differ and a common resolution must be found to represent all marginal models: Definition 17 (Multi-Resolution Causal Marginal Problem). Let ∗V^* be the set of high-level variables, and let ℳ1,…,ℳnM_1,…,M_n be SCMs together with associated mappings φi:ℳi→∗ _i:V_M_i ^*, find the space of joint causal models ℳ∗M^* over the variables ∗V^* consistent with ℳ1,…,ℳnM_1,…,M_n. The variable sets of the marginal models no longer need to overlap, but there must exist mappings from each marginal model to a collective set of high-level variables over which an SCM can be specified. This allows for different representations of the previously overlapping variables in each model; for example, a variable V∈∗V ^* may be represented by multiple variables in ℳ1M_1 and by only one variable in ℳ2M_2. Additionally, this framework allows for models to have different levels of resolution for each variable; for example, a variable V∈∗V ^* may be define on a discretized domain in ℳ1M_1 and on a continuous domain in ℳ2M_2. Interestingly, there is a close connection between consistency in embeddings and the marginal problem in that a set of consistent embeddings define a solution to the multi-resolution marginal problem (see proof in App.B.4): Theorem 5 (Consistent Embeddings as Solution to the Multi-Resolution Marginal Problem). Let :=ℳ1,…,ℳnM:=\M_1,…,M_n\ be a set of n SCMs and :=1,…,nA:=\ α_1,…, α_n\ be a set of n α-embeddings such that j:ℳj→ℳ′ α_j:M_j embeds into the same ℳ′M for all j. ℳ′M is a solution ℳ∗M^* of the multi-resolution marginal problem if for all j∈ α_j : (i) j α_j is ℒiL_i-consistent and (i) the set of relevant variables contains all variables =ℳjR=V_M_j. Notice that in the theorem we do not specify the level of consistency ℒiL_i; ℒ1L_1-consistency provides a solution to a statistical marginal problem, ℒ3L_3-consistency a solution to a causal marginal problem as proposed by Gresele et al. [2022], and an ℒ2L_2-consistency a solution to a causal marginal problem only up to ℒ2L_2. For an example of the use of embeddings in the multi-resolution marginal problem see the following. Example 4 (Multi-Resolution Marginal Problem). Recall the motivating example for causal embeddings in Ex.1-3. Models ℳ1M_1 and ℳ2M_2 (Fig.3) differ in resolution for overlapping variables: whereas ℳ1M_1 counts subspecies of deer separately ℳ2M_2 only has a total count of all deer, and similarly for the predator variables. This does not permit for a causal marginal problem to be defined, instead this needs to be framed in the multi-resolution causal marginal problem. Given the emdeddings 1 α_1 and 2 α_2, we want to find a specification of an SCM ℳ′M consistent with ℳ1M_1 and ℳ2M_2. Variable maps φi _i of the embeddings have been defined earlier, while the maps αV′:ℛ(φ−1(V′))→ℛ(V′) _V :R( ^-1(V )) (V ) for V′∈V can be defined simply as the sum of the pre-images. For 1 α_1 we define: αDeer _Deer :=Deer, :=Deer, αHumans _Humans :=Humans, :=Humans, αSquirrels _Squirrels :=Squirrels. :=Squirrels. and similarly for 2 α_2: αDeer _Deer :=Fallow Deer+Red Deer, :=Fallow Deer+Red Deer, αPredators _Predators :=Eagles+Wolves, :=Eagles+Wolves, αSquirrels _Squirrels :=Squirrels. :=Squirrels. It is immediate now to define a ℒ2L_2-consistent SCM that is a solution to the multi-resolution marginal problem. We now highlight further connections of our problem to previous marginal problems in the limit cases of reducing a multi-resolution problem to a single-resolution or using an identity embedding (proofs in App.B.5 and App.B.6): Lemma 6 (Multi-resolution Marginal Problem Reduction). Given a multi-resolution marginal problem, the application of α-embeddings reduces it to a single-resolution marginal problem. Lemma 7 (Identity Embeddings as Solution to the Marginal Problem). Let :=ℳ1,…,ℳnM:=\M_1,…,M_n\ be a set of n SCMs and :=1,…,nA:=\ α_1,…, α_n\ be a set of n α-embeddings such that j:ℳj→ℳ′ α_j:M_j embeds into the same ℳ′M for all j. ℳ′M is a solution ℳ∗M^* of the single-resolution marginal problem if for all j∈ α_j : (i) j α_j is ℒiL_i-consistent, (i) α′ _V and φ are identity maps, and (i) the set of relevant variables contains all variables =ℳjR=V_M_j. MERGING DATASETS Embeddings can also be applied to datasets. We propose a simple algorithm (see Alg.1), where we collect data from marginal models and we use embeddings to map the data to a single shared representation. As the marginal models are not necessarily defined on all variables, we might obtain a dataset with missing values. This constitute a case of structured missing data due to multi-scale linkage [Mitra et al., 2023], and Line 6 in the algorithm calls for a data imputation method to fill in the missing values. See the following two simulated examples as an illustration: Data: Datasets 1,…,nX_V_1,...,X_V_n from ℳ1,…,ℳnM_1,...,M_n, Embeddings ℳ1,…,ℳn α_M_1,…, α_M_n s.t. i:ℳi→ℳ′ α_i:M_i Result: Merged dataset ′X_V 1 ′←[]X_V ←[\;\;]; 2 for i∈0,…,ni∈\0,...,n\ do 3 i′←αi(i)X_V_i ← _V_i(X_V_i); 4 ′←[′,i′]X_V ← [X_V ,X_V_i ]; 5 end for ′←impute(′)X_V (X_V ) Algorithm 1 Multi-resolution datasets merging Example 5 (Merging datasets for increased statistical power). We continue Ex.4 and generate datasets of 20002000 and 40004000 samples for the marginal models ℳ1M_1 and ℳ2M_2, respectively (see App.C.2 for details). After collection, data from the marginal models is transformed into the shared resolution using the embeddings 1 α_1 and 2 α_2, allowing the merging of the datasets. We estimate the distribution P^(Deer,Squirrels) P(Deer,Squirrels) using the marginal and the merged datasets and computing the KL divergence between the estimation P P and true distribution P: ℳ1 _M_1 :DKL(P,P^)≈0.34 :D_KL(P, P)≈ 0.34 ℳ2 _M_2 :DKL(P,P^)≈0.77 :D_KL(P, P)≈ 0.77 ℳ1+ℳ2 _M_1+X_M_2 :DKL(P,P^)≈0.22 :D_KL(P, P)≈ 0.22 Relying on more samples, the merged dataset shows a clear improvement in the estimation of P P. Example 6 (Merging datasets to compute distributions undefined in the marginals). Suppose we want to estimate the distribution P^(Predators,Humans) P(Predators,Humans). This distribution cannot be estimated from the marginal models: ℳ1M_1 is not accounting for predators, while ℳ2M_2 ignores humans. However, by aggregating the data and imputing missing values we can estimate this quantity of interest (see App.C.3). CONCLUSION In this work we have introduced causal embeddings as a generalization of abstractions, and extended the notion of consistency. We illustrated with examples Ex.4 and Ex.5 that embeddings are not just a theoretical novelty, but serve a practical use: they express a multi-resolution marginal problem, they enable its reduction to the standard marginal problem in both the statistical and causal setting, and they allow for the merging of datasets or causal models with overlapping variables at different resolutions. 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Multi-Level Causal Embeddings (Supplementary Material) Appendix A Supplementary Definitions LATENT STRUCTURE PROJECTIONS Here we will introduce latent structure projections [Pearl and Verma, 1995], a seminal work in graphical models with applications in graphical abstraction. In the main paper we show that through a combination of abstraction and the extension of latent structure projections to SCMs we obtain causal embeddings. First we define latent structure models, as graphical causal models over a set of variables, some of which are unobservable: Definition 18 (Latent Structure Model). Given a set of causal variables U, a latent structure L:=⟨D,O⟩L:= D,O is a tuple of a DAG D with vertices U, and a set O⊆UO U of observable variables. Consider the case where some of the observable variables O have become unobservable, one may want to find a new model excluding that variable in the observable set whilst preserving all conditional dependencies implied by the original. Projections allow for reasoning about such cases: Definition 19 (Latent Structure Projection). A latent structure L[O]:=⟨D[O],O⟩L_[O]:= D_[O],O is a projection of another latent structure L if, and only if, the following holds: 1. Every unobservable variable of D[O]D_[O] is a parentless common cause of exactly two non-adjacent observable variables. 2. For every stable distribution P generated by L there exists a stable distribution P′P generated by L[O]L_[O] such that the independencies over the variables O implied by P[O]P_[O] equal to those implied by P[O]′P _[O]. It has been shown that latent projections are consistent in identifiablity and preserve topological ordering. Informally, given a model L and a its projection L[′]L_[V ] over a subset of variables ′⊆V this implies that any causal quantities over ′V representable in L[′]L_[V ] are equally representable in L. Appendix B Proofs PROOF OF LEMMA 1 Lemma 1 (Equivalence Def.11 and Def.14). Def.11 of the α-embedding in terms of projections is equivalent to Def.14 of the α-embedding in terms of explicit graphical constraints. Proof. In this proof we base ourselves mostly in the results of Schooltink and Zennaro [2025]. First, recall conditions 1. and 2. of Def.14: 1. a mediated adjacency X′↝Y′X Y wrt R is in ℳ′M iff there exists X∈φ−1(X′),Y∈φ−1(Y′)X∈ ^-1(X ),Y∈ ^-1(Y ) such that there is a mediated adjacency X↝YX Y wrt S. 2. a mediated confounder X′ Y′X to13.44pt to7.11pt -4.95274pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-4.16656pt0.0pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke $\;\;\;$ @invoke @endscope @invoke @endscope @moveto6.11934pt-2.35277pt @lineto2.51933pt-2.35277pt @lineto1.66576pt-3.05276pt @lineto-0.04141pt-1.65277pt @lineto-0.895pt-2.35277pt @lineto-6.11934pt-2.35277pt @stroke @invoke @beginscope @invoke @transformcm1.00.00.01.06.31934pt-2.35277pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm-1.00.00.0-1.0-6.31934pt-2.35277pt @invoke @invoke @invoke @endscope @invoke @endscope @discardpath @invoke @endscope Y wrt R is in ℳ′M iff there exists X∈φ−1(X′),Y∈φ−1(Y′)X∈ ^-1(X ),Y∈ ^-1(Y ) such that there is a mediated confounder X YX to13.44pt to7.11pt -4.95274pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-4.16656pt0.0pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke $\;\;\;$ @invoke @endscope @invoke @endscope @moveto6.11934pt-2.35277pt @lineto2.51933pt-2.35277pt @lineto1.66576pt-3.05276pt @lineto-0.04141pt-1.65277pt @lineto-0.895pt-2.35277pt @lineto-6.11934pt-2.35277pt @stroke @invoke @beginscope @invoke @transformcm1.00.00.01.06.31934pt-2.35277pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm-1.00.00.0-1.0-6.31934pt-2.35277pt @invoke @invoke @invoke @endscope @invoke @endscope @discardpath @invoke @endscope Y wrt S. Importantly, (i) the ℒ2L_2-distribution (in)equalities implied by a mediated edge X↝YX Y wrt R are equal to those implied by a directed edge X→YX→ Y, given X,Y∈X,Y . Similarly, a mediated confounding edge X YX to13.44pt to7.11pt -4.95274pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-4.16656pt0.0pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke $\;\;\;$ @invoke @endscope @invoke @endscope @moveto6.11934pt-2.35277pt @lineto2.51933pt-2.35277pt @lineto1.66576pt-3.05276pt @lineto-0.04141pt-1.65277pt @lineto-0.895pt-2.35277pt @lineto-6.11934pt-2.35277pt @stroke @invoke @beginscope @invoke @transformcm1.00.00.01.06.31934pt-2.35277pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm-1.00.00.0-1.0-6.31934pt-2.35277pt @invoke @invoke @invoke @endscope @invoke @endscope @discardpath @invoke @endscope Y wrt R implies the same ℒ2L_2 distribution constraints as a regular confounding edge X YX to13.44pt to7.11pt -4.95274pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-4.16656pt0.0pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke $\;\;\;$ @invoke @endscope @invoke @endscope @moveto6.11934pt-2.35277pt @lineto2.51933pt-2.35277pt @lineto1.66576pt-3.05276pt @lineto-0.04141pt-1.65277pt @lineto-0.895pt-2.35277pt @lineto-6.11934pt-2.35277pt @stroke @invoke @beginscope @invoke @transformcm1.00.00.01.06.31934pt-2.35277pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm-1.00.00.0-1.0-6.31934pt-2.35277pt @invoke @invoke @invoke @endscope @invoke @endscope @discardpath @invoke @endscope Y given X,Y∈X,Y . Secondly, (i) the following two conditions are known to together enforce graphically ℒ2L_2-consistency: 1. an adjacency X′→Y′X → Y is in ℳ′M iff there exists X∈φ−1(X′),Y∈φ−1(Y′)X∈ ^-1(X ),Y∈ ^-1(Y ) such that there is a mediated adjacency X↝YX Y wrt S. 2. a confounder X′Y′X to25.35pt to5.8pt -2.59998pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-6.94427pt1.7pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke \;\;\;\;\; @invoke @endscope @invoke @endscope @beginscope @invoke @setdash @temp @x @invoke @moveto-8.67729pt0.0pt @lineto8.67729pt0.0pt @stroke @invoke @beginscope @invoke @transformcm-1.00.00.0-1.0-8.87729pt0.0pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm1.00.00.01.08.87729pt0.0pt @invoke @invoke @invoke @endscope @invoke @endscope @invoke @endscope @discardpath @invoke @endscope Y is in ℳ′M iff there exists X∈φ−1(X′),Y∈φ−1(Y′)X∈ ^-1(X ),Y∈ ^-1(Y ) such that there is a mediated confounder X YX to13.44pt to7.11pt -4.95274pt to0.0pt @beginscope @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke @setlinewidth @invoke to0.0pt @beginscope @invoke @beginscope @invoke @beginscope @invoke @transformcm1.00.00.01.0-4.16656pt0.0pt @invoke pgfstrokecolorrgb0,0,0 @color@rgb@stroke000 @invoke @color@rgb@fill000 @invoke $\;\;\;$ @invoke @endscope @invoke @endscope @moveto6.11934pt-2.35277pt @lineto2.51933pt-2.35277pt @lineto1.66576pt-3.05276pt @lineto-0.04141pt-1.65277pt @lineto-0.895pt-2.35277pt @lineto-6.11934pt-2.35277pt @stroke @invoke @beginscope @invoke @transformcm1.00.00.01.06.31934pt-2.35277pt @invoke @invoke @invoke @endscope @beginscope @invoke @transformcm-1.00.00.0-1.0-6.31934pt-2.35277pt @invoke @invoke @invoke @endscope @invoke @endscope @discardpath @invoke @endscope Y wrt S. Therefore, by a combination of (i) and (i) it follows that conditions 1. and 2. from Def.14 must entail graphical ℒ2L_2 consistency. Similarly, Def.11 is necessarily graphically ℒ2L_2 consistent, as the CDAGs are known to be graphically ℒ2L_2-consistent [Anand et al., 2023] and the projections by definition preserve ℒ2L_2-distributional (in)equalities. Thus, both Def.11 and Def.14 define an α-embedding to be a non-surjective α-abstraction such that α is graphically ℒ2L_2-consistent. ∎ PROOF OF THEOREM 3 Theorem 3 (Graphical ℒ2L_2-consistent map ⇒ Functional ℒ2L_2-consistent embedding). Given a graphically ℒ2L_2-consistent φ:→ :R with ⊆ℳR _M and ⊆ℳ′S _M there exists a specification of ℳ′M and an α-embedding :ℳ→ℳ′ α:M with variable map φ s.t. α is functionally ℒ2L_2-consistent. Proof. Let ℳM be a SCM over the endogenous variables ℳV_M, ℳ′M an SCM over the endogenous variables ℳ′V_M , and φ:→ :R such that ⊆ℳR _M and ⊆ℳ′S _M . Note that φ is graphically ℒ2L_2 consistent and thus the causal graph G_R over R to the causal graph G_S over S is necessarily a CDAG [Anand et al., 2023, Schooltink and Zennaro, 2025]. Therefore, given an SCM ℳ[]M_[R] over R and an SCM ℳ[]′M _[S] over S, there exists a specification of ℳ[]′M _[S] such that ℳ[]′M _[S] is functionally ℒ2L_2-consistent with ℳ[]M_[R]. This leaves to show that (i) there is a model specification of ℳ[]M_[R] encoding the same distributions as ℳM and (i) there is a specification of ℳ′M encoding the same distributions as ℳ[]′M _[S]. (i) is given by the results of Schooltink and Zennaro [2025] Thm.23, and (i) is satisfied by the following construction of functions ℱℳF_M: We distinguish three types of endogenous variables in ℳ′V_M for which we specify a function respecting the causal graph: 1. Variables V s.t. V∉V and ∀S∈,V∉An(S)∀ S ,V∉ An(S). As these variables do not influence distributions of interest, arbitrary functions can be defined. 2. Variables V s.t. V∉V and ∃S∈,V∈An(S)∃ S ,V∈ An(S). These variables that do influence some S∈S , the function determining their value is taking the cartesian product of the parent variables including exogenous variables. 3. Variables V s.t. V∈V . These variables must produce same distributions as given in ℳ[]′M _[S]. By preservation of causal paths through projection the parents the function fVf_V must have access to the values of all its parents in ℳ[]′M_[S] either because the variable is in the signature of fVf_V, or if there is a mediated adjacency wrt S there exist a variable in the signature which carries its value (since all intermediate variables take cartesian products of the parents). Thus allowing the specification of the function fVf_V in ℳ′M to be equal to that in ℳ[]M_[S]. Specifically, this construction additionally allows for preservation of mediated confounding, as exogenous parents are also preserved in intermediate variables. Thus it follows that this model ℳ′M can represent the exact same causal effects as ℳ[]M_[S], and generate the same distributions. Showing that given an SCM ℳM over the endogenous variables ℳV_M, an SCM ℳ′M over the endogenous variables ℳ′V_M , and φ:→ :R such that ⊆ℳR _M and ⊆ℳ′S _M , we can always construct a specification for ℳ′M such that there exists a functionally ℒ2L_2-consistent α-embedding :ℳ→ℳ′ α:M . ∎ PROOF OF THEOREM 4 Theorem 4 (Functional ℒ2L_2-consistent non-surjective abstraction ⇏ embedding). A non-surjective ℒ2L_2-consistent α-abstraction does not necessarily imply an α-embedding. Proof. For this proof we will show that one can create a functionally ℒ2L_2-consistent non-surjective abstraction, that is not graphically ℒ2L_2-consistent. Thus by Thm.2 the proposed abstraction is not an α-embedding. We will define a α-abstraction :ℳ→ℳ′ α:M . Let ℳM and ℳ′M be SCMs with the following specification: ℳ: : :=X,Y,Z, with ℛ(X)=0,2,4ℛ(Y)=0,1ℛ(Z)=0,1,2,3,4,5 :=\X,Y,Z\, with casesR(X)&=\0,2,4\\\ R(Y)&=\0,1\\\ R(Z)&=\0,1,2,3,4,5\ cases :=UX,UY :=\U_X,U_Y\ ℱ :=fX(UX)=UXfY(UY)=UYfZ(X,Y)=X+Y := casesf_X(U_X)&=U_X\\ f_Y(U_Y)&=U_Y\\ f_Z(X,Y)&=X+Y cases P() P(U) :=UX∼0,2,4UY∼B(0.5) := casesU_X& \0,2,4\\\ U_Y& B(0.5) cases ℳ′: : :=X′,Y′,Z′, with ℛ(X′)=0,2,4ℛ(Y)=0,1ℛ(Z)=0,1 :=\X ,Y ,Z \, with casesR(X )&=\0,2,4\\\ R(Y)&=\0,1\\\ R(Z)&=\0,1\ cases :=UX′,UY′ :=\U_X ,U_Y \ ℱ :=fX′(UX′)=UX′fY′(UY′)=UY′fZ′(Y′)=Y′ := casesf_X (U_X )&=U_X \\ f_Y (U_Y )&=U_Y \\ f_Z (Y )&=Y cases P() P(U) :=UX′∼0,2,4UY′∼B(0.5) := casesU_X & \0,2,4\\\ U_Y & B(0.5) cases Then define α with =ℳR=V_M and =ℳ′S=V_M by the maps: φ :=X↦X′Y↦Y′Z↦Z′, := casesX& X \\ Y& Y \\ Z& Z cases, αX′ _X :=0↦02↦24↦4, := cases0& 0\\ 2& 2\\ 4& 4 cases, αY′ _Y :=0↦01↦1, := cases0& 0\\ 1& 1 cases, αZ′ _Z :=0↦01↦12↦03↦14↦05↦1. := cases0& 0\\ 1& 1\\ 2& 0\\ 3& 1\\ 4& 0\\ 5& 1 cases. One can easily verify α is functionally ℒ2L_2-consistent. Now we draw the DAGs entailed by ℳM and ℳ′M : ℳ:M:XXYYZZℳ′:M :X′X Y′Y Z′Z Notice that the edge X′→Z′X → Z is absent as X′X is not in the signature of the function fZ′f_Z . Thus the algebraic constraint P(Z′|do(X′))=P(Z′|do(∅))P(Z |do(X ))=P(Z |do( )) is in ℒ2(G[])G^L_2(G_[S]). In turn, P(Z|do(X))=P(Z|do(∅))P(Z|do(X))=P(Z|do( )) is in ℒ2(G[]−1)G^L_2(G^-1_[S]). However the graph of ℳM implies the constraint P(Z|do(X))≠P(Z|do(∅))P(Z|do(X))≠ P(Z|do( )), and thus ℒ2(G[]−1)⊈ℒ2(G[]).G^L_2 (G_ [S ]^-1 ) ^L_2 (G_ [R ] ). Showing a functional ℒ2L_2-consistent non-surjective abstraction does not necessarily imply graphical ℒ2L_2-consistency, and consequently not an α-embedding. ∎ PROOF OF THEOREM 5 Theorem 5 (Consistent Embeddings as Solution to the Multi-Resolution Causal Marginal Problem). Let :=ℳ1,…,ℳnM:=\M_1,…,M_n\ be a set of n SCMs and :=1,…,nA:=\ α_1,…, α_n\ be a set of n α-embeddings such that j:ℳj→ℳ′ α_j:M_j embeds into the same ℳ′M for all j. ℳ′M is a solution ℳ∗M^* of the multi-resolution marginal problem if for all j∈ α_j : (i) j α_j is ℒiL_i-consistent and (i) the set of relevant variables contains all variables =ℳjR=V_M_j. Proof. A solution to the multi-resolution causal marginal problem requires (i) a mapping φj:ℳj→∗ _j:V_M_j ^* from the variables of the low-level models into a shared set of variables ∗V^*, (i) a model ℳ∗M^* over the set of shared variables V∗V^*, (i) the model ℳ∗M^* to be consistent with all ℳj∈M_j . Requirement (i) is satisfied, since all model ℳj∈M_j embed into ℳ′M and thus have a mapping φj:ℳj→ℳ′ _j:V_M_j _M making ℳ′V_M the set of shared variables ∗V^*. (i) is then immediately satisfied as ℳ′M is the SCM over ∗V^*, thus ℳ′=ℳ∗M =M^*. Finally, (i) is satisfied by all embeddings being consistent. Therefore, ℳ′M is a solution ℳ∗M^* to the multi-resolution marginal problem at the resolution of ∗V^*. ∎ PROOF OF LEMMA 6 Lemma 6 (Multi-resolution Marginal Problem Reduction) Given a multi-resolution marginal problem, the application of α-embeddings reduce it to a single-resolution causal marginal problem. Proof. Let :=ℳ1,…,ℳnM:=\M_1,…,M_n\ be a set of n SCMs and :=1,…,nA:=\ α_1,…, α_n\ be a set of n α-embeddings such that i:ℳi→ℳ′ α_i:M_i embeds into the same ℳ′M for all i. For all ℳi∈M_i the application i(ℳi) α_i(M_i) applies the maps φ:→ :R and αV′:ℛ(φ−1(V′))→ℛ(V′) _V :R( ^-1(V )) (V ) for all V′∈V . Since ⊆ℳ′S _M and ℳ′V_M is the same for all embeddings i α_i, the transformed models i(ℳi) α_i(M_i) necessarily have equal representations and resolutions for any overlapping variables. Thus reducing to the single-resolution causal marginal problem of finding a model ℳ′M consistent with marginal models 1(ℳ1),…,n(ℳn) α_1(M_1),..., α_n(M_n) with possible non-empty overlapping variables. ∎ PROOF OF LEMMA 7 Lemma 7 (Identity Embeddings as Solution to the Marginal Problem) Let :=ℳ1,…,ℳnM:=\M_1,…,M_n\ be a set of n SCMs and :=1,…,nA:=\ α_1,…, α_n\ be a set of n α-embeddings such that j:ℳj→ℳ′ α_j:M_j embeds into the same ℳ′M for all j. ℳ′M is a solution ℳ∗M^* of the single-resolution marginal problem if for all j∈ α_j : (i) j α_j is ℒiL_i-consistent, (i) α′ _V and φ are identity maps, and (i) the set of relevant variables contains all variables =ℳjR=V_M_j. Proof. First note that an α-embedding with =ℳjR=V_M_j and α′ _V , φ identity maps does not alter the model if applied to it: (ℳ)=ℳ α(M)=M. Then as a consequence of Lem.6 it follows that if all j∈A α_j are such identity embeddings the problem is the same as finding a specification ℳ′M consistent with models ℳ1,…,ℳnM_1,...,M_n as in the single-resolution marginal problem. ∎ Appendix C Examples NON-UNIQUENESS OF DISTRIBUTIONS GIVEN CONSISTENT EMBEDDINGS – REMARK 1 Assume we have two SCMs ℳ1,ℳ2M_1,M_2 with respective graphs X→YX→ Y and Y→ZY→ Z, with all variables binary: X,Y,Z∈0,1X,Y,Z∈\0,1\. We then define an embedding into an SCM ℳ′M over the variables X,Y,ZX,Y,Z s.t. all maps of the embeddings are identity maps, as shown in the following diagram: ℳ1M_1:XXYYℳ2M_2:YYZZℳ′:M :XXYYZZ1 α_1:2 α_2:id.id.id.id. We will show through a concrete example that ℒ1L_1-consistency of both embeddings does not mean all ℒ1L_1 distributions of ℳ′M are uniquely determined: First, if the embeddings 1 α_1, 2 α_2 are ℒ1L_1-consistent, and the maps φ:→′ :V and α′:ℛ()→ℛ(′) _V :R(V) (V ) are identities all ℒ1L_1 distributions of ℳ1M_1 and ℳ2M_2 must be exactly equal to those distributions in ℳ′M . So the following distributions are fixed for ℳ′M : P(X),P(Y),P(Z),P(X|Y),P(Y|X),P(Y|Z),P(Z|Y),P(Y,X),P(Y,Z).P(X),P(Y),P(Z),P(X|Y),P(Y|X),P(Y|Z),P(Z|Y),P(Y,X),P(Y,Z). In the following we will show that this not enforce uniqueness of the ℒ1L_1 distribution P(Z|X,Y)P(Z|X,Y). First we set the following distributions that are enforced by consistency of the embeddings: P(X):0.40.6 := tabular[]R|C|CX:&0&1\\ P(X):&0.4&0.6 tabular P(Y|X=0):0.70.3P(Y|X=1):0.40.6 := tabular[]R|C|CY:&0&1\\ P(Y|X=0):&0.7&0.3\\ P(Y|X=1):&0.4&0.6 tabular P(Y):0.520.48 := tabular[]R|C|CY:&0&1\\ P(Y):&0.52&0.48 tabular P(Z|Y=0):0.60.4P(Z|Y=1):0.30.7 := tabular[]R|C|CZ:&0&1\\ P(Z|Y=0):&0.6&0.4\\ P(Z|Y=1):&0.3&0.7 tabular P(Z):0.4560.544 := tabular[]R|C|CZ:&0&1\\ P(Z):&0.456&0.544 tabular P(X) P(X) := :X01 P(Y|X) P(Y|X) := :Y01 P(Y) P(Y) := :Y01 P(Z|Y) P(Z|Y) := :Z01 P(Z) P(Z) := :Z01 Now we construct any number of distributions for P(Z|Y,X)P(Z|Y,X) as long as ∑x∈ℛ(X)P(Z|Y,X=x)P(X=x)=P(Z|Y) _x (X)P(Z|Y,X=x)P(X=x)=P(Z|Y) for example the following two: P1(Z|Y=0,X=0):0.60.4P1(Z|Y=0,X=1):0.60.4P1(Z|Y=1,X=0):0.30.7P1(Z|Y=1,X=1):0.30.7, := tabular[]R|C|CZ:&0&1\\ P_1(Z|Y=0,X=0):&0.6&0.4\\ P_1(Z|Y=0,X=1):&0.6&0.4\\ P_1(Z|Y=1,X=0):&0.3&0.7\\ P_1(Z|Y=1,X=1):&0.3&0.7\\ tabular, P2(Z|Y=0,X=0):0.40.6P2(Z|Y=0,X=1):0.733¯0.266¯P2(Z|Y=1,X=0):0.20.8P2(Z|Y=1,X=1):0.366¯0.633¯. := tabular[]R|C|CZ:&0&1\\ P_2(Z|Y=0,X=0):&0.4&0.6\\ P_2(Z|Y=0,X=1):&0.7 33&0.2 66\\ P_2(Z|Y=1,X=0):&0.2&0.8\\ P_2(Z|Y=1,X=1):&0.3 66&0.6 33\\ tabular. P1(Z|Y,X) P_1(Z|Y,X) := :Z01 P2(Z|Y,X) P_2(Z|Y,X) := :Z01 Thus showing ℒ1L_1-consistency of all embeddings does not uniquely determine all ℒ1L_1 distributions in ℳ′M . DATA GENERATION – EXAMPLE 5 For Ex.5 we generate data for both marginal models under the assumption they both measure the same system. As such, for data generation one large causal model is defined over which both models obtain samples. First, recall the marginal models: ℳ2:M_2:Red DeerFallow DeerSquirrelsWolvesEaglesℳ1:M_1:SquirrelsDeerHumansBerries We construct the data generation model as the following SCM. Important to note is that the functions and distributions are mostly chosen to generate reasonable looking distributions, they are not based in any real-world data. :=Wolves, Eagles, Fallow Deer, Red Deer, Squirrels, Humans, Berries :=\Wolves, Eagles, Fallow Deer, Red Deer, Squirrels, Humans, Berries\ :=UWolves,UEagles,UHumans,UBerries :=\U_Wolves,U_Eagles,U_Humans,U_Berries\ ℱ :=FWolves(UWolves)=100×max(UWolves,0.1)FEagles(UEagles)=10×max(UEagles,0.1)FHumans(UHumans)=15×max(UHumans,0.1)FBerries(UBerries)=max(UBerries,0.1)FFallow Deer(Berries, Wolves, Eagles, Humans)=max(300×Berries−Wolves−2×Eagles−3×Humans,0)FRed Deer(Berries, Wolves, Humans)=max(200×Berries−Wolves−3×Humans,0)FSquirrels(Berries, Eagles, Fallow Deer, Humans)=max(200×Berries−5×Eagles−4×Humans−12×Fallow Deer,0) := casesF_Wolves(U_Wolves)&=100× (U_Wolves,0.1)\\ F_Eagles(U_Eagles)&=10× (U_Eagles,0.1)\\ F_Humans(U_Humans)&=15× (U_Humans,0.1)\\ F_Berries(U_Berries)&= (U_Berries,0.1)\\ F_Fallow Deer(Berries, Wolves, Eagles, Humans)&= (300×Berries-Wolves-2×Eagles-3×Humans,0)\\ F_Red Deer(Berries, Wolves, Humans)&= (200×Berries-Wolves-3×Humans,0)\\ F_Squirrels(Berries, Eagles, Fallow Deer, Humans)&= (200×Berries-5×Eagles-4×Humans\\ &\;\;\;\;\;\;\;\;\;\;\;\;- 12×Fallow Deer,0) cases P() P(U) :=P(UWolves)∼N(1,0.20)P(UEagles)∼N(1,0.15)P(UHumans)∼N(1,0.25)P(UBerries)∼N(1,0.25) := casesP(U_Wolves)& N(1,0.20)\\ P(U_Eagles)& N(1,0.15)\\ P(U_Humans)& N(1,0.25)\\ P(U_Berries)& N(1,0.25)\\ cases Giving rise to the ground truth graph: WolvesEaglesRed DeerFallow DeerSquirrelsBerriesHumans For the purpose of avoiding fractional animals, we round the outputs of the functions up to integers. We generate three datasets by sampling the exogenous variables U and using the functions ℱF to find the values of the endogenous variables V. First, for the dataset for model ℳ1M_1, we generate 2000 samples, and omit the Berries and Humans variables. For the dataset for model ℳ2M_2, we generate 4000 samples, assign Deer=Fallow Deer + Red Deer, and drop the variables Wolves and Eagles. And finally, we generate 100000 samples for an ground truth dataset to compare against. All datasets are independently sampled, so there are no samples shared between the datasets. The implementation is available online111https://github.com/WillemSch/Causal-Embeddings-and-the-Multi-Resolution-Marginal-Problem. EMBEDDINGS FOR MARGINAL PROBLEM – EXAMPLE 6 Continuing App.C.2, we have two datasets with samples for ℳ1M_1 and ℳ2M_2, respectively. We illustrate by example how one can use embeddings to merge the datasets, and through imputation estimate distributions not represented in the marginal models. We will follow Alg.1 to construct a single dataset with missing values imputed. For the imputation step we choose a KNN-imputer with K=2K=2. The implementation is provided online222https://github.com/WillemSch/Causal-Embeddings-and-the-Multi-Resolution-Marginal-Problem. The application of Alg.1 provides one dataset of 6000 samples, with missing values imputed. Thus allowing for the estimation of P^(Humans,Predators) P(Humans,Predators), a distribution not available in either ℳ1M_1 or ℳ2M_2. We provide a visual comparison between the estimation P^(Humans,Predators) P(Humans,Predators) and the true distribution from the ground truth dataset in Fig.4. Figure 4: A visual comparison between the predicted estimation P^(Humans,Predators) P(Humans,Predators) and the evaluation distribution P(Humans,Predators)P(Humans,Predators). Imputation allows for approximation of distributions otherwise not available. Note we choose the KNN-imputer to approximate a solution to the statistical marginal problem in this case. This is a naive and illustrative choice, we do not claim the KNN-imputer to be an optimal approximation. A study of solutions to the marginal problem are out of scope for this work.