Causal Discovery in the Wild: A Voting-Theoretic Ensemble Approach
Authors: Vy Vo
Haoxuan Li
Mingming Gong
Venue: ICLR 2026 Poster (2026)
Keywords: Causal Discovery
Ensemble Learning
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Causal discovery is a critical yet persistently challenging task across scientific domains. Despite years of significant algorithmic advances, existing methods still struggle with inconsistent outcomes due to reliance on untestable assumptions, sensitivity to data perturbations, and optimization constraints. To this end, ensemble-based causal discovery has been actively pursued, aiming to aggregate multiple structural predictions for increased stability and uncertainty estimation. However, current aggregation methods are largely heuristic, lacking theoretical guarantees and guidance on how ensemble design choices affect performance. This work is proposed to address there fundamental limitations. We introduce a principled voting-based framework for structural ensembling, establishing conditions under which the aggregated structure recovers the true causal graph. Our analysis yields a theoretically justified weighted voting mechanism that informs optimal choices regarding the number, competency, and diversity of causal discovery experts in the ensemble. Extensive experiments on synthetic and real-world datasets verify the robustness and effectiveness of our approach, offering a rigorous alternative to existing heuristic ensemble methods.

Published as a conference paper at ICLR 2026
CAUSAL DISCOVERY IN THE WILD:
A TOTING-THEORETIC ENSEMBLE APPROACH
Vy Vol.2 Haoxuan Li3.4.
Mingming Gong
1 Monash University CCSIRO's Data61
PPeking University Thee University of Melbourne
*University of Oxford $Institute for Decentralized AI
MMohamed bin Zayed University of Artificial Intelligence
ABSTRACT
Causal discovery is a critical yet persistently challenging task across scientific domains. Despite years of significant algorithmic advances, existing methods still struggle with inconsistent outcomes due to reliance on untestable assump- tions, sensitivity to data perturbations, and optimization constraints. To this end, mesemble-based causal discovery has been actively pursued, aiming to aggregate multiple structural predictions for increased stability and uncertainty estimation. However, current aggregation methods are largely heuristic, lacking theoretical guarantees and guidance on how ensemble design choices affect performance. This work is proposed to address there fundamental limitations. We introduce a principled voting-based framework for structural ensembling, establishing condi- tions under which the aggregated structure recovers the true causal graph. Our analysis yields a theoretically justified weighted voting mechanism that informs optimal choices regarding the number, competency, and diversity of causal discov- ery experts in the ensemble. Extensive experiments on synthetic and real-world datasets verify the robustness and effectiveness of our approach, offering rigorous alternative to existing heuristic ensemble methods.
INTRODUCTION
Discovering the causal relationships among factors in system is active research interest across scientific disciplines. A plethora of causal discovery algorithms have been proposed over the years, which however tend to produce different graph predictions when applied to the same dataset. This is mainly because methods often rely on strong assumptions that are rarely testable in practice, while diagnostic tools for detecting violations of these assumptions are scarce (Prakash et al. (004).. Even algorithms with formal guarantees can suffer from various other issues such as sensitivity to data perturbation, finite-sampii estimation errors or inherent constraints in the optimization procedures due to the combinatorial search space of graphs. This consequently complicates the selection of algorithms for wallw problems when the true causal structure is unknown.
In response to these multifaceted challenges, ensemble- causal discovery has emerged as promising solution. This approach essentially aims to aggregate multiple output structures, either from heterogenous algorithms or from repeated runs of the same one different data partitions, into a single consensus graph. Structural ensembles help mitigate data variability and algorithmic instability, while offer scalability benefits for large datasets (Tang al. 2019; Guo et al., 1021 Sun et al. 2026 Ensemble learning is also known to effectively capture epistemic uncertainty (Kendall & Gal 2017 Hüllermeier & Waegeman 2021 In this context, it pertains to assessing the reliability of inferred causal structures, allowing practitioners to distinguish between confident causal links and more tentative ones that may require additional investigation. This approach to uncertainty quantification is well-suited for frequentist algorithms that produce point estimates of the graph, offering an alternative to Bayesian causal discovery approaches, which natively account for uncertainty yet relying on strong parametric assumptions to compute graph posteriors (Lorch et al.)
2021 Annadani et al. 2023).
uncertainty due lack of knowledge about the true model, which reduced with extra information

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Despite the advantages of ensemble-bast approaches, existing structural aggregation techniques remain largely heuristic and under-theorized. These methods can be broadly categorized as distance- based (Wang& & Peng 2014; Malmi et al., 2013) or ooting-based (Tang et al.. 2019; Zhang et al.) 2023). The former line of works treat graph aggregation as a clustering problem, searching for "centroid" graph of minimal distance all structures in the ensemble, whereas the latter view algorithms in an ensemble as voters, wherein weighted voting rules are designed to score individual structures and obtain the collective decision based on some decision thresholds. In either approach, the selection of distance measures or weighting schemes are made without theoretical justification as to how well the ensembled graph approximates the true causal structure. Furthermore, there is no principled understanding of how key factors such as the size of ensemble, the diversity performance of individual voters influence the accuracy of the aggregated graph. As result, practitioners are left without guidance on how to configure ensemble strategies for optimal performance.
Another source of uncertainty that often overlooked in the existing literature is the selection of the set of graphs used for aggregation. Typically, graph collections are generated from fixed data partitions or limited runs of causal discovery algorithms. This raises a fundamental question: how stable are the ensemble results given variations in the input graph composition? This oversight, coupled with the heuristic nature of current aggregation methods, has undermined the reliability of ensemble-based causal discovery in real-world applications.
Contributions. In this work, we address these key limitations in the literature by introducing a principled voting- based framework for ensembling the structures from heterogeneous causal discovery algorithms. We characterize the generative process underlying given voting profiles within a structural ensemble and formally analyze the conditions under which the aggregated structure aligns with the true causal graph. The analysis enables a theoretically grounded weighted voting mechanism that informs optimal design choices for the size, competency and diversity levels of causal discovery experts. We validate the theoretical advantages as well as the practical robustness and effectiveness of our approach through extensive experiments on synthetic and real-world system?
PRELIMINARIES
Notation. We use upper case letters (X) to denote random variables, lower case letters (x) for values and bold lower case (a:) for vectors. We reserve calligraphic letters (X) for spaces/sets, italic bold capital (X) for matrices and roman bold capital (G) for notations related to graphs. Finally, we use [d] to denote a set of integers {1,2, ... .d].
Structural Causal Model. A directed graph G = (V, E) consists of a set of (ordered) nodes V and an edge set E S V of pairs of nodes with (v, v) & E for any v E V (one without self-loops). For pair of nodes u, with (u, v) E E, there is an arrow pointing from to and write - Two nodes u and are adjacent if either (u. v) E E or (v, u) E E. If there is an arrow from u to e then is said to be parent of u and is child of u. Let pa, and ch, respectively denote the parents and children of node v in G. A directed acyclic graph (DAG) is a directed graph without cycles.
Let X Xpa; be the set of direct causes of X, (so-called parents), and {U. biei be mutually independent noise variables with strictly positive density. The data-generating process for set of d random variables X (Xi) is characterized by structural causal model (SCM, Pearl 2009), which is defined over a tuple (,, and consists of a sets of assignments
X = fo (Xpa, Ut), i [d].
(I)
The set of causal mechanisms F Lifeeed deterministically assign values to each X; respectively given its causes and the corresponding error term U.. An SCM without feedback loops gives rise to a DAG structure G where each variable X, corresponds to the i-th node in the graph. The recursive application of I induces a joint probability distribution over X, denoted as PPXX such that the Markov factorization holds: P(X)= = IIie[s P(X,)|Xpp)
Optimal Transport. Let a Ex ajōo, be discrete measure with weights a and particles m
I1, Zm E X' and 3 = j= byó,, be another discrete measure defined similarly. The Kan-
Coodes released at i thub.uubbo wtt.coooo/ii/o htw.bb..coo/iavyy//auusssaal Hayes- Knsemble
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torovich 1960) formulation of the optimal transport (OT) distance between and / is
We (a,, B) = min PeU(a.b)
(C,P),
(2)
where (.--) denotes the Frobenius dot-product; C E R the cost matrix of the transport:
P E IR is the transport matrix/plan; U(a.)) b) == [ Rmxm PPmm = a, P1 = b} is the transport polytope of and b: 1, is the m dimensionon column of vector of ones. Given a measurable map T X' + y. T denotes the push-forward measure of a which satisfies (B) = (B)) for any measurable set B C V. For discrete measures, the push-forward operation is equivalent to T#a Ei= a)0r(x,)-
3 STRUCTURE LEARNING FROM NOISY EXPERTS
Suppose that a set of algorithms (so-called "experts") of size n are selected to infer the causal graph G underlying system of variables. We use D to represent some fixed input accessible to all experts. For statistical causal discovery experts, which is the focus in our paper, D refers to a collection of data samples. In the general case with diverse types of experts, D can be arbitrary e.g., the combination of numerical data and semantic description of the variables, the latter of which is typically processed by human or LLM-based experts.
Let denote the space of possible DAG structures. In the general case, |9] can as large as 20(d)) However, our analysis here does not require G to contain all possible configurations; rather it is sufficient to consider purely non-degenerate structures i.c., G := {G P(G = G D) > 0), where P(G D) is the hidden prior distribution. Let the graph space & be arbitrarily indexed 1, .... G). with G₁ denoting the graph realization associated with an index k. Let G2 E G denote the random variable associated with the estimated graph from an i-th expert. Each estimation G2 can be viewed as noisy version of the true graph.
We then define for every i-th expert fixed (unknown) competence transition matrix T: [P(G, = Gj G = GR, D)) D) These quantities encode the noise rates the expert's prediction, which are also indicators of the expert's competence in predicting the causal graph. We denote T. GG A Ti G] :] as the j-th row vector, Ti, Tif: Gy] as the j-th column vector, and T.GGIGG & G, G.] as the k-th entry in the j-th row.
Without loss of generality, we assume the correct structure corresponds to index 1. Hence, the generative distribution for G; is determined by P(G, G G,, D). We denote Pa,c, := T.G (G. as the probability that the i-th expert makes a correct prediction given data D and oppositely, G.G,IG, T.G,, G. Vj > 1 as the probability that expert i makes incorrect prediction.
3.1 RECOVERY OF CAUSAL STRUCTURE VIA BAYES VOTING RULE
Essentially, we have a body of n decision decision-making experts, each voting for one alternative G out of [g] possible options. The voting setup can be cast as general problem of multi-class classification with unbalanced truth over an exponentially large space of |9] classes. Thus, recovering the true causal structure in our setting equivalent to finding the optimal decision rule mapping from the data of n noisy predictions to the ground-truth graph label, classic unsupervised learning problem now placed over a combinatorially large label space typical of causal discovery settings. We first introduce our proposed vote aggregation rule and establish the conditions for recovering the true causal structure from its noisy predictions.
[G.)iejm) Formally, a voting rule refers to function g(-) mapping set of votes õ = -G. Jiej to an estimate of the true state of G. In this work, we consider a weighted linear voting rule that aggregates the choices of the body of n experts into a collective decision by assigning weighted score to each alternative and selecting the one with the highest score. More concretely, the scoring function for candidate graph G E G is given
Sn Sn (7) D0=>wn wiG 1 [G G] bb:
(3)
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where wG "... W,.GG] R are weights assigned each expert reflecting their voting power or reliability, and be E R is the bias term towards a candidate solution G. The predicted structure G from this voting rule is given as G = arg maXGee Sa,wa (G).
The rule reduces to plurality voting in the unweighted case i.e., VG, G' E G: WG = wa and bg big. For brevity, define G P(G G
D).
It is well-established result (Nitzan & Paroush, 1982) that the error- auror-minimizing decision rule
corresponds to Bayes voting rule, where wn, (G) := log P(G = G G)D) where the weight of
each expert and the bias in (3) are given by:
GG log log G,
bg
Σ ==1
log G, + log TG.
(4)
See the detailed derivation in Appendix E. For the Bayes' voting rule to be defined, we adopt the convention that WI.G. G bg 0 where T A question of utmost importance is to what extent the collective decision under Bayes voting rule coincides with the true graph, that is
G G1- We now study the conditions under which this can be achieved, which first require the following assumptions.
Assumption 1 (Independence of Experts' Decisions). Experts in decision-making body make mutuallyy independent decisions conditioned the truth and the input D.
Assumption 2 (Bounded Competency). There exists a constant т < +oo such that Vi E [m]]
|| log Tilmmax S.
Condition 1 (Informativeness)) An i-th expert in the decision-making body is said to be informative if i'G.,,G. # T.,,,, vk +
concept eritical our analysis is informativeness, which refers to the ability to distinguish between any two structures Gj and Gg. Here the degree of informativeness of an i-th expert between two options ? k is quantified by KL (T,,G,; T.G where "being non-informative" is equivalent to T. = T. Finally, we assume that the truth is biased in the sense that TG, 2 T Vj > 1. As to be shown in the proof, this condition is not necessary and the result still holds as long the total informativeness level outweighs the log-odds TG,//G,,
and the collective decision from a group of n informative experts is correct under Bayes voting rule with probability at least
Theorem (Optimality under Bayes Voting). Under Assumptions 7
Σ exp -n -nn-Kin .T.KKT../// (xxp(nnKTi///223
1
where KT,,j = n- EE KL (T.,G.; (G, Vjj E (2....,19113
The proof is provided in Appendix F. Theorem provides an asymptotic guarantee for the recovery of the true graph under Bayes voting rule. It posits that the probability for the collective decision obtained from a group of informative experts to be erroneous decays exponentially in the group size and the average level of informativeness. For finite group size n, the more informative each expert in the group about the true structure G1, the faster the group arrives the correct decision.
Definition (Identifiability from Noisy Structures). causal graph G said to be identifiable from input D and a set of noisy structures G = {G G..... if there exists no other graph G' such that P( G, D) = P(G G)DD).
By Definition D Condition I is essentially a sufficient condition for the identifiability of causal structure from noisy counterparts. It is worth noting that informativeness is weak condition since it only requires every two rows in any matrix T; are distinct element-wisell which holds almost surely" Junder usual continuous models. fact, the result in Theorem | even allows for some, ideally in small number, non-informative experts in the group. More interestingly. it does not require the graph to be identifiable from any given data samples (c.f Definition 2 in Appendix A
"The more familiar term specific to the setting of binary choices is majority voting.
in theoreec sense; see discussion Appendix B
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3.2 RECOVERY VIA AGGREGATION OF NOISY SUB-STRUCTURES
The problem is now reduced to estimating the prior π and competence transition probabilities {T; /- of selected group of experts. The caveat however is that we are dealing with an exponential space of states, thus estimating these quantities of ultra-high dimensionality is computationally infeasible. In this section, we propose a strategy to decompose the graphs into sub-structures that can be aggregated through Bayes voting with more feasible state spaces.
Units of Aggregation. Given set of nodes V = 02. }. we define a feature level as an abstract type of sub-structure determined by a particular of grouping of nodes in V. A feature space (?(V) (or Ω in short) is the set of possible configurations associated with chosen feature level such that the union over all such groupings spans the set V i..., Uwen [w]- = V, where each element w E 1 is referred to as a featurd For example, a basic feature level is node pairs, where Ω = {(0,, 1y) :1 S j S [V] and an element (14,0y) E Ω is feature. Feature levels represent the granularity at which features are considered and define the basic units of analysis for the graph. Higher-order levels may be of interest such as node triplets (2), By, v&) or quadruples (14, 0j, Uks vt).
Given a feature level 52 each feature w is treated as random element with a finite space S(w) of mutually exclusive connective states among nodes in the grouping ([S(w)) 2 2), with some fixed labelling 1, 2, .... |S(w)). For example, one may consider adjacency between pair of nodes (u,, vy) - -equivalent to = {1 02- - Ujz 2 : no edge); if one further accounts for orientations, a possible domain is S = {1 02 -> Vi, 2: ea <- V; 3: no edge}. Let S := (S(w)) denote the set of state spaces associated with a feature space Ω2 A graph structure defined over a tuple (V, S2 S) can be viewed joint realization of mutually inclusive events of each 52 being assigned certain state in its domain S(w). Hence, the set of all possible graphs can be re-defined as G == Iw S(w).
Aggregation via Feature-wise Bayes Voting. Given a tuple (V,2,S) of feature space and domain under analysis, by construction, there exists a ground-true set of states corresponding to every feature E Ω. A feature w can be viewed as task subject to decision making, and an expert's prediction of a structure give rises to set of votes on a multiplicity of tasks. Our problem can thus be treated as a general weighted voting game
fixed set of tasks with finite choices (Baharad et al. 2011).
For a random feature w E Ω with domain S(w), let Xx(w) E S(w) be the random variable represent- ing the noisy prediction from the i-th expert for feature w. Let Y (w) S(w) denote the true state of a feature w and Y (w)) : (S(w))"* + S(w) be the collective decision from n experts about the state of every feature w E 12 based on some voting rule i.c., (w) is a function of (w), ... Xx,, (w).
Let C1,, S) return the structural pattern (e.g.. skeleton) associated with the true graph G2 induced by S), Let G) be the predicted structure by aggregating the collective decisions on all features E Ω over state space S from experts.
Corollary 1. G(2, coincides with G G if and only if the voting decisions on every feature E Ω are correct.
It immediately follows from the definition that a candidate graph coincides with the true graph if and only if every sub-structure is perfectly aligned. The formal analysis is provided Appendix E.2 which further shows that the probability of (22 S) being correct is upper bounded by the worst-case feature accuracy. This simple result enables to decompose the structure learning problem into series of sub-tasks of recovering the true state of every sub-structure w E Ω from its noisy versions.
By this re-definition of the graph space over (2, the prior π can be written as the joint distribution over the set of random variables (Y ((w))ee Let π (w) == [P = D)]jes(w) denote the true prior distribution associated with feature w, induced by marginalizing π over Y (w). We assume S(w) spans over the support of π (w)). Similarly, we define Vi E [n] : (w) := [P(X:(w) = result of marginalization on T; over w. Under the appropriate conditions on the experts' informativeness about individual feature w, the result of Bayes voting rule in Theorem
Y(w)= = j). (l..js((w)
1 directly applies to the task of predicting Y(w) over its state space S(w) of reduced cardinality. We
'The term is inherited from the classic works by Friedman & Koller 2013 Friedman et al. 2013 and to be
distinguished with features of data.
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here note that i-th expert is defined to be informative about feature w if Condition 1 is satisfied w.r.t the marginal transitions T; (w) ) for any two state options k,j E S(w).
4 ESTIMATING COMPETENCIES OF CAUSAL DISCOVERY EXPERTS
Direct graph-level Bayes voting is inherently intractable since it requires one to handle a (24 x 2dd) competence matrix for every expert. On the other hand, feature-level aggregation begins with specification of feature level I i.e., number of nodes per grouping of nodes in V. With careful selection of I, the recovery of the true causal graph can now be feasibly achieved by simultaneously recovering each feature's ' true state via the respective Bayes votes. Concretely, for a graph with fixed d nodes, a choice of level gives rise to feature space Ω of size (1). Suppose that all features in f have a shared space S cxhaustive m connective states (m 2 2). Feature-level Bayes voting instead requires handling (i) individual (m x m) matrices, which becomes significantly manageable for reasonably small (1, m)).
It now boils down to the estimation of the parameters 0 which consists of the prior π (w) and the set of transition probabilities (T((W)) for all sub-structures in a pre-defined space (V, S2,
0 [[(w)]wEO) ((w) = [T(W), (w)), (w))
In practice, no estimation procedure is perfect and the estimates would be inevitably noisy some extent. Before explaining how to estimate the parameters, we study the conditions under which Bayes voting remains optimal when applied noisy parameter estimates. is easy to see from Eq. (3) that there is still a chance for the experts to predict correctly as long as the score assigned to the true state remains maximal. Proposition [ characterizes the probability of having correct collective decision in this scenario, which will be called a "noisy" Bayes voting rule.
The following analyses apply to arbitrary feature E Ω with state space S(w). Let (ww) and (w) represent some noisy estimates of π (w) and (w) for a certain feature w E Ω. For ease of exposition, we again label as the correct state of the feature under analysis.
Proposition (Robustness of Bayes Voting). Suppose the estimates (T.(w) satisfy Assumption
2 w.r. some constant 7 <, +00, then under Assumption [ experts for a feature w is correct under "noisy" Bayes voting rule with probability at least
the collective decision from a group of
1
Σ exp j22
-
AKL (
(6)
as long as (KL >> 1/2 and log (क (#((w) (w)/H((w)) (w)) = O(1) where AKL
KL (Ta (Tin(w);; Tri, (w)) KI. (T. (w): T (w)) Vj e (2.,,,...,
The proof is provided in Appendix B. The probability of converging to the correct decision is thus driven by the deviation of the estimated competencies from the true transition distribution corresponding to the actual state. Specifically, the estimated row of the transition matrix for the true state should remain as close as possible to the ground truth, while the estimated rows for the incorrect options can vary more freely provided that they are sufficiently distinct from the latter. Fortunately, this condition is evaluated upon the total error across all experts, implying that the aggregation process tolerates the presence of bad estimates for some experts as long the group whole remains informative about the true state. If the estimates are perfect, then AKL1,, reduces to KT,,, (w) = n Σ KL (T. (w); T.,y(()) as equivalently defined in
Ensemble Design Choices. We now discuss the key constraints to be considered when designing a feature-level aggregation strategy. With experts, the total number of parameters for estimation is (f) x (nm² - (n 1)m- 1). The choices of the values for (I, m, n1) are thus critical performance factors. It easy to see that the number of possible states grows super-exponentially with the feature level. Furthermore, an insight from Proposition] iis that the probability of error is summed over all incorrect states. Since the goal is to have this error probability decay rapidly to zero, it is beneficial to minimize the complexity of the feature space. Regarding the number of experts, such choice comes

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with trade-off in computational efficiency. On one hand, is theoretically advantageous to have large n. since it would lead to a faster convergence to the correct decision ideally at an exponential rate should all experts be highly informative. On the other hand, the use of more experts introduce more quantities to be estimated, imposing higher memory and computational demands. Furthermore, there remains question about the identifiability of the parameters that also involves the decision on number of experts. We provide a detailed discussion on this matter in the next section.
Parameter Estimation with Optimal Transport. For a certain sub-structure w with unknown true state (w)) E [mx] we wish to estimate the parameters ((w), which consist of the marginal prior probabilities (w) E A m-1 and a fixed set of n competence transition matrices (e))))) where each (i))) E (Am-I)m with A m-1 denoting the probability simplex size m.
We observe set of noisy predictions about Let (()) := [m]] be the space over (w) and ((X(w)) be the set of probability measures on this space. When m, n are finite, we can treat X(w) as a discrete random vector taking finite values in {1,... ,,n"] and distributed according to some true distribution P. (X(()) D). We
X(w) {Xx(W))ee(m) with each Xi(w) [m]]
consider the parameter space (Ww) S (A mm+ ) endowed with a distance function de. Let M(w) == (X)((( D) P E P(X(W)), ((w) E ((w)} be the statistical models of interest.
Suppose that one can repeatedly query from each algorithmic expert and in total obtain N E N ii.d voting profiles [[r((W]N22 we denote the empirical data distribution as P (x(((w) D) =
- Ex 1 (w) = i] jEX(")"
We adopt the classic approach of minimum distance estimation (Bernton et al. 2019), which seeks an estimator over the parameter space ((w) that minimizes some distance between the empirical distribution N'N and the model distribution Pe. Under the OT distance, as defined in (2) for some cost
function c, the associated minimum Kantorovich estimator (Bassetti al. 2006 given as
@N(W)= A(W)EB(W)
arg min We [PN (X(w) : (
Let (.(w) TT. Po. (X(w) PX D). An immediate question arises as to whether the parameters 0, (w) identifiable from the observed data distribution. Another question is why the OT distance is ideal for our estimation problem. Theorem 2 addresses these two questions by first characterizing the condition for identifying the parameters, then establishing the consistency of the minimum Kantorovich estimator defined in 7 The proof is provided in Appendix E.4
[w), denote the true generative parameters that induce
Theorem (Consistency). For any sub-structure w with m possible states, let 0v(ww)
(w), Tw.,(w)... ... (.))....,,T,...( be the minimum Kantorovich estimates satisfying (7). As long as there are at least (2m 1) informative experts, then under Assumptions andd a.s. N-000
(i) 7 (W)
ne (w), and
(ii) Vi E [n] there exists a permutation 0, E Sm such that TN,(ww P PoTT,,(W).
N-000
where Sm is the symmetric group of degree m and Pa, is the permutation matrix associated with o,
Here we show that with at least (2m - 1) informative experts, the parameters can be identified up to some row permutations of every matrix (w) In the infinite limit, we can thus achieve the convergence of the estimates ((w) to an equivalence class of the true parameters under label permutation, with the empirical distribution P approaching P. at a rate of O(1/VN) (Agarwal et al., 2019) in the general case. Theorem 2 is adapted from a well-known result regarding the consistency of the minimum Kantorovich estimator in Bassetti et al.] (2006). which further requires two regularity assumptions to ensure the cost function is well-behaved. It highlights another design consideration: we should aim to keep the number of states reasonably small so that consistency can be achieved with a feasible number of experts. is worth noting this condition is generic to the rank of each matrix T, Given precise knowledge of the ranks, the number of sufficient experts can even be greatly reduced. Particularly, if all matrices are non-singular, then not only are as few as 3 expert necessary but also sufficient for identification for any m > 2 (Liu al. 2023).
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There remains caveat related to the partial identifiability result in Theorem 2 The presence of permutation indeterminacy implies that some state j could be mistaken for the true state under a misaligned permutation. In relation to Proposition this can have an adverse effect (w). To resolve such ambiguity requires additional knowledge beyond informativeness regarding the structure of the experts' competence matrices. A common assumption is that the diagonal entries of each matrix T. (w) are dominant i.c. Vi E [n], J E [m]] T5 (w) > Ti [w]] This implies that each expert more likely correctly label the true state than make an error. Therefore, obtain to those of diagonally dominance. As long as the majority of experts are truly competent, the robustness of Bayes voting can tolerate a small number of misspecified experts.
a stable solution structure, we impose an inductive bias restricting the search space of
Since the goal to minimize the number of features and state space, we simply focus on edge-level sub-structures (7 = 2) and consider shared state space over all features with m = 3 mutually exclu- sive states for each pair of nodes (v;,2) i.e., (v,, 21y) {1 : 02 - U/ 2 02 <- v,.3 : no edge). Ultimately, the scope of the parameter estimation task extends to all features and experts. In Ap- pendix B we consolidate the above discussions into a dedicated section that provides comprehensive guidelines for the optimal configurations of an structural ensembling strategy.
While minimum Kantorovich estimators are theoretically appealing, the remaining practical challenge lies in computing the OT distance in Eq. (7). This task can be viewed a parameter learning problem in a latent variable model, where X₃ (w),... (w) are the observed noisy label variable and Y (()) is the unobserved true label. To this end, adopt recent development proposed in Vo et al. 2024
which provides a tractable and likelihood-free formulation of the OT distance between the empirical and the model distributions over observables in a latent variable model. This framework additionally supports amortized optimization, allowing us to efficiently solve the parameters for all experts and features within a single training procedure. Appendix presents details on the algorithm along with illustrative pseudo-code. For the technical proofs, readers are referred to the original paper.
RESULTS AND DISCUSSION
We assess the structure learning performance in both synthetic and real-world settings, where the estimated causal structures are compared with the ground-true ones based on the common metrics: structural Hamming distance (SHD), structural Intervention distance (SID) and Fl score for adjacency and orientation accuracies. SHD measures the smallest number of edge modifications required to recover the true graph. SID assesses the number of interventional distributions in the true causal graph that are disrupted in the estimated one. While SHD favours sparse graphs, SID penalizes overly sparse ones, resulting in a more balanced and comprehensive performance evaluation. Lower SHD, SHD (4) and higher F1 (f) are better. We generate p graph predictions per expert, yielding up to N = p" possible voting profiles for a group of n experts. We estimate the parameters from this collection of voting samples by solving objective (7p. Bayes voting is thereafter applied on every voting profile to obtain an aggregated graph.
Simulations. We initially validate the theoretical results via simulations of the generative processes for noisy graph structures. We first define a prior joint distribution over edge states, where the probabilities for edge existence sampled from (O, 0.5). For each expert, we then initialize edge-wise transition matrices, which specify the noise rates for each edge and encode the probability that expert correctly predicts an edge's 's true state via the diagonal entries in the associated matrix. We examine three levels of expert competency strong, medium, weak based on the distributions of the diagonal probabilities: (0.55, 1.0), U (0.44 0.6) and (0.33, 0.5) respectively. is worth recalling that when experts are only moderately or weakly competent, unique identifiability of the parameters no longer guaranteed due to row-wise uncertainty. This setup aims to assess the robustness of causal discovery performance of noisy Bayes voting rule to this indeterminacy. For every edge, we iteratively sample its true state from the prior distribution and then select the corresponding transitional distribution to generate a noisy state. Repeating the procedure yields a set of noisy structures or voting samples for estimation. We evaluate the aggregated graphs from our "noisy" Bayes voting approach (Bayes Est) against the aggregated ones from the ground-true parameter values (Bayes True) and Plurality voting. The performance of the Best Expert (i.e., lowest in SHD, highest in F1) is also reported for comparative purposes.
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Adjacency
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Figure 1: Simulations for graph size 50 on 50"h voting profiles.
Real-world Systems. We additionally analyze ensembles of real-world experts on 9 popular continuous and discrete benchmarks. We consider selection of popular high-performing statistical algorithms of diverse classes of causal discovery approaches (see reviews in Appendix [A]. Besides Plura 1 i t voting, also explore a rank- ensemble method (Rank) proposed by Malmi et al. 2015) for comparison. In these experiments, either bootstrapping or data partitioning. In the main text, we summarize our key findings (F) and present representative empirical evidence. We refer readers to Appendix for the full results as well as the rationale behind the experimental design.
50 graph samples per expert are obtained via
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Figure Experiments on Sangiovese (continuous, 14) and Child (discrete, d 20) benchmarks. Causal experts are in black and ensemble-based methods are coloured.
Key Takeaways. Figures D and report the average structure learning performance over 2000 random aggregated graphs and 30 model initializations, (F1) When the true parameters known, the Bayes voting decision demonstrates its superior accuracy, which aligns with its theoretical optimality. In the absence of perfect knowledge, our noisy version Bayes Est secures the second-

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best performance, approaching Bayes True the number of experts increases long they are moderately to strongly competent. The performance gap widens with weakly competent experts, yet Bayes Est well surpasses P Plurality voting and confirms its robustness across varying competency levels of experts in the ensemble. This performance pattern is consistent throughout our ablation studies on different graph sizes d and voting samples p (see results in Appendix (F:2).
(F2) On the comparison with individual experts, the results are mixed, depending on the variation in experts' competency. This pertains to the effect of 7 a factor that so far remains undiscussed. In our simulations, the experts are homogenously competent, equivalent to keeping T reasonably small, thus allowing for large n to dominantly drive the probability of correctness to one. However, real-world experts tend to be heterogenously competent. Contrary to the common belief that ensemble performance is better than individual learners, our findings suggest that not always the case when experts' competence varies greatly. Bayes voting mitigates this situation by taking experts' competency into account and assigning more weights to competent experts, which Plural i y and Rank fail to do. Our empirical evidence demonstrates the accuracy of our noisy aggregated graphs in this scenario approach that of the best causal experts in the ensemble, which is in fact a good news when the true performance of each expert is unknown in practice. Across real-world experiments (see results in Appendix(F.] we further show that our Bayes Est performs no worse than other ensemble-based methods, and outperforms them in most cases with lower variance.
(F3) The key message is that given a random voting profile from the experts, the aggregated graph obtained under Bayes Est on average would yield more desirable and reliable performance. In many scenarios, one may be interested in deriving a single final graph for downstream causal tasks. For this purpose, we recommend two-phase ensembling strategy with two possible approaches: performing inter-algorithm aggregation first (via P Plurality or Rank) and then intra-algorithm aggregation. Tables present detailed numerical results, showing that applying Bayes Est on top of the set of "average" aggregated graphs produced within each expert achieves the strongest performance, where we find Rank + Bayes Est to the most effective and stable combination.
LIMITATIONS AND CONCLUSION
In this work, we have presented a rigorous approach ensemble- based causal discovery, comprehen- sively addressing key aspects of the problem from recovery of the true causal graph, to robustness of noisy estimates and computational efficiency. Our quantitative analysis sheds light on when and how aggregated graphs perform optimally, thereby strengthening the theoretical underpinnings of ensemble- methods.
Prior to aggregation, the proposed framework involves estimating experts' competence matrices using i..d. voting profiles conditioned on the input D, which can be generated via bootstrapping the experts' graph predictions. Although some data settings (e.g., time series) may violate the i.i.d. assumption, established methods such as block bootstrapping allow the framework to remain applicable.
Our theoretical results further extend learning general structures, including cyclic graphs. When the true graph is acyclic, under the conditions in Theorem Tor Proposition[]] the aggregated graph is expected to be acyclic. Notably, our experimental results reinforces the result: the final aggregated graphs remain acyclic even without explicitly enforcing acyclicity constraints during optimization. In practice, conflicts can however still arise due various computational complexities. In our setting, the aggregation procedure naturally produces posterior probabilities over the states of each feature (e.g., edges), which directly quantify the reliability of the experts' predictions. A common empirical strategy is to leverage this information post hoc: incorporating prior knowledge of acyelicity through thresholding (Zheng et al., 2020 [Bello et al., 2022), where low-confidence edges are iteratively pruned until acyclic graph is obtained.
Finally, due to the density of the paper, we currently focus on scenarios where the data inferred by the experts is sufficient, i.e., there are no latent confounders. Extending the model to account for confounders would involve introducing additional states in the features' state space, which comes with additional subtleties that require dedicated investigation. Fundamental considerations relate to, for instance, the number of confounders per pair of nodes or the selection of at least 7 reliable algorithms to deal with 2 4 states. We leave these challenges for future works to explore.
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ACKNOWLEDGMENTS
Haoxuan Li is supported by the Institute for Decentralized AI, project of the Cosmos Institute funded by the Al Safety Fund.
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Appendix
Table of Contents
Related Work
Causal Discovery Algorithms
A.2 Ensemble-Bare Causal Discovery
A.3
Theory Voting
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B
Structural Ensembling Guide
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B.1
Choice of Aggregation Units
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B.2
Characteristics of Experts
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B.3 Number of Experts
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Parameter Estimation Algorithm
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Summary of Notation
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E
Proofs
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F.1
Proof for Theorem
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Proof for Corollary
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Proof for Proposition
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Proof for Theorem
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F Numerical Results
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F1
Real-world Systems
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F2 Simulations
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USE OF LARGE LANGUAGE MODELS (LLMS)
The use of LLMs in this paper was exclusively limited to writing assistance and did not contribute to the conceptualization of the research or the implementation of the methodology.
A RELATED WORK
A. CAUSAL DISCOVERY ALGORITHMS
Classic causal discovery algorithms can be broadly classified constraint-based score-based methods. The former class of methods, namely PC (Spirtes & Glymour 1991) and FCI (Spirtes) al. 2000), detect edge existence and direction via conditional independence lests, while the latter consist of methods that optimize for DAGs based on a given objective function (Ott & Miyano 2003 Chickering 2002 Teyssier Koller 2012 Cussens al.) 2017). Research on continuous data in particular has seen significant advancements in recent years, largely due to the introduction of non-convex formulations for enforcing acyelicity. This has led to the emergence of numerous scalable DAG learning frameworks based on continuous optimization techniques, notably Lachapelle et al. (2019); Zheng et al. (2020); Yu et al. (211)) Bello et al. (2022). Constraint- based causal discovery also be extended discrete data with G-tests (Quine & Robinson 1985) x" tests (Cochran
1952). Score-based methods, on the other hand, can be applied on multinomial Bayesian networks
with BIC (Schwarz 1978 BDeu (Heckerman al. 1995) scoring functions.
Structure learning is computationally intractable in the general case, as searching over the combi- natorial space of DAGs known be NP-hard (Chickering) 1996 Chickering al. 2004). An
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important fact one can possibly exploits is that any DAG imposes at least one topological order and the ordering exists if and only if is a DAG, Ordering- -baied causal discovery is an emerging line of research that focuses the recovery of the partial orderings. The algorithm entails two stages: (1) determining a topological order and (2) subsequent post-processing to remove spurious edges. Research in ordering- -hanin causal discovery recently takes off with the use of score matching (Rolland et al., [2022, Montagna et al.) [023] Xu et al.) (024), in which a valid causal order can be estimated by sequentially identifying the sink/leaf nodes (i.e., nodes with no outgoing edges) based on the Jacobian of the data log- sigliikei
Definition 2 (Structural Identifiability). A causal graph G is said to be structurally identifiable from observed data X with distribution P(X G) if there exists no other graph G' + such that P(X G') = P(X G).
A data distribution IP is said to be faithful to causal graph G if every conditional independence in the data corresponds exactly to d-separation implied by G. is a known result that the causal structure is generally non-identifiable from observational data X (see Definition 22. There are however well- developed theories studying specific conditions under which identifiability is possible (Hoyer et al.) 2008 Peters& & Bühlmann] 2014, Zhang & Hyvarinen 2012). Particularly, when faithfulness holds, a graph G becomes identifialle from data up to its Markov equivalence class, which refers to a set of causal graphs that encode the same set of conditional independencies. and G can be represented by a completed partially DAG (CPDAG). We refer readers to (Glymour et al., 2019 Vowels et al. 2022 Kitson et al. 2023) for comprehensive reviews of the recent advances in the literature. In the following, we shift the focus to ensemble-base approaches to causal discovery.
A.2 ENSEMBLE- BASSE CAUSAL DISCOVERY
Ensemble- basel methods have actively advanced the field of causal discovery, moving beyond simple bootstrap aggregation to sophisticated, multi-layered frameworks designed for big data applications (Tang et al. 2019 Guo et al., 2021). These methods improve robustness and accuracy by integrating multiple causal graphs - often generated through data partitioning, algorithm diversity, or Bayesian sampling from some distribution over graphs into a single consensus structure.
Early ensemble approaches focus on enhancing the stability of causal graph estimation, which is notoriously sensitive to small perturbations in the data. The core idea is bootstrap aggregating (bagging), that is to apply causal discovery algorithms on multiple data subsets (often generated via bootstrap sampling) and then aggregate the resulting graphs. These approaches, namely in Hiidaan et al. (2002): Wang & Peng] (2014); Broom et al. 2012); |Friedman et al.) (2013), perform intra- algorithm aggregation, wherein the same causal discovery algorithm was applied across different samples. Recent developments extend this framework to multi-phase ensemble strategy that further aggregates output structures from heterogeneous algorithms, which can be ensembled graphs obtained from initial round of bagging. Such hybrid approach seeks to harness the complementary strengths of diverse algorithms (Tang et al. 2019 Aslani & Mohebbi [023) Guo et al.] 2021 Zhang
et al. 2023) Since aggregation can introduce cycles, most methods also include post-processing step to resolves violations of the acyclic constraint when specifically dealing with DAGs.
Structural aggregation methods generally fall into two categories: distance-based and voting-based approaches. In the former line of works, the goal is to find an "average" graph that minimizes defined distance all graphs in the ensemble. This includes methods such as MEDIAN (Jiang eet al. 2001]. which introduces the notion of "median" " graph selecting one with the smallest graph- edit distance, or DAGBag (Wang & Pengl 2014)). which leverages family of metrics based on structural Hamming distance. Alternativey. Lanking-basee methods (Malmi et al. 2015 Aslani & Mohebbij 2023) propose to use Kendall-'aa -Tau distance for aligning the partial orderings of causal DAGs. While promising, this aggregation problem is NP-hard particularly for DAGs. To address the computational challenges of high-dimensional data, several algorithms have been proposed: Wang & Peng (2014) develop score-based method using hill-climbing search algorithm for DAG learning; Malmi et al. (2015) use a heuristic greedy search method designed to provide approximate solutions with theoretical guarantees: Aslani & Mohebbi (2023) propose mixed-intedd integer optimization framework where each edge's 's inclusion is a binary decision variable, and acyclicity constraints are
handled efficiently through lazy constraint approach.
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Voting-based ensemble, on the other hand, is simpler alternative. This strategy is often based on weighted voting mechanism, which can vary in complexity of the weighting or scoring functions. Here each individual structure or its adjacency matrix is assigned weight, and this form of aggregation is typically performed at the edge level, where each edge inherits its weight from the same underlying mechanism. Weights can be based on edge frequency e-g-, across multiple bootstrap resamples (Broom et al. 2012 Friedman et al. 2013), or variability in edge frequencies measured by coefficient of variation (Zhang et al., (023). or goodness-of-fi Bayesian score like BDeu (Tang al.) 2019). Edge inclusion is then determined using threshold-basc criteria or majority voting, reflecting the level of consensus across models. Final edge inclusion decisions are made using either simple majority voting (e.g., above 50% consensus) or threshold- rules, allowing the ensemble method to balance confidence and consensus across multiple models.
Other directions include the work by Guo et al. (202T) that propose an ensemble causal discovery framework specifically designed for time-series data, or [Mio et al. (2025) that formulate DAG selection multi-label classification task, aiming to identify the DAGs that are closest to the ground truth by comparing the topologies of the reconstructed graphs.
A.3 THEORY OF VOTING
Voting theory is branch of social choice theory that analyzes how groups can make collective decisions from individual preferences and beliefs. The field emerged from both political philosophy and mathematics, seeking to understand when and how democratic processes can produce good outcomes. The Condorcet Jury Theorem (de Condorcet] 1785 Young 1988) provides a foundational result in voting theory: if each voter independently selects the correct option in a binary choice with probability above 50%, then the probability that the majority decision is correct approaches one as the number of voters grows infinitely large. This original formulation assumes identical competence. independent judgments, and majority rule.
Subsequent research has explored extensions of the theorem address more complex and realistic voting scenarios, such as dependent voters (Berg) 1993 1994. Berend & Sapir 2007; Peleg & Zamir 2012)or multiple choices (Goertz& & Chernomaz (201)). The theorem has also been extended to sequential voting where later voters observe earlier votes (Alpern & Chen 2022), and to weighted voting systems with major and minor voters having different influence levels (Lindner 2008). Modern works consider strategic voting and game- theoretic models, where voters behave according to Nash cquilibrium strategies rather than voting sincerely (Schoenebeck & Tao 2021).
An important extension of the Condorcet Jury Theorem is to the setting of epistemic Bayesian voting which incorporate heterogeneous voter competences and uncertainty. Notably, Nitzan & Paroush (1982) examine optimal decision rules in uncertain dichotomous choice settings, developing an optimal weighted majority rules where weights are proportional to the log-odds of each voter's s accuracy maximize the group's 's probability of correct decisions. This model allows for more nuanced aggregation, especially when experts' judgments are correlated or when their reliability varies across different issues.
Further advancements in Baharad et al. (2011) apply weighted aggregation across multiple issues, using performance on related questions estimate and assign Bayesian-inspired weights for improved collective accuracy. Built on this framework, the authors then introduce an expectation expectationmaxximizztion (EM) algorithm that infers optimal weights solely from voting records, without requiring knowledge of the correct answers (Baharad et al., 2012). By analyzing agreement patterns, this method ranks voter expertise and facilitates statistically robust aggregation rules that outperform simple majority voting in uncertain epistemic contexts. In comparison with these past studies, our work essentially deals with a more general setting involving multiple issues and multiple alternatives, where we leverage the novel estimation algorithm based on optimal transport to solve this complex problem.
B STRUCTURAL ENSEMBLING GUIDE
In this section, we present the details on how to configure an optimal ensemble strategy to maximize the probability of making the correct structural decisions under Bayes voting. The proposed guidelines directly informed by theoretical results in Proposition T
Theorems
[ and 2
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B.1 CHOICE OF AGGREGATION UNITS
As discussed in Section 4 it is ideal both theoretically and computationally to focus on edge-level sub-structures, which corresponds to the feature level / = 2. Generally, the state spaces are necessarily the same across edges. For example, the skeleton of a graph can be constructed from a shared domain over S that includes either absence or presence of undirected edges. CPDAG on the other hand has mixed domains where the state spaces of some features consider edge directions while others do not.
We here consider shared state space over all features with m = 3 mutually exclusive states for each pair of nodes (U,, 0y) e.g., S(u,, v2) [1 : + U/ 2 U/ <- Uy. no edge). We note that this design choice is purely for ease of implementation and it is entirely feasible to independently solve for each feature with different state spaces. For algorithms like PC or GES that only return CPDAGs, we handle undetermined edge orientations by assuming uniform probabilities to both directions when edge existence is confirmed.
B.2 CHARACTERISTIC OF EXPERTS
There are 4 critical aspects to be considered for desirable pool of experts. The most important conditions to satisfy are Assumption [and Condition [ The former pertains to the condition among experts, while the latter refers to the condition within experts.
First, Assumption I which reflects the diversity of experts in the prcision-making body, requires the experts be mutually independent given the ground truth. This condition can be reasonably satisfied by selecting causal discovery experts that operate on distinct modelling assumptions i.e., algorithms in diverse classes of approaches, for example those outlined in Appendix A
Second, by Condition I each of the experts to be informative in the technical sense that no every two row in every competency matrix is identical. Consider a competency matrix T; E (Al))-1)0 where each row is drawn from any distribution that has a joint density on the simplex A 19|-1 It is a classic result in measure theory (e.g.. see Hurder& & Mitsumatsu (199T), Folland (199)) that the set of events where every two point on probability simplex size K is equal has measure zero under any law that absolutely continuous w.r.t. the product Lebesgue measure on the simplex. This reassures us that Condition holdd almost surely and can be well justified for use in practice.
Third, Theorem introduces an important caveat in relation to the identifiability result: beyond informativeness. the experts must also be competent that is, they should be more likely make correct predictions than not. This corresponds the diagonal entries of each competency matrix being dominant in their respective rows, as formally defined below.
Condition 2 (Competency). An i-th expert in the decision-making body is said to be competent if T,GGIGG, Ti,,,IG, Vj E {1,..... (9))
Competency implies informativeness, thus all theoretical results hold when Condition 2 is satisfied. is worth noting that this condition is purely for the purpose of estimation, precisely ensure unique identifiability of the parameters for the reliable recovery of the true competency matrices. While this assumption is often unverifiable in practice, it can be approximated through prior knowledge or empirical benchmarking on datasets of related domains. A natural strategy to increase the chance of meeting this requirement prefer methods with few theoretical assumptions and/or strong empirical performance. We suggest realistic selection of such experts in Appendix F.1
Fourth, not only should the experts be principally heterogeneous but they should also be homoge- nously competent. In other words, the variation among experts' competence should be minimal, such that a small realization of T their competence can be achieved. This would contribute to minimizing the error probability, as suggested in Theorem
B.3 NUMBER OF EXPERTS
The final question is how many of such experts is need, A straightforward answer from our theoretical results is: the more. the better. Furthermore, Theorem[]sar (ayys that the minimum number of experts sufficient for identifiability is (2m - 1) where m is the number of feature states. Choosing m = 3 implies that need least 5 experts to be generally safe. As discussed in the proof in Appendix
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this requirement holds under the generic condition that the Kruskal rank of the competency matrices is at least With more specific knowledge about the exact rank of each expert's 's competency matrix, this number can be reduced to as few as 3 experts. In our empirical evaluation real-world discrete datasets, we show that such a reduced number of experts can still yield reliable performance in several cases.
E.4
Lastly, is worth recalling that a large number of experts introduces computational constraints related the estimation of the quantities necessary for applying Bayes voting. Fortunately, the parameter estimation framework introduced by Vo et al. (0224) has significantly contributed to alleviating this computational burden by facilitating an end-to- optimization procedure for all features and experts. We detail how this framework is applied in the next section.
C PARAMETER ESTIMATION ALGORITHM
Problem Setup.
We wish to estimate the parameters 0 that consist of
(0(w)) ((w) = [T)),, (w), .... (w),
Vo et al. (2024) proposes OT-based framework for estimating the parameters of general directed graphical models (of known structures) with latent variables. To understand how our problem fits into this landscape, we first examine the generative process of noisy sub-structures.
For every sub-structure w E S2 the noisy data generating process goes as follows:
1. Sample true state y(w) (at(TT(w)),
2. Sample noisy state x; (w) Cat (T.loww)). Vi E [n].
This generative setup can thus be represented by simple directed graphical model over the set of observed variables X X(w) = (W((w) [E[m]] and latent variable (w)..
Let Cy(w) be the one-hot representation of discrete value y(w) E [m]] Let uuw) P(U(w)) & Uniform( [0, 1)
If one employs the Gumbel-Softman reparameterization trick (Jang et 2016 Maddison al. 2016), the computation of z; (w) can be written deterministically as
xy (w) = the [u(w), y(w)] A Cat-Concred (eu(w) T T:(w)
(8)
Here the Cat-Conaten (ot-CConrett(p)) function is defined on probability vector p pp(),,(()
pp,ppmm) [m)]
exp ((ogg(() + G(0)//T]
Cat-Concrete(p) =
k= exp (log (kk) (k) G(K) jE[m]
with temperature т, random noises GO) = log( - log log and (6) Uniform 1).
For case of notation, compactly denote =ww) == [u(w), [(w]] [0, 1 [m] where z(w) Pe (w)) with
: P(U()) (Y (w) DD)
and the subscript 0 indicating the involvement of the parameters π (w)) in modelling the distribution of random variable Z(w).
Optimization Objective.
Let d(w) X()) + [0, 1l x [m] be a stochastic map that satisfies (P (X(P)((())) (w) () (Z))).
Then Vo et al.
2024 show that for fixed 0, the OT distance in can be tractably computed as
Wo (x((())
min E Ex c(o(w); (()))))((<()))
(9)
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Let define (Z)) Bn(w)~ [d(x(w))) This expression of W. renders objective constrained optimization problem, which can be relaxed into
min
min ((w)
cf ((ww))
(10)
(w)
A D Po(( |PP((
where any arbitrary divergence measure and is a regularization coefficient.
The key ingredient solving (10) is the construction of the push-forward map ((w) approximate the generative distribution for the noisy data. The authors further prove that the family of the valid maps ((w) have infinite capacity (i.e. they include all measurable functions), then (10) shares the same optimal solution with the primal objective 7 (see Theorem A.1 therein).
Practical Algorithm. By the above reformulation, our problem is now reduced to minimizing the reconstruction of the noisy data, which can now be solved with gradient-basee optimizers. Whereas the optimization objective appears complicated, Vo et al. (2024) suggests two simplifications of (TO) that are empirically shown to yield adequate estimates on problems dealing with categorical variables.
First, while in principle we need to additionally model a backward distribution align with the uniform noises, the authors show that it is empirically safe to ignore the noise variable, which means that in our case, is sufficient to only model ((w) N (w) D) = PP (w) (D) and apply the reparameterization trick with random uniform noises generate reconstructed samples of x(w)).
Second, the backward function & here, essentially, returns the probability vector parameterizing the fush-forward distribution Ps whose role is to mimic the model prior distribution T()). As for the reconstruction loss, the authors also show that sufficient to alternatively minimize cross-entropy loss between the target data samples a(w) and generative probabilities T(w). The practical optimization objective to our estimation problem is given as:
min
min BaNPR (o-(()) CB egi T) + CE( T) + + JS ((xx) T
(1I)
where we omit the notation (w) for readability. Here the Sensen-Shannor (JS) divergence chosen the divergence measure and the CE loss function computed
CE (a: p;
m log (p T.,)) -
(1)
log log Tp.T)
We refer readers to Vo et al. (2024) for formal algorithm (see Algorithm therein). To make the paper easy to follow, we only provide an informal summary the training procedure:
1. Initialize ((w) and ((w) (w() (w),,... (w)()).
2. Draw batch of samples (w) ~ PN DD).
3. Evaluate dhe (w))
4. Update ((w) and ((w) alternately by descending
Enforcing Diagonal Dominance via Continuous Sinkhorn Operator. To mitigate the effects of uncertainty arising from permutation indeterminacy, we impose an inductive bias on the structure of each competency matrix, based on the prior knowledge that each expert is competent in the sense that their diagonal entries should be the largest in their respective rows (see Condition 2
The goal is essentially to reorder the rows and columns of matrix in way that maximizes the total sum of its diagonal entries. Formally, this corresponds to a linear assignment problem: identifying the permutation that yields the highest sum along the diagonal. While this is classic combinatorial optimization problem typically solved using the Hungarian algorithm (Kuhn) 1955), these methods not differentiable and therefore unsuitable for gradient-based learning.
To impose this structure while maintaining differentiability for end-to-end training, we adopt the continuous Sinkhorn operator (Mena al., 2018), which offers differentiable alternative by
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relaxing the discrete of permutation matrices into the continuous space of doubly stochastic matrices-squueee matrices where each row and column sums to one. The algorithm iteratively normalizes the rows and columns to approximate this space, effectively yielding soft permutation matrix. Additionally, entropy regularization introduced to encourage smoothness and numerical stability. As result, the Sinkhorn operator produces soft assignment that aligns high-value entries near the diagonal, thus enforcing the desired diagonal dominance in a differentiable manner.
Training Configurations. We optimize [1 for all features w at once with shared backward function & modelled by 2-layer ML. of 192 hidden units. We use Adam optimizer at learning rate of 0.001, parameters with values drawn from the normal distribution. To reduce the uncertainty due to random initializations of the parameters and selections of the graph samples, we take an average of the estimated values obtained over 30 different runs the final estimates for 0.
0.001, 500 training steps and batch size of 10, 000. We initialize the model
D SUMMARY OF NOTATION
We summarize the key notations used throughout the paper provide quick reference for readers.
PP: probability measure
V: set of nodes in graph
d: number of nodes
n: number of experts
D: arbitrary input data for graph prediction
space of graph structures
G: true graph structure
G, noisy graph structure predicted by an i-th expert
π E [0,1]| prior distribution F(G) D)
E [o, 1|(6| competence transition matrix P(GG G, D)
Ω: space of graph sub-structures
arbitrary sub-structure (feature)
S(w): state space of w
Yw): true state of w
((w): noisy state of predicted by an i-th expert
(w) E [o, 1[w((: marginal prior distribution P(Y (w) [D]
(w))) [o, 1] ((W)))((()) marginal competence transition matrix P(X,(w) Y(w), D)
((w) = TTw)),
T, (w)): parameters to be estimated for w
(w): parameter space for w
N: number of empirical data samples
#: weak convergence.
For any E [n] and j
Pig A Pi,G, = -TGG, = P(ā: Gj G= G;
Kijkk A G. T.G. == P(õ. G, G Gt. D)
TG; = P(G = G, D). D).
For Oorany E [n] and j, E (1, (8(w))
(P)(ww) := P(X,(w)) [x)
j
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(.jl((W) -l (w) = ( = =,
Jy(W) = ((Y(Ww) Dj
PrslW) F.slk (w), (w(:::: noisy counterparts of Pi. pi,(w)), (w)) (yW)) respectively.
E PROOFS
E. PROOF FOR THEOREM
Under Assumptions [ the collective decision from a group of n. informative experts is correct under Bayes voting rule with probability at least
Theorem 1
Σ exp (-n-KTn.// /272
1
(12)
J22
where KLI,1 n
Ei= KL
i Vje (2, 12.....1911+
Proof. Let recall the scoring function for candidate graph G E G in 3
Sn u. (G) (G)= Σ Wi 1 [.. G ba:
and the collective decision is given as G = arg maX Sn,wa (G).
For ease of notation, we omit the subscript (n, wy) and simply denote S, (G) as the score assigned to a candidate structure Gj.
Let us also define the binary random variable G., := G. G,]. With slight abuse notation,
we sometimes treat each G. as a discrete random variable that takes index value corresponding to its structural realization. Assuming the true graph corresponds to index 1, have G2,1 Bernoulli(p,.). and Vj 2 2 G., ~ Bernoulli(q.,n).
In Bayes voting rule, the score function is given
Sj S,(ā) log P(G = Gj D), x log P( G,, D) + log (G,, D),
(13)
(14)
log P(द, Gj. D) + log (G, (D),
(15)
Σ G. log + - G.d) log 0, + log kj.
(16)
+ n
- log (ogg,,..)) Σ G. (logp.y-log
log 9, + log
(17)
The second equality 15) results from Assumption 1 This final equality gives rise to the weight and bias terms of Bayes voting in where
Wi.j = log.i. log 9, by = Σ log 9, + log T.
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The probability that the predicted graph G is correct is given
((10)>);())
Pr -
Pr
n
(18)
122
si(@) +
= -1-Pt Pr
U
(19)
J22
2 1 - Σ
Pr ( S,)
s
(20)
j22
Let Zi,- = Gig (logpt. log Q,0.115
+ log 3.1. + log Tj.
Tj. We have
B|Z2,] = Pi, log +
0t,k/1 log 0r,6 + log TI
(21)
h22
1 for j 2 2,
E Zc,]l = Ps.1 log I.1 +
R log 91, + log Ty
(22)
A2
where Pi,j & 46,i by definition. We thus have
Pt,1 El Ziol Pi,1 log
Σ
log
4r,A|1
+
log T
(23)
9r,15
[K,bj
222
9r,1|1
4;,A|1
11
9i log
9:
log
log TT
(24)
N22
K
4i,b/1
Di.k| log
+ log
(25)
qi,kij
= KL (Te Tyt:) + log
TT
(26)
n
TT.
The score difference between the true DAG and some candidate structure G, can be written as
S Sn
(Z,,1 Zz.j).
(27)
E [ Syl
KL (Tij,, + log Jjj B[Z,,1 Zrgl Σ i=1
(28)
Let KT,,, == n Σ KT GT, T. and 11. == / /mj. Since experts are informative, we
have vj 2: To # T. - It follows that vj 2 2 KL1,; 0.
By Assumption 2 there exists a constant < 00 such that Vi E [n] log Tillmax I T. Hence, we can establish that Vi [m]: Z Zz [-7, -|.
Since the truth is biased, that is 2 J > 0 then E[S; Sy] is strictly positive. Applying Hoeffding's 's inequality,
Pr S =P) - S - E(S; Si] 2 E(Si SzD)
(29)
2(n KL1,5 + log TI. )
< exp
(30)
4n72
n. KT. 3 + 2KTT, log Tıj + n log? MT.j
exp
(31)
272
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Substituting the above expression to
20 gives
Pr (G is correct) = Pr G
G1
(32)
n2 KL + 2KI,,; log 11, + n log Ti
2 1 - Σ exp
(33)
272
122
2 T
KL1.j
2 - Σ
exp
2r
(34)
J22
We obtain the desired lower bound due to the non-negativity of KL and log mi
E.2 PROOF FOR COROLLARY II
Corollary 1 w E Ω are correct.
S) coincides with ()) if and only if the voting decisions every feature
Proof. For a given feature space and domain (V, S), we denote Y(w) S(w) as the true state of a feature w E Ω and (ww) : (S(w))" + S(w) be the decision from n experts about the state of w based on some voting rule.
Structural Hamming distance (SHD) is a commonly used metric for comparing two graphs, which quantifies the number of edge additions, deletions, or reversals required to convert one graph structure into another. thus define the generalized SHD between the aggregated structure G(2,, and true structure G (2, as
SHD (Q
= Σ (w)) + Y(w))
(35)
WEB
We say that G)) coincides with (2,) when SHD (( = 00
By definition, we have
SHD () > 00 >0==
wER
Hence, the SHD equals 0 and only no feature error occurs. Consequently. the probability that G S) is correct is upper bounded by the worst-case feature accuracy:
( n -(()
Pr (G(2,)) S) is correct) ) Pr
(36)
WES
min Pr (Y(w)= ={(w)
(37)
well
The analyses of Proposition I and Theorem are conducted at the sub-structure level and applicable to arbitrary sub-structure w in pre-defined feature space Ω. For brevity, we will sometimes drop the notation (w) in the derivation whenever there is no ambiguity.
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E.3 PROOF FOR PROPOSITION II
Proposition [ Suppose the estimates (Ti(w) satisfy Assumption w.r. some constant
F +00, then under Assumption 7 the collective decision from a group of nn experts for a feature w is correct under "noisy" Bayes voling rule with probubility at least
Σ J22
exp 0 (n. AKT /
(38)
as long as AKL >> n 1/22 and log ( (w) (w)) = O(1) where AKL =
12 DE KT. TTin KT. 1()) T (w4) Vj e
Proof. Let 5y(w) denote the score function assigned to some candidate state j S(w). Under Bayes voting rule, we have Sy(w) = log PY (w) = =j X (w), D). The collective decision is given as ((w)= = arg (W)))(())) (w(w).
We will similarly label 1 as the correct state of w to avoid notation overloading. The probability that the collective decision on w is correct can be written as
Pr y (w) is correct) = Pr (w) = 1) 21
Σ Pr (Bi(w)
S 3,(w))
(39)
J22
Let us denote the binary random variable X (w) 1 [X)(ww) = jl. We similarly have X4,1 (w) (w)) and Vj 2 2 (w)(w) ~ (ernooull(q:,j1(000 (w)).
Following the same derivations in the Theorem (w) (logPt,(w) - log (w)) + log (w)))(())) log T. (w)
we can define Zzy(w)
In this case, we are dealing with the "noisy score" induced from from the noisy estimates (Ww) and (Ti Alternatively, we define (w)(w) = Xig(w) (log (ogg).)())) - log (w)) log (w) + n log (ww)
When the Bayes votes are obtained w.r.t the noisy estimates 7t (Tw) and (Ti the noisy equivalence of the expected score differences is given as:
E (w)) Zio(w)]
(40)
M
T log
45, log
+
(41)
T
M
qr.k/1
1
log
log
(42)
9;,b//1
kt,kl5
k=1
Σ k1
log
Σ 1t,b
log
1
log
(43)
n1
log (w)
KL (w), T. KL (w), (u,,,,,,,11
+
(44)
(w)
n
Let (w) = मे (w) (w) We have
E [[((w) [س(4ر] = > R (,] [cs(w))
(45)
Σ KL (w),
(w), Fnl1s) (w) - Σ
KL (T.ji((s) (w) R()l(w)) + log (w(
(46)
[,(w)²
1=1
1=1
, + log मं (w),
(47)
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Since AKL1,(() (w) > n n-1/2 and log (1g(1,(()) O(1), have A 1 (w) - log tT T1. + (n ( for sufficiently large n1 implying Vj 2 2: E [w((w) Sy Sy(w) > 0 eventually.
By Assumption there exists a constant F < oo such that vi [n] log T. S F. We can thus establish that Vi E [n]: (w,( Z E (-ř, 7]. Applying Hoeffding's inequality above, it can be shown that
(n AKLI, 1 + log T (wi)
Pr (P(w) is correct)
21 1 - Σ
exp
(48)
2n72
J22
n AKL 1, (w)
= 1 Σ
exp
θ
(49)
272
J22
We obtain the desired quantity since ( >> 1/2 implying n AKT (w) N-000
oo.
E.4 PROOF FOR THEOREM
The key ingredients to the proof for Theorem 2 are the results in Lemma | and Lemma 2 which respectively establish the conditions for the identifiabiliyy of the parameters (by Definition and consistency of the minimum Kantorovich estimator.
Lemma considers a setting as follows: Suppose that there is an unobserved variable Z that takes values in a s-sized discrete domain 21.2,... RR,, which has a non-degenerate prior Ty == (Z j) > O. We observe n variables (Xi)ee()) where each X; has finite state space {1,2 ... } with cardinality
Let T, be a matrix of size k x in which the j-th row is [P(X, = Z = j).... P(X; = Ti = j)l-
The latent parameters 0 contributing to the data data-generating process of the observations consist of:
0 [,t, T) π := [m]]æt)))
T = (T: )e)))
Definition 3 (Identifiability of Parameters). The parameters ((w) are identifiable if Po(((w)) * (X (w:),, v0(w) * 0' O'(w).
Lemma (Kruskal 1976; 1977; Sidiropoulos & Bro 2000 The parameters are identifiable up label permutation provided that
Σ Kr(T.) 2 2* n -
(50)
where Kr(T,) denotes the Kruskal rank of a matrix T,, which is defined as the largest integer R such that every set of R rows of the matrix is linearly independent.
Assumption 3 (Bassetti et al. (2006))
) Any cost function c: x2 is defined by h o d where d is a distance function on X and h H with denoting the class of all increasing, convex and continuous functions from + to IR+, which vanish at the original and satisfy Orlicz's 's condition.
Assumption 4 (Bassetti et al. 20066). There exists some fixed 0g contained in θ such that Po. = P. and To, Pa(x)) where (c( defined w.r.t cost function as
P(X): / clzz. ((z.)))(dx) < too for some y E x).
Pe(x) = {
(51)
Lemma 2 Bassetti et al. (006)). Let Ba, (t) = (0 O: E M, W. (Po: Poo) St). If
(i) there some T > 0 such that Ba, (T') is a relatively compact set and,
(ii) Poy = for any 0g and (0x)22) contained in O such that de 0p) - 0 as N + oc,
then under Assumptions andd 4 Ø - 0o P-almost surely N xc.
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We ready to prove Theorem
The goal is to obtain estimates for the true parameters 0, (w) := (T) (w)) (
[.n(w))
where Pe. (w)) (=))..).. The minimum Kantorovich estimator is the minimizer of the following OT objective over the parameter space ((w):
(x(W)= = min We [PN (X(w) D) Po (X(w) | ((w))B(w)
Theorem For any sub-structure (fr(w), (w), ....... Ty,n(u)} be the minimum Kantorovich estimates satisfying long as there are at least (2m - informative experts, then under Assumptions and
with m possible states, ler (N(w))
As
a.s (i) (N)
(w), and
N-220
(it) Vi E [n] there exists a permutationn o; E Sm such that TN,(w) Po,Tu,(w)
N-000
where Sm is the symmetric group of degree and Pa, is the permutation matrix associated with d.
Proof. We first prove the identifiability of ((w).
It is easy see that our setting resembles one under analysis in Lemma here K 71
Tn = m. This allows us to exploit Kruskal's condition to prove the sufficiency of at least (2m - 1) informative and competent experts.
By the definition of informativeness in Condition every pair of rows in T, (w) are distinct. Since any two distinet points on the (m - 1--dimensional simplex are linearly independent, one thus has that Vi E [n] (r())ww) 2 2.
Thus when 2m 1,
Σ Kr(T.) 2 2n n + n 2m - + n.
The inequality gives rise to Kruskal's identifiability result in 50 Invoking Lemma] 1 we can identify ((w) up to permutation in rows of every T (w).
We now prove the consistency 0x (w)) The observed distribution for noisy data X(w) = (Xi(w)) can be written as:
P ((((w) D)
Σ P(Y (w) = ( D)) II P(X((W) (() = j,D),
(52)
n
Σ ny
Tuxx (w) T(w) Tx(w),
(53)
where (x( = [T (w),)(()) IT T. T.x, is defined as the tensor product among the Xi(w)-th columns of each (w) respectively.
With finite n2). let recall that the space X((w) can be defined as a finite set 21, mmm)
This means any voting profile ((w) corresponds to some index j E [m*]. For any ((w) E e ( consider the induced discrete probability measure F& over m particles associated with weights }m=} where every value aj Pe (X(w)= j).
{ai, ...
By Assumption 4 there exists a fixed (n(w) E (ww) such that Pe, (X D) (X((W)) D).
Let the weights over m² particles corresponding to the discrete probability measure Po. given by {BI,... Bmm)
Let ((w)= = [w)] We define the metric de 0) = [|] To(w) T(w) T(w)l,, wet the usual absolute norm (A Am-1yam+1
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Our OT distance can be written according (2)
We (Fa.:..P & We Po. (X(w) Po (X(()) D) min Σ
(54)
J.ke|m"
where Cj. is the transportation cost from bin to bin ki and 7 (e((m))))) E [0, 1]m= xm" is the
1mm""
transport plan (or coupling) that minimizes the total transportation cost. The optimal 7 must satisfy the marginal constraints w.r. distributions Pg and Pe.:
W
Vj, k E [m"]:
Σ TUA -jj
Jisk = Bk.
j=1
As result, the maximum mass that remains not transported on the diagonal Σj=1 min(ay, 3j) since the total mass is i.c Σ, aj = Σ, Bg = Thus, the total mass transported off-diagonally, across pairs (j # k), must be at least Σ min(ay.8,)) We also have that
m
m
m
||a B|1
3,1
Σ 1=1
aj + Bg 2min (a,, = min(a,, 82).
(55)
1=1
Since (w) is finite and the cost e non-negative, there exists a value >> O where c(t, y) 2 y for every I, y E (()),, # y. Every unit of mass moved from some j to some * incurs a cost of least Hence,
Σ j.k
CATTs 2 " I1 $ = W.(PaoiPa) 2
1 "1 a - 311
(56)
If there exists a T > 0 such that W.PPo,;P Po) < T for some T < 0, then ||a 1o--81: B112 < 2T/n, This implies Ba, (T) is contained within closed L,ala of radius 2T /n. Then T be fixed in such way that the above sphere is contained in θ whenever On is interior to θ, proving its relative compactness. Furthermore, the mapping / + Pa is continuous w.r.t the L₁ metric on θ hence the weak convergence holds when de goes to zero.
Given that both conditions (i) and (ii) are obtained, invoking Lemma we can show the consistency of the estimate 0 (w), meaning that 0 (w) coincides with (w) when N + oo.
Finally, the partial identifiability result implies that (o(w) differs from (w) up label permutation. Putting all together, we have shown the convergence of (w) to स. (w) and each Tv,;(w) to some permutation of rows in (w,((w) in the infinite limit of sample size.
F NUMERICAL RESULTS
F.1 REAL-WORLD SYSTEMS
Experimental Design. We evaluate causal discovery performance using 7 real-world and 2 semi- synthetic systems with known ground-truth causal structures. The datasets are publicly accessible publicly accessible via the bnlearn repository.
For experiments on continuous data, we consider the real-world benchmarks: Artic (Huang et al. 202I). Sachs (Sachs et al. 2005) and Sangiovese (Magrini al., 2017). We further study semi- synthetic data generated from non-linear additive noise SCMs, where the structural assignment defined in ( takes the following form:
X; = fi (Xpa,) + U,, E [d]
We consider two types of systems where true structural equations f; are modelled either with Gaussian processes following Erdos-Renyi structure (GP-ER), ML. following Scale-Free
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structure (MLP-ER). The noise variables have unequal variances, drawn from Uniform and Laplace distributions respectively. We refer to the systems as romi-synthetic since they are relatively general and widely studied in causal discovery literature. In these experiments, consider the following causal discovery experts:
Constraint-based: PC (Spirees Glymour 1991).
Score-based: GES-BIS GES with BIC score (Chickering 1996).
Linear, non-Gaussian models: ICA-LiNGAM (Shimizu al. 2006).
Continuous optimization for non-linear models: NOTEARS with mean-squared-basset loss; DAGMA (Bello al. 2022) with likelihood-based loss,
Ordering-based: SCCRE (Rolland al. 2022).
For experiments discrete data, analyze the datasets: Asia (Lauritzen 1988). Earthquake (Korb)
Nicholson 2010), Child (Spiegelhalter et al., 1993). and Insurance (Binder et al. 1997) networks. In these settings, consider only minimum number of 3 search-bases algorithms:
Constraint-based: PC (Spirtes & Glymour 1991),
Score-based: GES BTC GES with BIC score (Chickering 1996).
Hill climbing search HCS Koller & Friedman 2009
This goal is to study to what extent 3 experts are sufficient for effective ensemble learning practical scenarios. It is worth recalling that theoretically, 3 experts are sufficient for (partial) identifiability of the parameters when each expert maximally informative i.e., each competency matrix has full rank.
For continuous data, we adopt a setting where a fixed dataset is used. To generate diverse voting profiles for parameter estimation, especially for the deterministic causal discovery algorithms that generate single graph predictions, we consider the "bootstrapping" version of the expert, where the algorithm is repeatedly run on bootstrap samples i.e., subsets of the data sampled with replacement. For discrete data, we use a different approach: large dataset is partitioned into non-overlapping subsets, and each algorithm is repeatedly run on a different partition. For both cases, we fix the 50.
number of graph samples per expert
is important to emphasize that our selection of statistical algorithms is based purely on empirical performance. The above groups of algorithmic experts selected because they among the most popular and consistently high-pen gegh-performng methods across a variety causal discovery benchmarks. They also representative methods of diverse classes of causal discovery algorithms, each based independent modelling assumptions and thereby meeting our criteria for expert diversity as discussed earlier. In fact, our framework agnostic the type of expert used as long as the conditions are satisfied. This means that our approach does not exclude test LLM-based methods on the benchmarks, they unfortunately yielded suboptimal performance and were therefore excluded from our analysis.
experts. However, while we did
Regarding the ensemble-based baselines, although several ensemble- based causal discovery methods have been proposed, only Malmi et al.] (2015) provide public codes for their rank-bar based method, which motivates its inclusion in our experiments.
Results. Figures to 11 report empirical results for the real-world experiments. Table repoots the performance of final aggregated graphs from the proposed wwo-phase ensembling under different combinations of approaches, where the first step of inter-algorithm aggregation be done by either P Plurality or Rank. The methods with the superior accuracies of DAG estimation are bolded.
F.2
SIMULATIONS
38 report the simulation studies that assess the performance of Bayes voting across Figures 12
various graph sizes d E [10, 15, 20, 25, 30, 35, 40, 45, 45,50) and number of graph samples per expert for estimation p E {10, 30, 50).
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SID
SHD
Adjacency
Orientation F1
KINGAM
+
-0
GES-8IC
+
NOTEARS
DAGMA
Boyes Est (Oun)
+
60
100
0.5
0.1
0.2
0.3
Figure 3: Experiments on Sachs dataset (continuous, d = 11).
SID
SHD
Adjacency Fl
Orientation FL
KAALINGAM
GES-BIC
SCORE
NOTEARS
DAGMA
Muraliy
Boyes Est. (0un)
125
130
135
30
40
50
0.6
Figure Experiments on Artic dataset (continuous, 12). SID value is upper-bounded by
d(d - 1) and the first plot shows that all experts score the worst SID exactly of 132 at d = 12.
SID
SHD
Adjacency
Orientation Fl
KAALINGAM
M
(D)
GES-8IC
40
+
NOTEARS
H
DAGMA
11
11
1
Murality
Bayes Est. (Oum)
+
+
140
160
40
60
0.00
0.25
0.50
0.75
0.1
0.2
0.3
0.4
Figure 5: Experiments on Sangiovese dataset (continuous, d = 14).
SID
SHD 4
Adjacency F1
Orientation F1 1
KINLAN
I
SCORE
MOTEARS
LH
Furaliy
Boyes Est. (Ours)
280
250
0.1
0.8
Figure 6: Experiments on GP-ER model (continuous, d 20).
Adjacency
Orientation
LINGLNG
GES-BIC
HH
PC
+
+
SCORE
DACMA
+
Rank
18:
Plursiiy
+++
44-
++-
Bayes (Oun)
1.000
0.75
Figure 7: Experiments on MLP-SF model (continuous, d= 40).
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SID
SHD 4
Adjacency' F1 T
Orientation F1 T
GES-8IC
H
HCS
Bank
Puraliy
Biyes Est. (Dun)
1.00
200
0.925
0.550
0.975
1.000
0.6
Figure 8: Experiments on Child dataset (discrete, d 20).
SID
SHD &
Adjacency F1
Orientation F1
HI+
HHH
HCS
Bank
Furally
Bayes Est. (0un)
200
300
500
40
50
0.8
0.9
0.6
Figure 9: Experiments on Insurance dataset (discrete, 27).
SID
SHD 4
Adjacency F1
Drientation F1
HCS
Rank
Puraliy
Bayes Est. (0un)
10
20
2.5
10.0
Figure 10: Experiments on Asia dataset (discrete, d 8).
SID
SHD 4
Adjacency F1
Drientation F1
HCS
Bank
Hurslly
Boyes Est. (0un)
3
15
0.7
0.8
0.9
3.0
0.25
0.50
0.75
1.00
Figure 11: Experiments on Earthquake dataset (d 8).
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Table 1: Results of the final aggregated graphs from two-phase ensembling.
Methods
SID 4
SHD
Average F1
GP-ER (continuous, d 20)
Plurality Rank
204
38
0.435
Plurality Plurality
205
39
0.412
Plurality + Bayes Est. (Ours)
199
35
0.500
Rank + Rank
205
39
0.412
Rank + Plurality
205
39
0,412
Rank + Bayes Est. (Ours)
199
35
0.500
MLP-SF (continuous, d 40)
Plurality + Rank
186
20
0.844
Plurality Plurality
187
20
0.851
Plurality Bayes Est. (Ours)
182
19
0.851
Rank + Rank
187
20
0.851
Rank + Plurality
187
20
0.851
Rank + Bayes Est. (Ours)
169
18
0.865
Sachs (continuous, d
11)
Plurality Rank
99
26
0.167
Plurality + Plurality
99
27
0.162
Plurality Bayes Est. (Ours)
99
27
0.205
Rank + Rank
99
26
0.167
Rank Plurality
99
27
0.162
Rank + Bayes Est. (Ours)
99
27
0.205
Artic (continuous, d
12)
Plurality Rank
132
27
0.211
Plurality + Plurality
132
27
0.211
Plurality Bayes Est. (Ours)
132
26
0.241
Rank + Rank
132
27
0,211
Rank + Plurality
132
27
0.211
Rank + Bayes Est. (Ours)
132
26
0.241
Sangiovese (continuous,
14)
Plurality Rank
172
41
0.256
Plurality + Plurality
172
41
0.263
Plurality Bayes Est. (Ours)
170
39
0.296
Rank + Rank
172
41
0.256
Rank + Plurality
172
0.256
Rank + Bayes Est. (Ours)
170
39
0.293
Child (discrete, d
20)
Plurality Rank
20
1
0.960
Plurality Plurality
44
2
0.920
Plurality + Bayes Est. (Ours)
0.960 1
Rank + Rank
20
1
0.960
Rank + Plurality
20
1
0.960
Rank Bayes Est. (Ours)
20
1
0.960
Insurance (discrete, d 27)
Plurality + Rank
349
20
0.725
Plurality Plurality
374
22
0.697
Plurality + Bayes Est. (Ours)
367
22
0.703
Rank + Rank
349
20
0.725
Rank + Plurality
349
20
0.725
Rank + Bayes Est. (Ours)
342
20
0.731
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SHD
Adjacency F1
Orientation
STRONG
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10 No. experts (n)
No. experts (n)
No. experts (n)
Bayes (Ours)
Bayes True
Expert
Plurality
Figure 12: Simulations for graph size d = 10 on 50" voting profiles.
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Orientation
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No. experts (m)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 13: Simulations for graph size d 15 on 50"h voting profiles.
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SHD
Adjacency F1 T
Orientation F1
1.0
1.0
STRONG
30
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10
80
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MEDIUM
60
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20
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WEAK
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60
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12
12
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 14: Simulations for graph size d = 20 on 50" voting profiles.
SHD
Adjacency F1 1
Orientation F1
1.0
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40-
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MEDIUM
100
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50
0.4
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12
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 15: Simulations for graph size d = 25 on 50"h voting profiles.
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SHD
Adjacency F1 T
Orientation F1
1.0
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MEDIUM
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250
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12
14
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 16: Simulations for graph size d = 30 on 50" voting profiles.
SHD
Adjacency F1 1
Orientation F1
150
1.0
1.0
STRONG
100
0.9
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0-
300
0.8
MEDIUM
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50
330
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300
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150
10
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 17: Simulations for graph size d = 35 on 50" voting profiles.
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SHD
Adjacency F1 T
Orientation F1
200
1.0
1.0
STRONG
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100
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MEDIUM
300
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100
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WEAK
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0.4
10
12
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 18: Simulations for graph size d = 40 on 50" voting profiles.
SHD
Adjacency F1 1
Orientation F1
1.0
1.0
STRONG
200
0.9
1.50
100
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50
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0-
500
MEDIUM
400
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600
500
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12
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 19: Simulations for graph size d = 45 on 50"h voting profiles.
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SHD
Adjacency F1 1
Orientation F1
300
STRONG
200
0.9
0.9
0.8
0.7
01
MEDIUM
400
0.6
200
0.6
0.7
600
WEAK
500
0.6
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300
0.3
0.2
10
14
12
14
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 20: Simulations for graph size d = 50 on 50" voting profiles.
SHD
Adjacency F1
Orientation F1
10
1.0
1.0
STRONG
0.9
0.9
0.7
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20
0.8
MEDIUM
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15
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20
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15
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10
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0.4
No. experts (m)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 21: Simulations for graph size d = 10 on 30"h voting profiles.
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SHD
Adjacency F1 f
Orientation F1
25
10
20
STRONG
0.9
15
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10-
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0.6
50-
0.8
MEDIUM
40
0.8
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20-
10-
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50
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WEAK
0.6
0.5
20-
8
10
12
6
8
10
12
24
8
14
experts
experts (n)
experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 22: Simulations for graph size d = 15 on 30" voting profiles.
SHD
Adjacency F1 1
Orientation F1
1.0
1.0
40
STRONG
30
0.8
20
20-
0.9
MEDIUM
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60-
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40-
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60.
0.5
40
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30
12
14
30
14
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 23: Simulations for graph size d = 20 on 30"h voting profiles.
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SHD
Adjacency F1 T
Orientation F1
1.0
1.0
60
STRONG
0.8
20
150
0.9
0.8
MEDIUM
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100
75
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25
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75-
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10
12
12
14
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 24: Simulations for graph size d = 25 on 30" voting profiles.
SHD
Adjacency F1 1
Orientation F1
1.0
1.0
100
STRONG
0.9
60
40
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0-
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MEDIUM
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0.2
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12
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 25: Simulations for graph size d = 30 on 30"h voting profiles.
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Adjacency F1 T
Orientation F1
1.0
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1.00
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MEDIUM
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150
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No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 26: Simulations for graph size d = 35 on 30" voting profiles.
SHD
Adjacency F1 1
Orientation F1
200
1.0
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STRONG
150
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0.-
100
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12
No. experts (n)
No. experts (n)
No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 27: Simulations for graph size d = 40 on 30"h voting profiles.
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SHD
Adjacency F1 T
Orientation F1
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No. experts (n)
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No. experts (n)
Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 28: Simulations for graph size d = 45 on 30" voting profiles.
SHD
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Orientation F1
300
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Bayes Est. (Ours)
Bayes True
Best Expert
Plurality
Figure 29: Simulations for graph size d = 50 on 30"h voting profiles.
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SHD
Adjacency F1
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STRONG
0.9
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MEDIUM
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No. experts (n)
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Bayes (Ours)
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Expert
Plurality
Figure 30: Simulations for graph size d = 10 on 10" voting profiles.
SHD
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Bayes True
Best Expert
Plurality
Figure 31: Simulations for graph size d 15 on 10" voting profiles.
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SHD 4
Adjacency F1 T
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Bayes True
Best Expert
Plurality
Figure 32: Simulations for graph size d = 20 on 10" voting profiles.
SHD
Adjacency F1 1
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Figure 33: Simulations for graph size d = 25 on 10"h voting profiles.
44

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Published as a conference paper at ICLR 2026
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Figure 34: Simulations for graph size d = 30 on 10" voting profiles.
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Figure 35: Simulations for graph size d = 35 on 10"h voting profiles.
45

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Published as a conference paper at ICLR 2026
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Figure 36: Simulations for graph size d = 40 on 10" voting profiles.
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Figure 37: Simulations for graph size d = 45 on 10"h voting profiles.
46

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Published as a conference paper at ICLR 2026
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Figure 38: Simulations for graph size d = 50 on 10" voting profiles.
47


Reviews:
[{"review_id": "EduKcBmyYE", "reviewer": "Reviewer_239W", "summary": "This paper proposes an ensemble framework for causal discovery task. More specifically, the authors analyze the conditions under which the predicted graph matches the true graph. They also provide theoretical justification for different size, competency and diversity levels of causal discovery experts. Finally, they show performance on synthetic and real-world datasets.", "questions": "Below I share my questions. \n\n## **Question**\n* What does convergence mean in the “converging to the correct decision”?  \n* Line 346: at least $2m - 1$ experts are needed. For a three-node directed graph having three features {(v1, v2), (v2, v3), (v1, v2)}, what will be the value of *m* and how many experts do we need?  Is it number of experts always feasible in real-world setups?\n* If $\\mathbf{G}_1$ is the true graph, does the proposed algorithm work with a vector of probabilities $T_i[:, G_1]$ instead of the matrix?  \n* How large is the competence transition matrix $T_i$? Is it of size $|\\mathcal{G}| \\times |\\mathcal{G}|$? Is it very large and hard to learn?  \n* In Condition 1, the authors consider that two rows are distinct element-wise. To my understanding, each index of $T_i$ represents $T_i := [{P}(\\tilde{G}_i = G_j |  G = G_k,  D)]$. A row represents the probability distribution of different true graphs while the prediction is fixed. On the other hand, a column represents the distribution of different predictions while the truth is fixed. \nNow, when the true graph is different, if the algorithm provides different probability distributions over the predicted graphs, that should indicate the algorithm’s capability to distinguish between two graphs, right? Why then do the authors use rows in condition 1 and not columns of the matrix?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "The proposed method offers a strong contribution as it provides practitioners with a guide on how to configure ensemble strategies while combining outputs of multiple causal discovery experts for optimal performance. It’s interesting how the authors avoid searching the exponential space by decomposing the graphs into sub-structures.", "weaknesses": "Below I provide my comments:\n\n## **Major**\n* The paper is little convoluted. The steps of the proposed algorithm can be organized in a better way.   Section 3.1 is notation-heavy, with lots of notations, making it quite hard to keep track of all notations.  \n* A simulation of the voting mechanism is needed, i.e., how $\\hat{Y}(w)$ is constructed and what its relation is with $\\hat{G}(\\sigma, \\mathcal{S})$, how the transition matrix looks like and how it contributes.\n* Some of the discussion from Appendix D can be transferred to the main paper (Section 2) to provide readers with a better understanding of the concepts. Many concepts in the main paper cannot be connected without proper preliminaries or background.  \n* How parameter estimation with optimal transport works is not clear. The authors should focus more on explaining the main algorithm intuitively.\n* The authors did not compare with other existing distance-based or voting-based structural aggregation algorithms. For example, the authors cited papers such as Mio et al. (2025), Guo et al. (2021), (Tang et al., 2019), Aslani & Mohebbi, 2023. They should compare their performance with these ensemble frameworks.\n* In figure 2, are the causal experts (black) executed once to record their performance? In such case, the proposed ensemble framework should perform better always. Would that be a fair comparison? I would request the authors to point me if I misunderstood this.\n\n---\n\n## **Minor**\n* There are too many versions of the graph: $G$, $\\mathbf{G}$, $\\mathcal{G}$, $\\tilde{G}$, $\\tilde{G}_i$, $\\hat{G}$.  \n* Some visualization or example of $T$, $\\theta$ would be nice.  \n* Line 140: shouldn’t the incorrect prediction be a sum over all $q_{i, G_j | G_1}$?  \n* It is not clear intuitively what the bias term indicates in Equation 3.  \n* Lines 200–201: should it be an equal sign?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jL9EVzhfZo", "reviewer": "Reviewer_kwwE", "summary": "This work aims to address three fundamental limitations of ensemble-based causal discovery. They claim existing methods are largely heuristic, lack theoretical guarantees and guidance on how ensemble design choices affect performance. The paper proposes a principled voting-based framework for structural ensembling, establishing conditions under which the aggregated structure recovers the true causal graph. They concluded that their analysis yielded a theoretically justified weighted voting mechanism that informs optimal choices regarding the number, competency, and diversity of causal discovery experts in the ensemble. Their model was tested on both synthetic and real world data to verify effectiveness and robustness.", "questions": "Q1: When learning the structure from noisy experts or base learners, how is the competence of an expert measured, before creating the competence transition matrix?\n\nQ2: Different experts/base methods use different parameters to identify causal links, how did you reconcile and consolidate these parameters? \n\nQ3: The proposed solution using the Optimal Transport framework is not clear.\n\nQ4: Can this proposed approach identify nonlinear causal links, i.e often found in dynamic systems like climate systems with feedback loops?\n\nQ5: Did you encounter situations were weighted linear voting rule say false positives, i.e the experts voted the presence of an edge in a graph when there exist no edge in the true graph? How did you reconcile such situations?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "S1: Extends the novel estimation algorithm based on optimal transport in a clearly non-trivial way.\n\nS2: Solid mathematical formulation with proofs or guarantees, with clear connection between intuition and formal results.\n\nS3: Comprehensive empirical validation with experiments on diverse, well-chosen benchmarks", "weaknesses": "W1: The solution approach is hard to follow with very limited intuitive explanation of complex maths used.\n\nW2: It is not clear how the paper addressed the well-defined problems with existing ensemble methods.\n\nW3: Authors did not share the code therefore reproducibility is questionable.\n\nW4: The problem the paper is trying to solve (existing methods are largely heuristic, lack theoretical guarantees) is not clearly motivated and lacks practical importance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lqfBzpiJT4", "reviewer": "Reviewer_94Bz", "summary": "The authors study ensemble-based causal discovery, where the results of several causal discovery algorithms are aggregated into a single final graph. In particular, they divide the graph into different \"features\", such as possible edges, and perform feature-wise aggregation. Following a typical aggregation approach, they assume there are $n$ independent experts, and seek to approximate the Bayes optimal voting rule by estimating the transition probabilities of each expert on each feature and the prior over each feature. To estimate these quantities, they assume iid samples from each expert, and use minimum distance estimation.\n\nThe authors provide two sets of experimental results. In the first set of results, they use a simulation where, instead of using causal discovery algorithms, they specify each expert by a deterministic (but unknown) transition matrix from the true graph to its predicted graph, and show that their procedure outperforms both the best expert and the plurality voting rule. In the second set of results, they consider actual causal discovery algorithms such as ICA-LiNGAM, GES-BIC, PC, SCORE, NOTEARS, and DAGMA, on 9 benchmarks from the bnlearn repository, with the proposed method achieving the best results on the SHD metric and competitive results on Adjacency F1 and Orientation F1 metrics.", "questions": "1. Is the SID plot in Figure 1 an error?\n2. Since you aggregate at the feature level, there could be issues with combining the aggregated features into a single graph. For example, with plurality voting, if expert 1 gives the graph (1->2, 2->3), expert 2 gives the graph (2->3,3->1), and expert 3 gives the graph (3->1,1->2), then the aggregated features would be 1->2, 2->3, and 3->1, so combining these features gives a cyclic (rather than acyclic) graph. How do you handle such issues? If you don't, what is the justification?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "**Originality:** To the best of my knowledge, the use of more sophisticated aggregation techniques for causal discovery is relatively novel.\n\n**Quality:** The paper is of good quality, with a good balance of theoretical motivation and experimental investigation.\n\n**Clarity:** The motivation for going beyond plurality voting and other simple aggregation techniques is well-articulated.\n\n**Significance:** The results are promising and may help push causal discovery towards more practical reliability.", "weaknesses": "**Theorem presentation/novelty:** Theorem 1 in this paper seems to be an extension of Theorem 1 in [1] to multi-class classification and unbalanced truths. Overall, I think the nature of both Theorems 1 and 2 are obscured by an early overcommitment to the causal discovery setting: the nature of the results is really about general aggregation of experts with categorical outputs. I'm not an expert in the area, but aggregation is a very well-studied subject in statistical learning theory, and the paper would benefit from contextualizing itself within this field before turning the attention to causal discovery. A better understanding of aggregation would also address some issues with the theoretical assumptions, discussed in the next point.\n\n**Theorem assumptions:** A key assumption (Assumption 1) is that the experts make independent decisions (conditioned on the truth). However, in practice, each of the causal discovery experts would be using the same dataset (or, you would have to reduce the sample size and divide a dataset of size $M$ into $n$ datasets of size $M/n$), i.e., their decisions are only independent when conditioning on both the ground truth *and* the dataset. Further, to estimate the expert transition matrices, the theory requires another $N$ independent voting profiles from each expert. Overall, these assumptions are far different from how any aggregation method would be used in practice, and seem to be too strongly based on intuitions from crowdsourcing/voting theory. At the very least, I think it is essential that the paper emphasizes these limitations and better describes the mismatch between theory and practice (e.g., the experiments use bootstrapping to get different graph samples for each expert, which would not be independent). Ideally, the authors would be able to use more sophisticated analysis to handle these issues, but in reality I think it's fair to leave that direction for future work.\n\n[1] Berend et al. (2015), A Finite Sample Analysis of the Naive Bayes Classifier", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XTwJBfe0Me", "reviewer": "Reviewer_mcKH", "summary": "This paper proposes a principled ensemble framework for causal discovery by treating multiple causal structure learning algorithms (or multiple runs of one algorithm) as noisy experts. It introduces a voting-theoretic model based on Bayes-optimal weighted voting to aggregate DAG structures. Theoretical analysis  of the proposed method (including the version under estimated parameters) are provided.", "questions": "- How is $\\mathbb{P}(G | \\mathcal{D})$ or $\\mathbb{P}(\\widetilde{G} | G,\\mathcal{D})$ defined? What is the mariginal distribution $\\mathbb{P}(G)$ and the data generating process $\\mathbb{P}(\\mathcal{D} | G)$? Why is $\\mathbb{P}(G | \\mathcal{D})$ called \"prior\" rather than \"posterior\"?\n- Why does switching the focus to \"feature\" level from graph level make the computation for feasible? The \"feature\"s are essentially sub-structure in the graph, e.g. edges. How to compare the size of two levels quantatively?\n- In proposition 1, it seems we do not need $\\widetilde{\\pi}$ to be close to $\\pi$?\n- In Theorem 2, what is the intuition that we need at least $2m-1$ experts? More experts leads to more parameters to estimate. Does the estimation/identification of $\\pi$ come from number of experts?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This paper provides first voting-theoretic error bounds for structure ensembling in causal discovery, particularly useful as many algorithm candidates are available.\n- The error bounds offer insights into how ensemble size, diversity, and competency for algorithm design affect the performance of the ensembling.\n- The experiments combine synthetic and real datasets, showing consistent performance and wide algorithm coverage.", "weaknesses": "- the motivation of this work (or ensembling) is not about the structure learning is bad (low $\\mathbb{P}(\\widetilde{G} | G,\\mathcal{D})$) and we want to aggregate them. But the focus is on the fact that causal dscovery methods rely on very different identifiability assumptions, thus different data generating process $\\mathbb{P}(\\mathcal{D} | G)$. So the \"competency\" is often zero once the identifiability assumptions are violated. How do we evaluate them under the same probability measure? and how does this change the problem setup and estimation of $\\theta$'s?\n- It is helpful to include a runing examples when introducing \"feature\" level to illustrate how thinking in \"feature\" level instead of graph level helps reduce the complexity of the problem.\n- See questions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)