Causal Mixture Models: Characterization and Discovery
Authors: Sarah Mameche
Janis Kalofolias
Jilles Vreeken
Venue: NeurIPS 2025 poster ()
Keywords: causal discovery
mixture modelling
Primary area: general_machine_learning
TL;DR: Given a mixture of samples from unobserved subpopulations with distinct underlying causal mechanisms, we give results on identification and discovery of causal graph with latent mixing variables.

Abstract: Real-world datasets are often a combination of unobserved subpopulations that follow distinct causal generating processes. In an observational study, for example, participants may fall into unknown groups that either (a) respond effectively to a drug, or (b) show no response due to drug resistance. Not accounting for such heterogeneity then risks biased estimates of  drug effectiveness.  
 In this work, we formulate this setting through a causal mixture model,
 in which the data-generating process of each variable depends on latent group membership (a or b). Specifically, we model each variable as a   mixture of structural causal equation models, where latent categorical (mixing) variables index assignment to subpopulations. Unlike prior work, the approach allows  for multiple independent mixing variables, each affecting distinct sets of observed variables. To infer both the graph, mixing variables, and assignments jointly, we integrate mixture modeling into score-based causal discovery; show theoretically that the resulting scoring criterion is consistent; and demonstrate  empirically that the resulting causal discovery approach discovers the causal model in synthetic and real-world evaluations.

Causal Mixture Models:
Characterization and Discovery
Sarah Mameche CISPA Helmholtz Center for Information Security
Janis Kalofolias CISPA Helmholtz Center for Information Security janis, www.oolialscisspeeeii.eleesll
Jilles Vreeken CISPA Helmholtz Center for Information Security jv@cispa.de
sarah. mamechh@@cispa.ee
Abstract
Real-world datasets are often a combination of unobserved subpopulations that follow distinct causal generating processes. In an observational study, for example, participants may fall into unknown groups that either (a) respond effectively to drug, (b) show response due drug resistance. Not accounting for such heterogeneity then risks biased estimates of drug effectiveness. In this work, we formulate this setting through a causal mixture model, in which the data-gene process of each variable depends on latent group membership (a or b). Specifically, we model each variable as a mixture of structural causal equation models, where latent categorical (mixing) variables index assignment to subpopulations. Unlike prior work, the approach allows for multiple independent mixing variables, each affecting distinct sets of observed variables. To infer both the graph, mixing variables, and assignments jointly, we integrate mixture modeling into score-based causal discovery; show theoretically that the resulting scoring criterion is consistent; and demonstrate empirically that the resulting causal discovery approach discovers the causal model in synthetic and real-world evaluations.
Introduction
A central part of scientific investigation is understanding cause-effect relationships, which the field of causal discovery [Pearl, 2009] aims to discover directly from observational data. Many existing causal discovery approaches, however, rely on idealized assumptions, among others assuming that no relevant unmeasured variables exist and that all samples come from a homogeneous distribution. Real-world applications might violate both assumptions, for example, when observations come from heterogeneous populations or environments.
Take, for example, nationwide study of antimicrobial resistance in hospitalized patients, focusing on a resistant pathogen such as Methicillin- Rethicillin-Resistant Staphylococcus Aureus (MRSA) [Hasanpour et al., 2023). As patients come from different regions, their individual medical histories differ, including prior exposure to pathogens such as Enterococcus [Li et al., 2022] with known cross-resistance to MRSA. The regional plasmid profiles of Enterococcus largely determine its susceptibility, say to Vancomycin [Boumasmoud et al., 2022). in turn influencing MRSA cross-resistance [Arredondo- Alonso et al., 2020). Although well documented, this variable is not routinely measured, and is hence a latent variable that defines the mechanism under which the presence of Enteroccocus affects MRSA cross-resistance [Cong al., 2019]. Consequently, observations across regions effectively arise from a mixture of distinct causal mechanisms.
More generally, observational data can be combination of multiple subgroups with distinct gen- crating processes. To illustrate, consider a simplified, synthetic setting in Fig. [ where the causal mechanism for Y is mixture of two functional relationships X + Y. Treating all samples one population can then cause artifacts during causal discovery, such as spurious relationships or reversed
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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MIXING (A: 1)
MIXING (A 2)
Grop Group
as good as random s-split better than a random A-split
0.8
AMI(Y)
0.4
0.2
MM
CMMM
MM
CMM
Figure 1: Causal Mixture Models. Left: Example of a mixed causal relationship X + Y. Right: Recovering the class assignments with (Gaussian) mixture models (MM) compared to causal mixture models (CMM) on simulated bivariate datasets.
causation [Huang et al., 2020J. While a line of research known as multi-context and interventional causal discovery [Mooij et al., 2016 Huang et al., 2020 Squires et al., 2222) addresses similar setting, these require access to multiple datasets where all causal mechanisms remain fixed, which unknown in the motivating example.
Recent works attempt to separate a single given dataset into multiple interventional datasets or environments; for instance, Kumar et al., (10.22224combbl combine Gaussian mixture modeling (GMM) with interventional discovery (UT-IGSP), addressing. mixtures arising from atomic interventions (Kumar et al., 2024). In examples such as Fig. D the underlying mixtures are better captured through conditional relationships, such as in mixture-of-rreressin
models (000n, To illustrate the difference between unconditional mixture model (MM) and the conditional (causal) counterpart (CMM), we draw synthetic datasets over X, Y and evaluate the discovered class assignments for Y via the Adjusted Mutual Information (AMI). While in the case without mixing (k: group), both methods correctly reject the existence of groups, with mixing (k 2 groups), the MM reports no better than random assignments, whereas the CMM improves considering the causal mechanism for Y as conditional mixture subject to a latent variable Z.
the baseline. This suggests
When extending the scope from the bivariate case to multiple observed variables, our interest then also extends from a single to multiple, independent latent variables. Returning to motivating example, for instance, suppose each medical center chooses different collaborating laboratory for antibiotic testing. This introduces batch effect, a latent factor independent of regional susceptibility. To address this, different from previous formalizations that assume single mixing variable affecting all observables [Mazaheri et al., 2023 Kumar et al., 2024]] we allow multiple independent mixing variables that each affect different subsets of observed variables (ef. Fig. 2 For each variable, we model its generating mechanism as a conditional mixture given its direct causes and its corresponding latent factor, referred to Causal Mixture Model (CMM). The central question in the remainder of this work is how to infer the structure of such models from data.
Causal Mixture Models To summarize, we propose basing mixture modeling on functional relationships within a causal graph, where we consider the causal mechanism for each variable as a mixture of conditional relationships given causes and an associated latent variable. Unlike two-stage approaches that separate cluster and graph discovery, our approach integrates mixture inference directly into causal discovery by extending local score- based criteria, where we focus on linear
(MLR) models inferred through Expectation Maximization (EM) and extend the BIC score [Chickering, 2002). We show that under oracle access to the MLR parameters, can guarantee the identification of the causal model under mild assumptions, and hypothesize that this also holds for their EM estimates in practice. We propose integrating this approach into score-based causal discovery algorithms such as Greedy Equivalence Search (GES) [Chickering, 2002] or TOPIC [Xu et al., 2025). To demonstrate empirically we can recover the CMM components in practice, we consider simulated mixed data, a mixture of interventions Kumar al., 2024). as well as a real-world benchmark on causal cell signaling pathways [Sachs et al., 2005
We include all theoretical justifications and experimental details in the supplement.

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2
Causal Model
Given set of continuous random variables X (X1....,
Xn). we interested not only in causal relationships among them, but especially in unknown changes of their causal mechanisms. Assuming that the mechanisms are linear functiins of fixed subset of the observed random variables, we allow their coefficients to be chosen from a finite set of vectors conditional on an external latent variable.
To formalise this, we also consider a set of discrete, unobserved random variables Z = -Z,..... 2m J. with m 5 n, each following a categorical distribution Z; ~ Categorical(?") with Z, E fl...., K(). That is, each 7 lies on K--dimensional probability simplex Y = (?i.....7kk with Ek= 7k 1, so that PZZ k) % We call these random variables mixing variables.
The causal mechanism of each observed random variable X depends, besides on the set of observed causal parents denoted Paj, on exactly one of the unobserved, latent Z, as determined by the surjective map La : X + Z X, + Zi, which case we simply write Lay Z, The mixing variable directly affects the parameters of the causal mechanism, which we therefore express as a function by [K,] + RIPa,
+ B)= mapping each value z of Z; to a linear coefficient vector Azz E IPall Hence, the parameters of the functional dependency f is the collection of vectors B, (B)1...., 3rK,); )) this consists of one coefficient vector 35 for each mixing coefficient 1 S k S K; and each such vector has dimension equal to the number of parents Pa, of Xj-
Summarizing, we can now model each random variable X, as generated from its observed causes Paj S X by the causal function f and the coefficients by that depend on the latent Zi Laj E Z, where we recall that Z X- 0. Then, have
X,- ((Pay + Nr
with
33, + $
(1)
where N, 1L Pay is additive Gaussian noise, N2 (N(
This construction implies that for a random variable Xy for which Lay = Zz we get for k E {1, (1...., Ki} and
(X, PPa y, Lay k) N (B
(2)
(X, y)
MLR (B),
where MLR (B, 7,57) 0²) is the conditional distribution of a mixture of linear regressions with density
K
113,TTy y x=1133)
MLB (x,
(x
The
exp
(4)
2
20
and xx (z; μ, 0²) is the density of the normal distribution with mean and variance o². Note that in case Pay = 6. the formulation reduces to a standard unconditional GMM.
Graphical Model We represent the above causal model as Directed Acyelic Graph (DAG) G = (X Z, F. over both observed X and latent Z random variables, between which we add edges X, - X; whenever X, a cause of X,, as well as Z, - X, whenever La, = Z,; since the Z were assumed exogenous and independent, we allow incoming edges toward the Z themselves. Of special interest is the subgraph G (X, E) that is induced on G by the node subset X. with E = [X, + X2 X, + X₁ E } Considering only the latter subgraph, we denote the set of all observed direct predecessors of X, in G by Pay C X {X We also consider a surjective map т X + N which induces a topologiaal ordering on & (and thus partial ordering over X) that we call a causal order under G. and assigns such values on X that X, Pre; direct and indirect predecessors of X, in G. Fig. 2 depicts example of such graph G (colored and black) and the resp. induced subgraph G (black). We then model edges such as Z, + +X, where X, is a source node of the causal graph a GMM and edges such as Z1 X2 through an MLR.
Z1
Zn
X1
Example Causal Mixture Model. The observed variables are affected by two latent mixing variables Zı, 72
Figure 2:
T(j) (r), for Pre, C -X (X,) the set of all
We assume the causal Markov Property to hold in the full causal model G², which results in the following factorization over the observed variables.
3

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Assumption (Mixture-Marken Property). The distribution of X has (marginal) density
(x
II
MLR PX|Pa, (z. Pa,aaBBBy..
(5)
X,EX
where we implicitly use the independence of all Z (see Appendix A.3 on relaxing this assumption).
The objective in this work is to infer the CMM from finite observations, as follows.
Problem 1. Gäven ii.d. observations D == (x1.-..,X,, Xx),, of the random variables X generated from a distribution compatible with our assumptions, we aim to infer the structure of G
1. the causal dependencies X; + Xi E E encoded in the edges E' of the induced graph G,
2. the linear coefficients of the corresponding causal mechanisms 7y' for all X; E X,
3. the set of latent variables, i.e, their number m, and each Ze Z the domain K; and mixing probabilities -y', as well as
4. the mapping Lay= E Z for each X, X.
We address this problem in the following section.
3 Theory
As the first contribution of our work, we study conditions that allow inference on G as formalised in Problem [ using score-based causal discovery. To this end, it is of interest determine whether existing local scoring criteria can be adapted for our setting, so as to use them consistently in a score- based framework, such the well-known Greedy Equivalent Search (GES) algorithm [Chickering. 2002] and related approaches [Xu et al., 2025).
GES performs greedy search over the set of all Markov equivalent classes (MECs) of causal graphs with nodes X. More specifically, each iteration updates the current MEC by the best hypothesis among all those that differ from the current one by a single edge modification (i.e., edge reversal, addition, or removal). This process allows for the inference of the most likely MEC, as expressed in the form of the edges of Partially Directed Acyclic Graph (PDAG) with nodes representing the observed X.
Importantly, GES can be shown to be asymptotically consistent long the search is guided by scoring criterion that satisfies the appropriate criteria given shortly. We write Lx E Ig for graph with nodes X whenever the (conditional) independencies implied by the PDAG G are also true for the distribution Lx.
Definition 3.1 (Consistent Scoring Criterion [Chickering, 2002 ). Given data D of size r sampled from distribution Lx and two graph hypotheses 9 9 then the score S is consistent whenever
1. Lx E Tg and Lx Igz S(G):D) S(G:; D)
2. Cx E Igp O Igg and G has fewer parameters than Gh it S(C); D) S(C:; D).
One score that satisfies this criterion is the Bayes Information Criterion (BIC). Definition 3.2 (Bayes Information Criterion). Given model hypothesis assuming distribution Lx (0) with parameters 0 E θ C Rd and observations D [xI..... x,} of X, the score of 7 is
BIC(H) -2 log Px ( +dd dlggr,
(6)
where the first term is the scaled (log) likelihood of Lx evaluated at its maximiser θ.
This score has the important property of decomposability [Chickering, 2002). dictating that when f factorises product the BIC decomposes as sum
(x) II Px,IPa, (3,(pa,) X,EX
BIC(X) Σ BIC(XX X,EX
To transfer these properties to our own model, we need to further account for the latent mixing variables. Hence, each step the GES algorithm, and according to our model of Eq. (5) we need to compute the likelihood PX (z. pay; ; B,y, o2), which requires an estimation of the Mixture of

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Z'
Z"
X
X
Y
X;
X₂ = X
X X (A3)
X, X (Xi)
Figure 3: Identifiable Cases. Under mild assumptions, when Z has least two mixing components, all of the shown DAGs are identifiable.
Linear Regressions (MLR) parameters. Unfortunately, the computation of the Maximum Likelihood Estimate (MLE) estimates for both Gaussian Mixture Models (GMMs) and MLR models is NP-hard in the general case. However, an efficient algorithm to compute well-performing local maximisers of the corresponding likelihoods the Expectation Maximisation (EM) framework [Wu, 1983) and its variants [Bishop, 2006]; we therefore employ these algorithms to approximate the MLE parameters.
Although certain conservative claims can be made for the consistency of EM, we postpone them for later on. Instead, for the first part of our analysis we assume the availability of an oracle that can compute the maximum likelihood estimates for the MLR model for a given number of mixtures K,
i (B.
MLR arg max PX (y.y; (B.r,))BRk
(8)
where Øх (IR K SK-1, R4 is the space of the model parameters for K mixtures and SK the K -Kddimensional probability simplex. Given the availability of such an oracle, we can adapt the GES algorithm way that both preserves its asymptotic consistency and takes into account any possible latent variables. For this, we also incorporate the asymptotically consistent estimation of the number of components K by maximising the BIC score over of allowed values K S K Kmax- This gives rise to the latent-aware BIC scores
BIC 2 (H) = max ISKSK
BIC BIC 2 (H, K)
and BIG (H) 2
max 1SK K.K.K Kaax
BIC (H, K), (9)
where BIC 2 uses the oracle estimate and BIC 2 the EM one. By appropriate use of the decompos-
ability property,
show that the criterion of Def. 3.1 holds for the former score, BIC
Note that known result shows that linear additive Gaussian noise models are identifiable [Shimizu et al., 2006, Pearl, 2009). in the sense that the graphs X + Y and X + Y are then Markov equivalent. In our general case, however, and under mild conditions (see Lemma B. 1, our model deviates from this problematic case enough for the identifiability of all models in Fig. 3 to become possible.
2 Theorem 3.3 (Local Consistency of BICMI, ) Let D = {xI....., x, be observations of random
variables X, Y, such that X|Y MLR o²), with general parameters 0 (see Lemma B. Then, out of the structural hypotheses depicted in Fig. the BTC score of the ground truth hypothesis 9h is asymptotically larger than any of the alternative ones, Gh and Ghe almost surely.
As corollary, GES with the BIC can identify between these models.
Corollary 3.4. The latent-aware score BICML a consistent scoring criterion.
In our analysis, however, we have yet to address the discrepancy between the BIC that we assumed to be tractable, and the one actually above, the BIC based the EM algorithm. Of interest one particular result of Balakrishnan et al. (0117), which shows that for a Euclidean ball around the optimal parameters of the MLR model, the EM algorithm finds the MLE estimate if it is initialised within this ball. The radius of this ball depends on how well the components of each mixture can be separated, and holds both asymptotically and for the finite sample case.
At the same time, we intuit that when the observations do not fit well with an MLR model, the algorithm will fail to find the correct estimate, resulting in an underestimation, BIC & BICML This assumption would steer the GES algorithm away from selecting the said model, which would instead favor the selection of correct model. We formalise this intuition follows.
5

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Algorithm 1: DISCOVER A CAUSAL MIXTURE MODEL (CMM)
Input: Dataset X, max. number of mixture components maax
Output: Set of latent variables Z, causal graph g
Initialize Z <- 0, G 0, g 6, T [];
// Discover local mixing and graph
while not all nodes are ordered do
X; + INFERSOURCE(T: 9);
G EDGEADDITIONS 9)
G < EDGEPRUNING(XX, 9);
for each k in 1..... K max do
Using EM, fit (X, Paj y) MLR (B,, (,))),,²2) with & components;
Zj <- mixing assignment with best BIC 2 maxigKsKma. BIC T.APP
(K): (K):
// Infer global mixing 10 9 INFERMIXING(S, {22});
I1 return G, Z;
Conjecture 3.5. The latent-aware score BIC is a consistent scoring criterion.
The analysis justifies the use of the latent-aware score BIC in score-based algorithms for consis- % tent structure estimation. Inspired by this, we complement our theoretical results with practical implementation using the BIC which we present and evaluate in the remainder of this work.
4 Algorithm
Here, outline algorithm for discovering i) the mixing variables Z and ii) the causal graph G². Building on the score consistency that our theory establishes, we propose a joint inference procedure that discovers both components within score-based framework. At each scoring step, we fit an MLR under a given candidate parent set. Choices for the score-based algorithm include, for example, the GES algorithm [Chickering, 2002] other score- based causal discovery frameworks. As main proof-of-ore proof-of-concept implementation, we describe how to integrate the latent-aware BIC within the topological order- framework TOPIC [Xu et al., 2025
Causal Mixture Inference and Scoring Given a node Y with parents pa(Y) C X, we score the causal relationship using the latent-aware BIC of Eq. (9). That is, the score accounts for an unknown number of mixtures, where the likelihood takes the form of Eq. 4 To compute the BIC in practice, we need to consider each < < Kmax up to a given hyperparameter K Kmagand and use the Expectation Maximisation (EM) [Bishop, 2006 algorithm
obtain estimates for the MLR parameters; finally, we then use the BIC score of the best such in our algorithm and the corresponding estimates for the rest of the model parameters. In the special case of source nodes, we infer GMM similarly using EMM Regarding the number of components kt, we assume that a reasonably high maximum number of Kmax is given that caps the number of mixture components. When MLR model selects K < Kmax. is (asymptotically) sure that this estimate of is the true number of mixture coefficients. When 2 Kmax < K*, for K" the true number of clusters, the causal structure would still be correctly inferred, as the capped MLR would still offer higher likelihood versus any alternative model.
Causal and Mixing Structure Search We discover the causal DAG & over the observed X by embedding the above inference step into the TOPIC algorithm. Considering candidate edge X -YY,, we also define the score difference g as the difference in BICs before and after adding X, O( G) BICz (Y pa(Y') U (X)) BY pa(Y)). TOPIC then constructs the DAG in topological order the following steps.
(i) Source Identification: Identify X3 = arg max miny [x(X, Y, G) gY,
(X; i Edge Pruning: Remove any redundant incoming edges Z X, using
+ Edge Additions: For each Y * X,, add an outgoing Y

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Following this strategy, we discover topological order T. DAG G as well as mixture assignment Z; for each variable X, along the topological order. We finally consolidate these. With the reasoning that pairwise estimated 7;, 7- exhibit significant overlap when they correspond to the same mixing variable, we measure the similarity of the assignments, here using Adjusted Mutual Information (AMI), and merge variables when the AMI exceeds its expected value [Vinh et al., 2010].
We summarize our approach in Alg.
5
Related Work
Discovering causal graphs from observational data is well-studied problem. Among constraint-baser methods is the PC algorithm [Spirtes et al., [200]] using conditional independence (CI) testing, among
score-based approaches is GES [Chickering, 2002 using local scoring criteria [Huang et al., 2018 Both search over the space of Markov Equivalence classes (MECs) over DAGs. LINGAM [Shimizu et al., (2006) and RESTT [Hoyer et al., 2008) assume onnGGaussianity and independence of regression residuals to orient edges. CAM [Bühlmann et al., 2014) and SCORE [Rolland et al., /022]/DAS
[Montagna et al., 2023] combine topological order estimators with pruning discover DAG, while TOPIC [Xu et al., 2025 uses Bayesian or anformation-theorett scores for both steps. NOTEARS [Zheng et al., 2018 frames DAG discovery a continuous optimization problem. Given the potential variance- resp. R² -ortaabbiyyy problems of causal discovery benchmarks [Reisach et al., 2021]. Reisach al., propoos
propose simple baselines exploiting these scoring criteria, VARSORT and R-SORT. Finally, some works relax the common assumption of causal sufficiency and allow for latent confounding variables, such FCI [Spirtes al., [993]. In comparison, with discrete-valued latent variables and special interest in their inference, our setting is more similar to multiple environments.
Multiple Environments and Interventional Data Towards addressing non-i.i.d. data, several works consider different datasets (called environments, contexts, or interventional datasets) that arise from interventions or causal mechanism shifts, but within which all causal mechanisms are fixed [Schölkopf et al., 202T]. Extensions of common causal discovery algorithms to this setting exist [Mooij et al., 2016; Huang et al., 2020) Squires et al., 2020 Mameche et al., 2023). Common to these methods is that they consider causal discovery in an augmented graphical model with a single latent variable that can be interpreted as the (known) dataset index, respectively, with intervention variables, allowing identifiability of the causal model up to the interventional MEC (iMEC) [Hauser and Bühlmann, 2013 Wang et al., 2017a) A different line of work, addressing causal discovery in time series [Runge, 2020). considers discovering temporal regimes with changes in causal mechanism, such as RPCMCI [Saggioro et al., 2020). - This approach differs from ours in that it discovers changes in causal mechanisms along the temporal dimension, assuming a temporal ordering of observations.
Mixture Modeling and Causal Discovery Gaussian Mixture Models as well as their conditional counterparts, Mixtures of Linear Gaussian Regressions, are well studied [McLachlan and Peel, 20000) Hennig, 2000). The idea of reinterpreting clustering and mixture modeling within a causal framework is not new; most works in this area assume global mixing variable, also termed latent-class confounder [Mazaheri et al., 2023). Some works study for example the bivariate cases [Hu et al., 2018). discrete variables [Mazaheri et al.. 2023) and causal inference settings [Kim et al., 2024 Mazaheri et al., 2025). Most closely related to ours is the work of Kumar et al., 2024 which proposes Mixture-UT-IGSP (MIXUTIGSP), combining a GMM with adequate component selection with UT-IGSP. Kumar et al., 2024 give insight into the sample complexity of identifying interventional mixtures, as well as establish identifiability of the iMEC in this setting using the two-stage approach. We will evaluate against this approach in the next section.
Causal Identifiability Methods exist to identify linear additive Gaussian noise models with equal variance, under further conditions, such as non-perfect dependencies, and what is more, also when other constraints are added (e.g., that the coefficients are all less than the unit) [Peters and Bühlmann, 2013). Our identification results do not rely on these specific assumptions; instead, we base identifi- cation on the effects of the latent mixing variables and the asymmetries they introduce in the causal directions. The results are thus independent of the approach of restricting the functional model with further assumptions, and can address scenarios where these assumptions do not hold.
7

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Mix. NODES
MIX. EDGE DENSITY
MIX. PRACTION
MIX. COMPONENTS
OBS. NODES
0.8
CMIMGS
Jace
0.6
CMM(C)
0.2
Pi (turgess)
CMME
0.6
CMM:
GIM
0.2
KMEANS
DBSCAN
0.8
HDBSCAN
AMI
SPECTRAL
0.4
0.2
00 0
3
N2
P2
9g
K
Nx
Figure 4: Discovering Mixing Structure. In synthetically generated CMMs, we evaluate the quality of the recovered mixing structure, evaluating the affected observed variables (F1 (target)), variable sets affected by the same latent variable (Jacc) as well as per-node mixing assignments (AMI).
6 Evaluation
We evaluate our approach on two main questions,
(i) Discovering Mixing Structure:
Given causal graph, can our approach accurately discover the underlying mixing structure, including the numher of mixing variables, the number of their components, assignments, and sets of targeted observed variables?
(ii) Discovering Causal Structure: When the causal graph is unknown, can our approach recover the mixing structure as well as the causal graph?
address the above questions on synthetic and real-world data.
Experimental Setup We generate data according to our assumptions by drawing Erdós Rènyi DAGs using X,[Pa, ~ MTR (B, 7", ,²) with Gaussian additive noise N, ~ ~N(0 1) and ensuring that the linear coefficients B, are bounded away from zero, f, E [-11, 0.255] [0.25, and similarly from one another. To avoid issues related Var-sortability and Re-sortability [Reisach et al., 2021 generate internally standardized structural causal model (iSCM) [Ormaniec et al., 2024
The experiments address the effect of several parameters: the number of observed Nx {5, ...... .0)
and latent variables N2 E (O...., 10), number of latent classes K E (2..... 5}. fraction of observed variables Pz E [0, 1] affected by mixing, dag density Pg E [0, 1], sample size {200, 10000) and parameter controlling the size of the samples in each group S2 E {2..... 10]; for example, for K=2, we uniformly at random draw S.
3z samples belong class O, otherwise assign class 1. By
default, we show results for Nx 10, Nz 4, K 2, 0.5, Pg = 0.4, S = 500, Sz 5. For detailed description of the data generation setup, see Appendix D
6.1 Discovering Mixing Structure
We assess the quality of mixing assignments Z; using the Adjusted Mutual Information (AMI) averaged over the estimated vs. true assignments for each observed node X, in a simulated graph G. and the mixing structure in GD using Fl scores (binary edge existence Z, X3) and Jaccard indices (set overlap of each Z, observed targets). We include simple baselines that apply mixture modeling without causal information for comparison.
In Fig. we observe reliable recovery of mixing assignments (AMI) and their targets (F1, Jacc) for our approach given the true graph (dashed green) and for the full framework (solid green). Applying a clustering baseline such as GMM over all nodes in the graph, as in [Kumar al., 2024 (blue, x), is misspecified in case Nz 1, per-node assignments distinct, hence not surprisingly,

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MIX. NODES
MIX. EDGE DENSITY
SAMPLE SIZE
OBS. NODES
OBs, EDGE DENSITY
0.8
CMM
SD
0.6
CAM
0.4
0.2
LINGAM
VARSOKT
K² SORT
FCI
0.6
GES
0.4
0.2
2
200
400
600
800
1,000
10
N2
Pz
s
Nx
Pg
Figure 5: Discovering Causal Graphs. synthetically generated CMMs, we evaluate the quality of the causal DAG over the observed variables in terms of the separation distance (SD) and true positive rate over directed edges (TPR).
performance worsens as Nz increases (left), Given this, we also apply the models to each node in turn (purple, gray), where gray variants show different instantiation choices. These perform not much better than random splitting in most cases (AMI, F1) and with accurate set recovery (Jacc).
6.2 Discovering Causal Structure
Next, we evaluate the learned causal DAG resp. CPDAG G against the ground truth G*. We consider the separation-bas bbasd distance metrics proposed by Wahl and Runge, 2024 measuring among others the Separation Distance (SD) over graphs. To demonstrate the strength of our approach in discovering edge orientations in particular, we also compute orientation F1 scores and report true positive rates (TPR). Additional metrics of interest postponed to the Appendix. Baselines include a range of causal discovery methods, CAM, SCORE, LINGAM, FCI, GES, PC, VARSORT, and R² SORT.
While some baselines perform reasonably well in mixed settings, CMM maintains high accuracy across both structural and causal direction evaluations (Figure 5). We note that Mixture-UTIGSP is not meaningfully applicable here, as there may be multiple mixing variables and no well-defined "observational" part of the dataset as required as input to UTIGSP; thus we will instead consider an experimental setting with single mixing variable.
Case Study: Mixtures of Interventions Next, we repli- cate the experimental setup Kumar et al., 2024 us-
OBs. NODES
SAMPLES
ing their data generators to evaluate performance in the mixtures-of -niteveeetnteeeniiinnss setting. As Figure shows, the GMM used in Mixture-UTIGE performs well in recov- ering mixture assignments as expected, given that its mod- eling assumptions
0.8
0.6
0.4
0.2
0
met in this setting. Our method performs comparably in mixture assignment recovery and recovers the causal graph more reliably (Figure 6 bottom). We also noticed competitive performance of VARSORT in this dataset suggesting potential Var-sortabity which may inflate the performance of the discovery approaches.
0.8
0.4
0.2
200400600880..om
Nx
S
Case Study: Flow Cytometry Data Finally, we in- vestigate the real-world flow cytometry dataset curated by Sachs et al., [2005] The dataset originates from single- cell pootei-siignaliig samples of the human immune re- sponse system, where each of the 11 observed variables corresponds to the activity one compound of inter- est: either protein or phospholipid. The dataset con- tains (among others) experimental conditions, in each of which particular molecular modifier has been applied
CMM
CAM
LINGAM
GES
MIXUTIGSP
SCORE
VARSORT
FCI
PC
Figure 6: Mixtures of Interventions. For data generated from an interventional mixture, we show the quality of mixture separation (AMI. cf. Fig. and causal DAG discovery (SD, cf. Fig 5

---

MIXED/IV. SAMPLES (AMI)
CAUSAL GRAPH
TARGET
CMM
CMM
MIXUTIGSP
METRIC
CMM
MIXUTIGSP
Akt
0.02
0.04
0.04
TP
PKC
0.26
0.43
0.49
FP
10
PIP2
0.10
0.10
0.03
FN
10
Mek
0.20
0.20
0.14
SD
0.25
0.75
PIP3
0.00
0.00
0.00
S/C
0.27
0.57
Table 1:
Mixing discovery as in Fig. for the data by Sachs et al., 2005
Table 2: Causal discovery as in Fig. 5 for the data by Sachs et al., 2005
to the cells, such that the activity of exactly one of the 11 compounds of interest is affected; this results in known change of the measured activity for the corresponding compound. As previous analyses [Wang et al., (017],, we combine the data from all experimental conditions into larger dataset of size 5846, and do not disclose their origin to the algorithm. Hence, in the pooled data, each variable is affected by latent one.
For example, considering the node "'Akt", there are two mixture components: one is the experimental condition where the so-called Akt inhibitor was applied, directly inhibiting Akt activation; the other mixture component comprises the remaining samples where Akt is in its baseline condition. In particular, the variables Akt, PKC, PIP2. Mek, and PIP3 were manipulated. We test how well, for each given target node, its intervened subsample can be recovered (AMI). The dataset split found by Mixture- Mixture-UTIGSP appears to match PKC best, similar for our per- variable splits which match PIP2 and MEC slightly better (Table TT. Still, neither method reaches convincing AMI, consistent with the observations [Kumar et al., 2024; Squires et al.. (0220) that the interventional targets are difficult to identify in this dataset. In terms of causal discovery, our approach performs better in terms of separation distances (SD, S/C) albeit discovering a number of spurious (FP) edges (Table 2 All other baselines report similarly high false positive directions (10+ FP) with the exception of PC (2 FP). Additionally, some edge directions in the ground truth such as Raf - Mek are subject to debate. We discover the Mek - Raf direction which has been previously discussed as agrecing better with the given data [Mooij et 2016).
In summary, the experiments give empirical support for effectively recovering the mixing structure and the causal structure. In Appendix E we additionally (i) compare the GES to the TOPIC instantiation; (ii) study the effect of latent mixing variables on causal discovery algorithms; and (iii) compare functional forms both in the data generation as well as during scoring.
7 Conclusion
We address the problem that real-world datasets rarely come from fixed data- Bata-generating process, but could be combination of subpopulations with distinct causal mechanisms. This gives rise to a causal mixture model in the sense that each effect given its causes results from a finite mixture of linear regressions. We characterize this model in our work and establish theoretical results showing that consistent scoring criteria, such as the aitent-aware BIC, allow causal identifiability. Complementary to these insights, we propose a practical implementation for causal discovery using this score which we demonstrate to have strong empirical performance various settings.
In the real-world case study [Sachs et al., 2005 while the approaches recover the causal graph structure an extent, we observed limited ability of both the CMM and Mixture-UTIGSP to discover the intervention targets, likely due to the strong assumption of lincarity. As our proposed approach can in principle easily incorporate richer models, future work could study scoring criteria and theoretical guarantees for non-linear mixtures, encouraged by proof-of proof-of-concept experiment that we provide in Appendix[E] Future work could also study how to further interpret the meaning of a given Z, especially when it points to previously unknown states, for example, by exploring explainability approaches such as LIME [Ribeiro et al., 2016) or Shapley values [Lundberg and Lee, 2017). Furthermore, invoking Expectation Maximization at every scoring step comes with drawbacks regarding the scalability of our method, especially when relying on random restarts, therefore subsequent analysis could devise algorithmic approaches to reduce this overhead.
10

---

Acknowledgements
We would like to thank the anonymous reviewers for their helpful feedback.
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A Preliminaries
In this section, we provide brief overview of helpful preliminary concepts that, although relevant to our analysis, we assume to be widely familiar in the main target audience of our work; for brevity and clarity, we therefore chose to postpone them away from the main manuscript and into this appendix.
A.1 Structural Causal Models
Given random variables X = (X1.... X, we can encode the underlying data generating process (DGP) as a structural equation model (SEM) [Koller and Friedman, 22009; this model encodes a set of hypotheses this process in the form of one functional dependency fj for each random variable X, X, so that
= (X,,) with X, S X (x₂).
(10)
Of special interest is structural causal model (SCM) [Bollen, 1989], which particular kind of an SEM with additional assumptions that allow it to also model the causal mechanisms of the DGP. Here, the set of random variables X is extended also include random unobserved variables
U = {UI..... }n), which play the role of noise. Hence, each functional dependency takes the form
Xj = (,)
with X; S X {X3} and U3 E U.
(11)
To further study SCMs, we need to establish their correspondence with causal graphs [Pearl. 2009
A.2 Causal Graphs
Consider set of random variables X = (X1,.... X, } that follow distribution Lx that has a joint probability density Px with respect to some appropriate measure, and an (arbitrary) total ordering
X1 < X2 < ... Xn. Then the joint probability density factorises as
xx = II
Px,IX, (x, (z), X,EX
where Y, S (XI,, X,,
(12)
and y, are those values out of x corresponding to the same indices as y, This comes as a direct result of the chain rule and the conditional independence rules. Any such factorisation can be represented as (fully) directed acyclic graph (DAG) G = (X, E) with nodes the random variables X and edges the set E. - (i > j|x; E ), In other words, in this graph we add an edge to the dependent variable X, from each variable in the corresponding conditioning set Y, that appears in each factor Px, of Eq. (12) We further make this relation explicit, by instead writing Paj = Y, to indicate that the conditioning set Y, serves the set of direct parents of node X, in G. Such graph is called a Bayesian network [Koller and Friedman, 2009) and allows for a visual representation of all those independencies that are implied solely from this factorisation of the joint density and irrespective of the form of each factor.
These independencies can be read from the graph in terms of the d d-separation [Pearl, 2009
Definition A.1 (d-separation). For any pairwise disjoint subsets U, V, W C X, and U, V * 0, it
U IL V O all paths from any variable in to any of V are blocked by W.
(13)
We call path blocked if it either
traverses a section + V -,, -, v - for some variable V E W. or
traverses a section - V +- where neither V nor any of its descendants are contained in W.
Hence, Bayesian network G can be seen description of entire family of distributions that fulfill a given set of conditional independencies. When a distribution Lx exhibits all the conditional independencies that one read from the graph G, we call an I-map of Lx and write Lx Ig-
In other words, the I-map defines an equivalence relation among all DAGs via the relation G = G' - Ig Tg., of which each equivalence class is called the Markov equivalence class (MEC). When, in addition, each of the factors in the factorisation of Eq. (12) correspond to functional dependency of an SCM, we call the resulting DAG causal.
20

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A graphical representation of MECs can be given through the common notion of completed partially directed acyclic graphs (CPDAGs) [Chickering, 2002] also known under other names such as maximally oriented graphs [Meek, 1997]. A partially directed graph (PDAG) PP contains both undirected and directed edeges, and can be associated to an equivalence class M(P) with 9 E .M(P) if and only if G, P have the same skeleton and v-structures. The notion of completion of PDAGs allows for representing equivalence classes uniquely. To this end, for a given equivalence class M one distinguishes between compelled edges with the same directionality in every member of M, and reversible edges otherwise. The completed PDAG P for M is then the one having a directed edge for every compelled edge in M, and undirected edge for every reversible edge in M.
A.3 SCMs and causal graphs
We now return the assumptions implicit in SCM.
Assumption (Causal Interpretation). Each functional dependency fy corresponds to a true causal mechanism in the data, with X, being the direct causes of the direct effect Xj.
As corollary, the corresponding causal graph can have no recurrence. Assumption 3 (Orientation and Acyclicity). The causal graph of an SCM is DAG.
Hence, we once again identify the direct causes of each effect Xy with its parents in the corresponding DAG, X5= = Pay.
Assumption 4 (Exogeneity of Noise), The random variables U are exogenous; that is, there are no edges Xj + U, for any X, and any U,. Assumption (Independence of Noise). The random variables are mutually independent.
In our work, in particular, we consider that part of the noise for each effect X; is the latent variable Laj. in addition to the typical U.. Hence, even though for the set U do make Assumption 5 note that this fails to hold for the entire set of exogenous noise sources, which, in our case, is further extended include all Z. As result, to be more rigorous, make claims to find the CPDAG corresponding to the conditional distribution L rather than the marginal Lx.
We remark that, if one is interested in the entire marginal Lx, this can be achieved as two-level algorithm, in which we first use our presented method to infer the Markov equivalence class of Z and then the inferred values of each La, identify which variables correspond to the same underlying latent Zi, and finally performing causal structure inference over the values of Z to complete the Markov equivalence class of Lx.
In the scope of our work, we hence focus on the conditional Lx|z- In the following section we formally show that we can recover the corresponding underlying Markov equivalence class, as represented by CPDAG G. Specifically, we next move to formally showing our main claims in Corollary 3.4 on using the proposed latent-aware BIC as consistent scoring criterion for this purpose.
B Proof of Theorem 33
In this section we elaborate on the theoretical justification of our method. For this, we make use of the relaxed space constraints to break down analysis in finer steps. We first show that under mild conditions, the MLR distribution does not degenerate to a Gaussian.
Z
X,
We hence treat the parameters 0 as random variables and assume each MLR distribution in the data generating process to drawn as follows. First, finite number K > 2 of components is arbitrarily fixed; then the tuple
x
(B.7,' o²) E k is drawn, where θx = (R²) x - R4 is the parameter space of the model with K components and s the -ddimensional probability simplex.
X R N vIX ~ MLR(0),
(B. E Or
R22
Out of this tuple, the linear coefficients assumed to be drawn from a prior B ~ 1 the mixture coefficients from r ~ Hyy and a positive variance x² > from an arbitrary distribution.
0R (R")* x SK R+
Figure B.I: The parameters of the MLR distribution.
Also let X be the push-forward of the Lebesgue measure on m
through the homcomorphism of that space with (Ra)* K and X the Lebesgue measure on SK-1
21

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Lemma B.1 (Non Gaussianity of Direct Effect), Let X E Rd, Y R be random variables such that Y|X- MLR (B, Y v.)²) with parameters 0 (B
Uk Let the parameter prior peo be a a-additive; let for each Øg the marginal priors of the linear coefficients and the mixture coefficient [ he absolutely continuous with respect to the corresponding Lebesgue measure, that is, I & XB and poy & Ny.
"B
Then the distribution of Y is almost surely not Gaussian.
Proof. Fix a K 2 2. We first show that the ML. distribution with density
K
MLB PY (U, x: B, 7.
xx
=
(14)
1=1
almost never degenerates into a Gaussian.
The alternative can only happen under either of the following two conditions:
1. there is only one (active) component in the mixture, or
2. each active component has the same linear coefficients.
For the former condition to hold, it must be that for the drawn 7 it 7/k 1 for some 1 S k S K. The probability of drawing such y is equal to My (Ex) where
K
Ex = {r E S -1 3k; i K, 7 =
(15)
is the set of the K extremal points of SK-1, However, EK is a finite subset of SK and so X (Ex)= O; this, implies that / (Ex)= = O, since 2 < X
To study the latter condition, we treat the domain of the linear coefficients B R=*K which is homeomorphic to (IR") Denote Vx the set of matrices with all columns equal. This set forms a linear subspace VK S Ra as it contains the zero matrix and closed under scalar multiplication and addition. Additionally, this linear subspace VK is proper subspace of R as the latter contains matrices with unequal columns, which are not part of VK. Therefore, the Lebesgue measure of this subspace vanishes, XB (Vx) = O, which in turn implies that #g (Vk) O, due to #B
K < XB.
We now consider the pre-images VK- TBiK (Vx) and EK " (Ex) of these sets, where TB|K Øx + R K and TylK θạ + S are the Cartesian projections from the space of parameter tuples θş to that of the space of linear coefficients and mixing coefficients, respectively. Further, since θ = Uk22 Ox. it is also that VK, EK E O. However, since the measure of both Vx and Ex vanished in the corresponding marginal measures, / n respectively, it must also be that peo (Vx) e Ex)
Hence, the probability that for a given K the distribution degenerates is equal to (Dx),, which we define DR == VK U Ex. Now, by the union bound (D(Dx) S (V()) + ue (ER) = 0.
Finally, we can bound the probability that the MLR distribution degenerates into a Gaussian for an arbitrarily drawn number of components. Let D C θ be the parameters of the MLR distribution that correspond to the cases that degenerate into a Gaussian, for which it is D = Uk22 Dx. By the o-additivity of μe we get
IP (MLR degenerates into Gaussian)) = (D( = pe UDx) Σ Ho(Dx) =0 (16)
K22
K22
which concludes the proof.
To show identifiability, we need to be able to distinguish, under the true hypothesis Y|X MLR, between all competing hypotheses of Fig. B.2
Theorem 3.3 (Local Consistency of BICMLL ). Let D = fxI....,, be observations of random 2
variables X. Y. such that X|Y MLR (B. 7, o²), with general parameters 9 (see Lemma B.1 Then, out of the structural hypotheses depicted in Fig. 3 3the BIC score of the ground truth hypothesis 9h, is asymptotically larger than any of the alternative ones, Gh and Ghe.. almost surely.
Note that a weaker condition is needed for the mixture coefficients, namely that at least two components have positive probability. This more general requirement, however, is already general enough and easier to treat.
22

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X
X₁
X
X1
Xj
X
X, X (Xi
X; E (X)
X, = X (X)
(a) Correct Structure Po
(b) Wrong Sink G
(c) Missing edge Ii
Figure B.2: Identifiable Cases. Under mild assumptions, when has least two mixing components, all of the shown DAGs are identifiable.
Proof. This claim builds on the established properties of the vanilla BIC score. Here, we treat X {Xi} as nuisance parameters, and we focus on single edge modifications, between the sink of each structural hypothesis and the node on the right of each depiction.
We first consider the pair Gh and gh.s- this case, since the BI based on the Maximum
Likelihood Estimate (MLE) estimates O of the true parameter values and the MLE estimate is asymptotically unbiased, then the the correct model I and the one arising from the alternate hypothesis gh have the same number of parameters, while the same time the likelihood under Gh larger than that of S Hence, in this case the BIC value an increasing function of only the likelihood, and hence it must be that also BIC 2 (G.)) > BIC (Ghs).
For the pair 2h and Gh the number of parameters in the latter hypothesis is smaller than that of the Gh. Under similar reasoning as in Lemma [B1] we can claim that Y A X, almost surely. The rest follows from established asymptotic behaviour of BIC as special case of BIC, due the decomposability property that BIC inherits from BIC.
2
Using this result, we extend the local consistency of BIC to its global consistency. Corollary 3.4. The latent-awand score BIC % is a consistent scoring criterion.
Proof. By considering any sequence of appropriate single edge modifications between adjacent struc- tural hypotheses as in Fig. B.2 can extend the global consistency of BIC to that of BIC Chick- ering, 2002
C Implementation Considerations
We note that our main theory covers the Greedy Equivalent Search (GES) algorithm. In our imple- mentation, however, have used TOPIC [Xu et al., 2025). more recent greedy score-based search that has similar guarantees to GES, and when similar requirements are met by the used score. Hence, we replace within TOPIC our proposed BICEM 2 score, to thus derive TOPICBIC. and here analyse the two algorithms.
We first assume access to the MLE oracle. Then, although the output of both algorithms lies on the domain of all CPDAGs over X, TOPICHIC builds on TOPIC, which is both asymptotically and practically more efficient than GES. Indced, in each iteration, it first limits the candidate hypotheses from the set of all representatives of the Markov equivalence classes which perform single-edge modification from the current best, to only those which differ with respect to the most likely modified source. Formally, the combination of these two steps two subsequent maximisations over exactly the same domain (of all hypotheses with single edge modifications), only performed first over the possible sources, and then over the rest of the hypotheses, conditioned on the chosen source. Hence, at each iteration, the asymptotic greedy optimum is the same in both algorithms.
To see the asymptotical consistency, in the particular assumptions of our causal model, we treat two different cases. First, when no latent variable affects the result, TOPICaiC would have similar ease to detect edge additions as in the case of TOPIC/GES. In the case that a the true structure is an Mixture
23

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Xj
X, SgeX)x (XL)
X₁ XX
X,EXX (A))
X, e
X
X
(a) Correct Structure Gh
(b) Missing edge Giee
Figure C.I: Decomposability of the MLR model in the intermediate stages of TOPICBIC. The ---fa--undiscoveeed edges in X' C X are akin to noise that affects both cases equally.
of Linear Regressions (MLR) model, we can consider the sequence of edge modifications that the TOPICBIC algorithm would produce. Within this sequence, we can revisit the cases of Fig. [B22] and notice that when subset X' c X is not yet discovered in the structure of the algorithm, the effects of the far-undicovered variables X' can be seen as added noise, which equally affects an appropriate hypothesis Gh and G shown in Fig. CI.
Hence, intuitively, we expect a point at which one of the edges of the type X, -> Y would be added the model, until all parents X will be discovered. Finally, posit that the practical use of BIC in lieu of BI is equally affecting both frameworks, as long as Conjecture holds. 2
D Detailed Evaluation
Experimental Setup
We give a more detailed description of our synthetic data generation here.
In iteration (1....Ni) Nr} of each experiment, we randomly sample DAG G over Nx |X] observed variables under an Erdōs Rěnyi model with edge density Pg [0. 1]. In addition, we draw N2 == Z| latent mixing variables Z; Categorical(""") with Z; E {l...., :K,...,,,,, Ki}. where we fix all K; K to the same hyperparameter K. We then sample a set of so-called mixing targets T (X, 32, Tay 7.} C X where X, E with probability pz [0. 1]. We distribute the effect of the Nz mixing variables equally across these targets, resulting in 0 S i < Nz many disjoint sets T, = {X; La,- 7.}. For example, we have T (X,, X2} and T2 = {X4] in Fig. 2.
To generate samples, we traverse X in topological order of the induced G. For each Xj. we sample B, = [3 [1.1...... ,3K,} coefficient vectors, where 3₁x E R PPal for all k E {l...., K,} with K, = K if X, E T and K, = 1 otherwise. We draw By E [--,-- [e, 1] to avoid causal effects close to zero, and if possible also ensure that 13 - Bax-I > for all pairs k; k' to create sufficient class separation, where E 0.25 by default. We then draw S samples from (mixture of) linear regression model(s) PP, y) ~ MLR (B,, o²) and standardize the resulting samples to generate an stendally-ta structural causal model (iSCM) [Ormanice al., 2024].
In the case studies, we consider a mixture of interventional datasets, as well as the flow cytometry dataset by Sachs et al., 2005 under the experimental setup in Wang al., 2017 For both cases, use the same scripts as Kumar et al., 2024 For the mixture of interventions, we have N = 1 with K Nx x + 1 classes which defines a split into one observational and K interventional datasets. Under a so-called diagonal or atomic setting, one node at a time undergoes intervention, resulting in disjoint sets 1 C X, here with hard interventions that fix Byx if X, E g- A similar structure applies to the real-worle dataset with 5846 samples and known manipulations on 5 of the 11 variables, namely I [Akt], I2 [PKC). I3- (PIP2). I1 [Mek]. Is (PIP3).
To evaluate the discovered number of mixing components and assignments, standard metrics clustering evaluation are appropriate, where we show the v-measure and the Adjusted Mutual Information (AMI) as two examples (e.g., Vinh et al., 2010). We average each score over X since we can associate each variable X2 to a fixed assignment with K; components La, Z,, else K; = 1. To validate statements on whether observed variables are mixing targets,
Evaluation Metrics
using the implementation of Fixture-UTIGST httph::t///eett //ththu comm/BBigaagg0772/mixturrr
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Mix. NODES
MIX. EDGE DENSITY
MIX. PRACTION
MIX. COMPONENTS
OBS. NODES
0.8
CMM
Jace
0.6
CMM(
0.2
CMM
0.6
CMM(
1
GIM
KMEANS
DBSCAN
V-mecuuure
HDBSCAN
SPECTRAL
0.8
ARI
0.6
0.2
AMI
0.6
2
4
6
0.20 0.200.4.60. 40.60.8
2
8
10
2
3
4
5
10
13
20
Nz
pz
8z
K
Nx
Figure D.I: Discovering Mixing Structure. In synthetically generated CMMs, we evaluate the quality of the recovered mixing structure, evaluating the affected observed variables (F1 (target)), variable sets affected by the same latent variable (Jacc) as well as per-node mixing assignments (AMI, ARI, v-measure).
compute F1 scores (called F1-target) the statement X, T averaged over X. To validate results on which mixing variables affect which mixing targets, we compute the Jaccard index (called those returned by our algorithm.
Jacc) comparing the sets {T 1......, Tz.,
also compare induced DAG G against the discovered DAG or CPDAG g'. To give intuitive directed edge counts
insight into correct incorrect edge orientations, show F1 scores E in G' (called Fl-dir), where we note that in the case CPDAGs, we only count edges that are directed with certainty. As this is simplistic score mainly included for illustrative purposes, also consider classical distance metrics. A common distance measure for two DAGs or CPDAGs G. is the Structural Hamming Distance (SHD). More suitable for graphs G, g' with causal interpretation are the scores proposed in Wahl and Runge, 2u24 The s/c-metrics (S/C) are based on counting separation statements and comparing their validity in G, G'. Scalable variants thereof are the separation distances (SD) that associate each pair of separable nodes in G' with a separation set S under given separation strategy, and validate whether S remains separating in G. We report the SC (without randonization) and the SD (with the 'parent' resp. 'pparent' type), which are defined both when the output DAG CPDAC
Baselines As method is the only to discover the full G², show (i) AMI and F1- target scores over X for all clustering baselines and wherever applicable for Mixture-UTIGSE (ii) metrics for all causal discovery baselines, and (iii) Jaccard scores over (T:); only for our method. We note that we apply all baselines without optimization of their hyperparameters using their implementations available in the causal-learn. causal discovery toolbox (cdt), causalDisca and dodiscover Python libraries. For all conditionallindeeeeedee Fisher-Z test given the linearity of our functional model. We ran the evaluations on 11th Gen Intel Core 19 CPU.
tests, we use the
using the implementation of the metrics ht https htt///gtttut wwt.t/u.boom////o////o/ooo//o/ao http://w...oooooooooooaa/aoooeoo
25

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MIX. NODES
MIX. EDGE DENSITY
SAMPLE SIZE
OBS. NODES
OBs. EDGE DENSITY
0.8
CMM
SD
0.6
CAM
0.4
0.2
LINGAM
R
S/C
0.6
GES
0.4
0.2
0.8
SHD
0.6
0.4
0.2
IPR dit
0.4
0.2
TPR. dir
0.6
0.4
0.2
0.8
11-irr
0.6
0.4
0.2
0.20.40.60.8
200100600800,000
0.20.40.60.8
Ng
Pz
Nx
Pg
Figure D.2: Discovering Causal Graphs. In synthetically generated CMMs, we evaluate the quality of the causal DAG over the observed variables in terms of separation distances (SD, S/C), structural hamming distance (SHD) and directed edge counts (F1-dir, TPR-dir, FPR-dir).
Discovering Mixing Structure Fig. D.I shows an extended variant of Fig. 4 the main manuscript. The parameters are Nx 10, N2 = 2, P, 0.4, Pg = 0.4, S 1000, Sz = 5, which are held fixed while changing one parameter of interest (columns Fig. D. where we run N, = 10 iterations for each parameter configuration. Different choices of the clustering algorithm besides the GMM, here shown color for better readability, perform either worse on recovering targeted nodes (KMEANS, SPECTRAL) mixing assignments (DBSCAN. HDBSCAN). We observe noticeable differences between the clustering metric choices, so we report the AMI in the main manuscript.
Discovering Causal Structure Fig. D.2 shows an extended variant of Fig. 5 in the main manuscript.
The parameters are Nx = 10, Nz = 4,K K 2,Pz = 0.5, Pc 0.4, 500, Sz = 5. N, 10. All structural metrics (SD, S/C, and SHD) show stable performance of the CMM across the settings. The intuitive score TPR-dir furthermore suggests that our method performs well distinguishing causal edge directions, improving sample size S increases, and the likelihood of mixing pz increases. particular, the results suggest that "sparse" mixing 0 < pz < 1 is most beneficial. We connect this to previous findings in the multi-environmen setting [Perry al., 2022) showing that identification of edge orientations is possible under the sparse mechanism shift hypothesis. This could inspire a future work direction generalizing this property from the nulti-envvionment to the latent-mixed setting.
Discovering Interventional Mixtures Finally, Fig. D.2 extends Fig. 6 to show both mixing structure (Jacc, Fl-iv) and causal structure (SD, S/C. SHD) discovery for the mixture of interventions. As well-defined split into observational and interventional datasets exists for this setting, also
26

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OBS. NODES
SAMPLBS
OBS. NODES
SAMPLES
0.8
CMM
0.8
CMM
Jace
0.6
MIXUT.
S/C
0.6
CAM
0.2
0.2
R"SOKT
GES
MIXUTIOSP
SCORE
0.6
SD
0.6
VARSORT
0.4
0,2
0
0.8
0.8
AMI
Fi-dir
0.4
0.4
0.2
200-006600800.0 ,000
200400600000,00
Nx
Nx
Figure D.3: Mixtures of Interventions. For data generated from an interventional mixture, we show the quality of discovered mixing (left) and graphs (right).
include Mixture-UTIGST UTIGSP the presentation (dark blue). The parameters Nx E {4, 6, 8} are as in Kumar et al., 2024 while restrict the setting to up to S 1000 samples, given that the true positive rates over G for the CMM (and VARSORT) already approach for S = 1000; we refer to Kumar al., 2024 to the performance of -Uxture-UTIGSP with more samples. Compared to Fig. [ the CMM performs much better on discovering whether (X, E T) and which (X, E T variables are mixing targets. A likely reason for this the fact that the hard interventions B
create more distinct separation than re-sampling of 3 with B3 B E.
E Ablation Studies
As additions our main experimental evaluation, perform ablation studies questions.
(i) Choice of Score-Basee Based Algorithm. We address the choice of the score-based causal discovery algorithm used together with the latent-aware BIC, showing GES alongside TOPIC.
(ii) Effect of Latent Mixing Variables. We take a closer look at how latent mixing affects causal discovery algorithms in practice. We hypothesize that spurious edges will appear between mixing targets, and study the extent to which the latent-aware BIC can prune these.
(iii) Choice of Functional Forms. Given that the results presented far are restricted to linear settings, we test a proof-of-concept extension to nonlinear functional forms.
E.1 Choice of Sccoe-Baased Algorithm
CAUSAL GRAPH
METRIC
CMM (TOPIC)
CMM (GES)
GES
SHD
0.17 0.01
0.31 0.02
0.36 0.04
S/C
0.13 + 0.03
0.19 + 0.03
0.26 0.03
SD
0.17 + 0.02
0.33 + 0.03
0.48 0.04
Table E.1: Choice of Score-Based Algorithm. We combine the CMM with different acore-based algorithms (TOPIC, GES) well GES itself
on causal discovery (cf. Fig. D.2)
While we used the latent-aware score BIGEM within the topological-orderigg--ber framework in the main evaluation, we can also use it within the GES algorithm, compared GES with vanilla BIC. We compare these three variants in Table E. for the basic experimental parameters (as in
27

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MIXING ASSIGNMENTS (AMI)
FUNCTION f
CMM-LINERR
CMM-NATURAL--PLINN
GMM
linear
0.7624 + 0.0378
0.7659 + 0.0296
0.3598 = 0.0431
quadratic
0.6465 = 0.0935
0.7163 + 0.0507
0.3116 = 0.0452
cubic
0,5984 = 0.0893
0.6876 + 0.0589
0.2841 = 0.0554
exp
0.6345 = 0.0860
0.7025 + 0.0544
0.3007 = 0.0604
log
0.6628 = 0.0849
0.7100 + 0.0527
0.3100 = 0.0615
sin
0.6452 = 0.0884
0.7035 = 0.0561
0.3144 = 0.0603
Table E.2: Choice of Functional Forms. Under different generating processes f for the synthetic data, we compare the linear and nonlinear instantiation of our model, in bold to indicate the best per-row model.
Regarding the choice of the algorithm, the topological-orddeing--oooo variant (CMM (TOPIC)) appears to have better practical performance in our experiments. This experiment also confirms that regarding the score, BIC with MLR fitting (CMM (GES)) provides benefit over plain BIC (GES).
Fig. D.2
E.2 Effect of Latent Mixing Variables
We are also interested in how exactly the pres- ence of latent mixing variables influences the graphs G' returned by causal discovery methods unaware of mixing. As the latents Z introduce dependencies between the mixing targets T, presume that the reported g' will contain addi- tional spurious (FP) edges.
SCORE-DIC
CAM-BIO
incorrectly removed
TPR där
0.2
correctly removed
In Fig. (light gray) we observe that the FPR in g indeed increases the probability pz of X, increases (shown for SCORE and
0.8
FPR dir
0.6
0.4
0.2
CAM). Given this trend, we investigate whether we can correct the result by pruning any FPs that arise from mixing. Thus, we apply the CMM to each node X3 given its causes in G", fit an MLR, and use the BIC to remove any redundant 2
0
0.20.10.60.8
0.20.44.60.8
Pæ
Pz
Figure E.1: Effect of Latent Mixing.
parents of X, under this model. This results in a graph g". also shown in Fig. (dark gray).
The shaded regions indicate the extent to which FP edges are removed correctly (green), respectively, TP edges are removed incorrectly (red). The BIC prunes some of the spurious and almost no causal edges. However, there still remain additional FPs in g" when Pz increases. This is perhaps due to practical limitations of EM in estimating the correct mixing, leaving room for future improvements.
E.3 Choice of Functional Forms
While consider it reasonable to limit the scope of our work to linear models, also conducted proof-of-concept case study to demonstrate the integration of nonlinear models.
For this, we replace the linear regression mixtures with nonlinear variants, specifically, natural spline. We again use the EM algorithm to infer the mixture components and the BIC criterion to pick their number. Table E.2 shows how well we can reconstruct the mixture components (cf. Fig. 4 in the main text), measured by Adjusted Mutual Information (AMI) averaged over each node 10-node graphs. We show the CMM with the MLR in as our main presentation (left) and the one with mixture of natural splines (right) under different true generating functions (rows). The experimental setting corresponds the same base parameters that base Fig. 4 on.
28

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The results match the expectations, where the models perform very closely in the linear case, but the MLR degrades in performance under nonlinearity. Replacing with the mixture of splines allows covering the nonlinear cases reasonably well.
Conclusions We reach a similar conclusion from our questions in Sections and E.2 using the latent-aware BIC, we can expect not a substantial, but at least some improvement causal discovery be it via correcting the outputs causal discovery algorithms (E.2). as an scoring criterion GES replacing vanilla BIC (E.1. or similarly within TOPIC (Fig. D.2 while in addition being able to discover the latent structure Z that can point us to subsamples the data with a distinct causal generating process. Finally, the proof-of-concept experiment using nonlinear functional forms in Section E.3 also supports exploring our algorithm with more flexible regression mixtures.
Limitations
One limitation regards the assumption of oracle MLR parameters; result, Alg. iis only consistent under appropriate conditions, such as good-enough EM initialization regime, and otherwise should be viewed reasonable approximation the problem. Studying the consistency of the estimates derived by the EM algorithm is a difficult problem, and the literature is limited to rather strict scenarios, which is why we cannot easily provide strict guarantees.
Given this, we can still note that the experimental evidence supports the connection we make in Conjecturo There is also reasonable mechanism that could theoretically explain the observed behaviour (asymptotically). Simply put, when the data truly follow the MLR model, it is more likely for EM implementation align with the oracle; when this does not happen, it is likely due to a faulty model, in which case our downstream process would anyways (asymptotically) reject the specific MIR hypothesis in favour of the correct alternative.
Furthermore, we limit the scope of the exposition mainly to linear additive noise models. While our guarantees for the latent-aware oracle BIC can in principle be transferred to nonlinear setting, this would however require access to oracle mixture parameters, and the connection to estimates obtained using EM is less clear under nonlinearity. This point, while we supplement it with a small empirical analysis, requires a deeper theoretical analysis, which we postpone to future work.
29

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Reviews:
[{"review_id": "YWXJonbfNv", "reviewer": "Reviewer_yqWB", "summary": "The authors consider the problem of performing causal discovery using observational datasets where samples come from multiple unknown distinct environments. They note that if samples come from heterogeneous subpopulations but this heterogeneity is ignored, applying traditional causal discovery methods may lead to spurious results. They observe that existing methods for addressing this setting, which use Gaussian Mixture Models (GMMs) to first split the dataset into estimated sub-populations, may lead to sub-optimal recovery of the subpopulations; in particular, GMMs may perform no-better than a random partition of the samples into subgroups. To address this issue, the authors propose Causal Mixture Modeling (CMM), which replaces the GMM-based subpopulation clustering with clustering based on a causal mixture of linear regressions (MLR).  \n\nThey consider a setting where (observed) random variables X have distribution defined by a linear function of (unknown) causal relationships, with coefficients determined by (unobserved) latent categorical variables. Conditioned on its causal ancestors, this means that X is distributed following an (MLR) model. These distributions can be associated with a DAG containing nodes for both observed variables X and latent variables Z; edges between observed variables with causal relationships; and latent variables -> observed variables. Latent variables are assumed to have no incoming edges (no observed variables or other latent variables have causal effects on latent variables). The target problem is then to discover the causal subgraph on the set of observed random variables X, as well as the set of latent variables Z and the set of edges from Z to X.\n\nOne fundamental obstacle to solving this problem is that computing the Maximum Likelihood Estimate (MLE) of an MLR model is NP-hard. The authors first show that access to some oracle for computing the MLE of the MLR model for any number of mixtures, existing score-based frameworks for causal discovery (in particular Greedy Equivalent Search (GES)) can be extended to handle the latent-variable setup described above, and that the consistency results for GES hold under this extension.\n\nTo derive a practical algorithm motivated by the above observation, the authors propose using Expectation Maximization (EM) to approximate the MLE. They note that under certain conditions, the EM algorithm can find the MLE for MLR models if initialized appropriately. They replace the oracle rule with the EM algorithm, and then use the extended score-based framework developed based on GES to discover a topological ordering over the nodes of the DAG corresponding to X’s. They call this framework Discover a Causal Mixture Model (CMM).", "questions": "Is my understanding that the authors have not derived any guarantees on Algorithm 1 correct? If the components of the mixture are assumed to be well-separated, could the good-initialization regime for the EM algorithm described in lines 183-185 imply consistency for Algorithm 1, or would additional assumptions be necessary?\n\nIt is my understanding that the maximum number of latent variables Z identified by Algorithm 1 is capped at K_max, the user-specified maximum number of mixtures. Is this correct? Are there any heuristic rules that the authors would propose to help guide practitioners in selecting K_max and/or given multiple outputs from Algorithm 1 with different values of K_max, deciding which output is the most useful model of the data?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\nThe authors do a good job of motivating the problem of causal discovery with unknown heterogeneous subpopulations, and they clearly point out why existing methods (based on GMMs) do not adequately address this setting.\nThe MLR model studied and set of assumptions made seem pretty typical within the field of causal discovery, and none seem too unrealistic.\n\nWeaknesses:\nThough the oracle-MLE method describes inherits consistency results from GES, it is my impression that the authors do not establish any guarantees for Algorithm 1. If my understanding is incorrect, please correct me. However I consider this a fairly minor weakness, as the connection to the oracle-rule version is made very clear in the paper, and there are strong computational barriers to guaranteeing good approximations to the MLE so the authors’ framework is ultimately a reasonable approximation to a consistent method.", "was_revised": true, "final_justification": "I maintain my score. In addition to the strengths I outlined in my initial review, I also endorse the paper because the authors discussed the main weakness I identified (the lack of consistency results for Algorithm 1) and provided a satisfactory discussion. I thus endorse this paper for acceptance.", "rebuttal": "We thank you for your response and appreciate your time in the review process.\n\nPlease consider our response to your comments below.\n\n\n# Response to Questions\n * **Consistency of Alg. 1.**\\\n Indeed, studying the consistency of the estimates derived by the EM algorithm is a difficult problem, and the literature is limited to rather strict settings (see the references in our manuscript). Therefore, it is correct that we cannot easily provide strict guarantees for Alg. 1. \\\n    Having mentioned this, we can further note that the experimental evidence supports the connection we make in our conjecture. There is also a reasonable mechanism that could theoretically explain the observed behaviour (asymptotically).\\\n\tSimply put, when the data truly follow the MLR model, it is more likely for our EM implementation to align with the oracle; when this does not happen, it is likely due to a faulty model, in which case our downstream process would anyways (asymptotically) reject the specific MLR hypothesis in favour of the correct alternative.\\\n\tIn summary, we confirm your observation that Alg. 1 is only guaranteed to be consistent under appropriate conditions, such as a good-enough EM-initialization regime, and should otherwise be viewed as a reasonable approximation to the problem. \n\n* **Choice of $K_\\text{max}$.**\\\nThis is a fair point; indeed, we assume that a reasonably high maximum number of $K_\\text{max}$ is given that caps the number of mixture components.\\\nHere, the following comments may be made.\n   \n   * First, even when $2 \\leq K_\\text{max}<K*$, for $K*$ the true number of clusters, the causal structure would still be (likely) correctly inferred, as the capped MLR would still offer a higher likelihood versus any alternative model (under appropriate assumptions, that is, non-zero additive noise and that the total back-door noise being higher than the remaining variance due to the mixture collapse.)\n   * Further, if the MLR model ever selects $K < K_\\text{max}$, it is (asymptotically) sure (under the oracle assumption, assuming the latter to be capped, too) that this estimate of $K$ is the true number of mixture coefficients. When $K=K_\\text{max}$, one could keep increasing the bound and repeat. Hence, we can consider this bound as more of an implementation issue than a truly limiting factor. \n   * For practitioners, we consider small values of $K_\\text{max}$ (in the order of 10 or less) sufficient, with motivational examples in mind where $K$ counts distinct biological states, subpopulations, or conditions with distinct causal mechanisms, so that very large $K$ are most likely not meaningful. We recommend that users inspect the BIC values across different $K$ and increase the maximum if necessary. \n    \n\nFeel free to let us know whether we sufficiently addressed the questions and if any concerns remain."}, {"review_id": "N4o5pUpxOJ", "reviewer": "Reviewer_qK58", "summary": "This paper considers the problem of causal discovery in linear Gaussian structural equation models (SEMs) when the data consist of multiple subpopulations. Specifically, unlike previous work where data can typically be partitioned into multiple domains or environments, all observed data here are collected within a single environment. The authors first define the causal mixture model (CMM), where unobserved mixing variables $Z_i$ are introduced to index different linear coefficients or weights. Subsequently, they present identification results and algorithms. The authors show that, given an oracle maximum likelihood estimator, the structure can be uniquely identified using an adapted Bayesian Information Criterion (BIC) score. Furthermore, they propose an algorithm based on greedy equivalence search that employs the Expectation-Maximization (EM) algorithm to compute the BIC score. The effectiveness of the proposed method is demonstrated on both synthetic and real datasets.", "questions": "1. In Figure 2, what does the edge Z_1 -> X_1 mean? According to Equation (1), the mixing variable is introduced to distinguish the linear coefficients. Since $X_1$ is a source variable, I am not sure how different values of $Z_1$ would affect $X_1$.\n2. Do the identification results and the proposed algorithm rely on the prior assumption that all observed variables have equal noise variance? It is known that while linear additive Gaussian noise models are generally not identifiable, linear Gaussian models with equal noise variance are identifiable and can be identified using maximum likelihood estimation.\n3. Regarding Theorem 3.4:\n\n    (a) What does $Y$ represent? According to the notation in Section 2, $Y$ should be the parents of $X$. If it represents some type of \"outcome variable\", then I am unclear about the meaning of the conditional distribution $\\boldsymbol{X}|Y$.\n   \n   (b) In line 179, I recommend that the authors avoid using the term “Markov equivalence class” if the intent is to state that the models can be distinguished based on the observational distribution.\n\n    (c) Is Theorem B.2 the same as Theorem 3.4? If so, then it seems that Lemma 3.3 is derived from Theorem B.2 and should be considered as a corollary.\n\n4. In lines 210–212, how is the BIC score computed? Does it account for the unknown number of mixtures by employing the EM algorithm, or does it follow the conventional BIC formulation that assumes a single mixture component?", "limitations": "The authors discuss the limitations of the proposed model and identification algorithm.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**:\n1. The presentation of the paper is clear and easy to follow, although adding more explanations and examples may further aid in understanding the technical details.\n2. The problem is well-motivated and clearly defined.\n3. The authors provide a clear description of the background on greedy equivalence search (GES) and clearly state the limitations and conjectures.\n\n**Weakness**:\nThe presentation of certain technical details and theoretical results (e.g., Theorem 3.4) is difficult to understand (see the questions below).", "was_revised": true, "final_justification": "I recommend the paper for acceptance, but would like to maintain my current evaluation. \n\nWhile the overall quality of the paper is good, there is still room to improve the clarity and presentation.", "rebuttal": "We appreciate your time and thank you for the detailed feedback. \n\nPlease consider our responses to your points below.\n\n# Response to Questions\n\n**1. Source Variables**\n\nThis is a good question; we will clarify the meaning of edges such as  $Z_1 \\rightarrow X_1$ (Fig. 2), whenever $X_1$ is a source node of the causal graph.\\\nNote that the GMM used in the sources can be seen as a special case of the MLR model if we use an implicit input variable with a constant value of 1. Indeed, re-visiting Eq. (4) we would then have $\\mathbf Y= \\{\\mathbb 1\\}$, where $\\mathbb 1$ is a deterministic input variable $\\mathbb 1 = 1$.\\\nThen, we would obtain the GMM as \n$$ p_{X|\\{\\mathbb 1\\}}^{\\mathrm{MLR}}(x;\\mathbf\\beta,\\mathbf\\gamma,\\sigma^2)=\\sum_{k=1}^K\\gamma_k p^{\\mathcal N}_{X}(x;\\beta_k\\cdot 1,\\sigma^2),$$\nwhich the reader can easily verify to be the p.d.f. of the GMM model.\n\nThis also gives a seamless approach: We use EM and the BIC to infer latent mixing for both sources and non-sources, where we fit a GMM in the case of sources and an MLR otherwise.\n\nWe will include this explanation after Eq. (4) to be clear on its meaning for source nodes.\n\n**2. Equal Noise Variances and Identification**\n\nIndeed, methods exist to identify linear additive Gaussian noise models with equal variance (under further conditions, e.g., non-perfect dependencies), and what is more, also when other constraints are added (e.g., that the coefficients are all less than the unit).\nOur identification results do not rely on such assumptions. Instead, we base identification on the effects of the latent mixing variables and the asymmetries they introduce in the causal directions.  \nThus, our results can be seen as orthogonal to the approach of restricting the functional model with further assumptions, and can therefore address scenarios where these perhaps unrealistic assumptions do not hold.\n\n\n**3. Formulation of Theorem 3.4**\n\n* **(a)** We thank you for spotting this typesetting error. Here the correct distribution is $X|\\mathbf Y$, in alignment with the notation of Eq. (4).\\\nWe have made the correction $\\mathbf X\\to X$ and $Y\\to \\mathbf Y$ in the improved manuscript.\\\nThe intended meaning is as you correctly assumed.\n\n* **(b)** We thank you for and agree with this recommendation. Following your suggestion, we refrain from using 'Markov Equivalence Class' here to avoid confusion.\n\n* **(c)** Indeed, the two are the same. We have corrected the theorem numbering in the updated appendix.\\\n  As for Lemma 3.3, we agree and will make the appropriate change.\n\n**4. Specifying the Number of Mixtures**\n\nExactly, the BIC in lines 210-212 accounts for an unknown number of mixtures. That is, we rely on the (theoretical) latent-aware BIC as in Eq. (6), where the likelihood takes the form of Eq. (4).\\\nTo compute the BIC in practice, we need to consider  each $1\\leq K\\leq K_\\max$ and use the EM algorithm to obtain estimates for the MLR parameters; finally, we then use the BIC score of the best such $K$ in our algorithm (and the corresponding estimates for the rest of the model parameters). We updated our manuscript text to make this explicit. \n\nWe are happy to dispel any remaining doubts and clarify any persisting source of confusion."}, {"review_id": "YzRlhUBM0Y", "reviewer": "Reviewer_Vdb6", "summary": "This paper addresses statistical causal discovery under the assumption of unobserved subgroups. While prior methods have incorporated such assumptions, they typically rely on a single latent environment variable that affects all observed variables. In contrast, this work proposes a more realistic setting in which multiple latent mixing variables independently influence distinct subsets of observed variables. The authors present a novel method based on this assumption and demonstrate that it can be implemented as an extension of score-based causal discovery, specifically Greedy Equivalence Search (GES), using the Bayesian Information Criterion (BIC).\n\nThey show that while causal discovery is known to be non-identifiable under simple linear additive Gaussian noise models, the introduction of a mixture structure improves identifiability. The paper further proposes the use of the Expectation-Maximization (EM) algorithm for likelihood estimation within the mixture model, and argues, both intuitively and theoretically, that EM yields sufficient identifiability in practice.\n\nExperiments on simulated data demonstrate that the proposed method outperforms existing approaches in both recovering the mixture structure and the underlying causal graph. Additionally, evaluations on real-world cell signaling data show that the method achieves strong causal structure recovery performance.", "questions": "1. Could the authors provide a discussion on how the inferred mixing variables in the Sachs dataset relate to known experimental conditions or biological states?\n\n2. Could the authors comment on possible directions for more general research into the interpretability of inferred mixing variables?", "limitations": "As mentioned above, there is room for improvement in the discussion of the interpretability of the inferred mixing variables.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\nThis paper presents a causal model based on a more realistic assumption in which multiple latent mixing variables independently influence different observed variables. The authors make a valuable theoretical contribution by demonstrating that this formulation can be addressed using an extension of score-based causal discovery with the Bayesian Information Criterion (BIC). This significantly enhances the practical applicability of statistical causal discovery methods. Moreover, the proposed method incorporates a practical inference procedure using the Expectation-Maximization (EM) algorithm, and the authors provide theoretical justification suggesting that EM yields sufficient identifiability in practice. The effectiveness of the proposed approach is convincingly demonstrated through both synthetic and real-world data.\n\nWeaknesses:\nWhile the assumption that each observed variable may be influenced by a distinct latent mixing variable is realistic and well-supported by both theoretical analysis and empirical evaluation, it comes at the cost of producing highly complex inferred causal structures. As a result, interpreting the estimated mixing variables in a meaningful and actionable way becomes challenging. In the context of causal inference, even when the structure is correctly recovered, models that lack interpretability tend to have limited value in practical applications. Addressing this issue, either by proposing a strategy for interpreting the inferred latent variables or by providing concrete examples from real-world datasets (e.g., the Sachs dataset) where the inferred mixing variables align with known experimental conditions or biological states, could greatly enhance the credibility and practical utility of the proposed method.", "was_revised": true, "final_justification": "The rebuttal provided a detailed explanation of the Sachs dataset results, convincingly showing that the inferred mixing variables can be meaningfully related to known experimental conditions and biological states. For the question on interpretability, the authors proposed plausible future directions (e.g., LIME, Shapley values), but these remain potential approaches without concrete implementation or results. Given this, I lean toward recommending acceptance, but will keep my original rating unchanged.", "rebuttal": "We thank you for your time and thought-provoking comments.\n\nPlease consider our response to your questions below.\n\n# Response to Questions\n\n**1. Experimental States in Sachs et al.**\n\nYes, there  is a known correspondence of the mixing variables to experimental conditions for this data.\n\nThe dataset arises from single-cell protein-signalling data of the human immune response system, wherein each of the $11$ observed variables corresponds to the activity of one compound of interest: either a protein or a phospholipid. The dataset contains (among others) 5 experimental conditions, in each of which a particular molecular modifier has been applied to the cells, such that the activity of exactly one of the $11$ compounds of interest is affected. This results in a known change of the measured activity for the corresponding compound.\n\nIn our experiments, we combine the data from all experimental conditions, and we do not make their origin available to the algorithm. Hence, in the pooled data, each variable is affected by a single latent one. The usefulness and relative prevalence of this dataset as a real-world case study among causal inference methods arises from both\n\n 1. the well-studied underlying causal mechanisms, and, in our case, also\n 2. from the known ground-truth values of the resulting latent variables.\n\nFor example, considering the node \"`Akt`\", there are two mixture components: one is the experimental condition where the so-called Akt inhibitor was applied, inhibiting the activation of Akt directly; the other mixture component comprises the remaining samples where Akt is in its baseline condition. Similar interventions have been applied to each of the remaining $4$ variables shown in Table 1 (PKC, PIP2, Mek, and PIP3) which have been intervened upon in a different experimental condition.\n\nWe hope that this explanation makes the setting more clear; we will expand appropriately the description of the Sachs dataset in the improved manuscript. \n\n\n\n**2. Interpretability**\n\nWe agree that interpretability is a valuable property of any causal model. Like many statistical models, the full causal structural model may contain information that can overwhelm a human researcher, more so when it includes latent variables. Below we give some pointers on how to interpret their meaning.\n \n* *Interpretation of mixing variables:*\\\nThe categorical nature of the mixing variables has the benefit that a domain expert can directly inspect the components and their number. For example, someone interested in a specific target node $Y$ with cause $X$ and affected by a latent $Z$ can readily visualize the $k$ components of $Z$ as \"clusters\" and inspect how the causal mechanism $X\\rightarrow Y$ differs in the respective groups, similar to in Fig. 1 of the manuscript.\n\\\n When  $Y$ has more than one cause, a pair-wise analysis or a dimensionality reduction technique could allow for a similar visual analysis. It is also interesting to see which nodes $Z$ jointly influence a subset of nodes, as this is a strong indication of latent subgroups influencing a part of the system. When sufficient background domain knowledge is available, one could match the $k$ components to known genetic subgroups or biological states. If no such expert knowledge is available, one might gain novel insights into the existence of possible causal mechanisms that were previously unknown.\n \n* *Follow-up work:*\\\nYour question inspires some ideas for future work where we further interpret the meaning of a given $Z$, especially when they point to previously unknown states.  To this end, one could explore explainability approaches such as LIME or Shapley values. For this, one could see the latent mixing variables $Z$ as classification targets, and then test whether any known or background knowledge features are important predictors of $Z$ using LIME or Shap. For example, one might find whether the latent components arise from a population group of a certain age, demographic, or other background.\n\nPlease let us know if we can address any remaining questions or concerns."}, {"review_id": "Yz6SjL9jyq", "reviewer": "Reviewer_PSSz", "summary": "The authors study the problem of causal discovery using data from different distributions/environments which they frame as a causal mixture model using potentially multiple discrete mixing variables. These mixing variables are assumed to be unobserved. They propose a score-based EM-style method using BIC that jointly finds the causal ordering and the mixing variables. They provide a theoretical analysis of their approach as well as numerical experiments.", "questions": "- In the motivational example you provided (with the antibiotics), it is not a setting with multiple independent latent mixing variables, right? Can you give a real-world example where there are such independent variables? I think this would strengthen the motivation for this work.\n- In the description of the algorithm, what do you mean by the edge $Pa_j \\rightarrow X_j$? How does it relate to the formula in line 206? Do you mean the edge $X \\rightarrow Y$?", "limitations": "There is no specific section addressing limitations. Some limitations are mentioned in the conclusion.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n- The authors study the relevant problem of causal discovery with data from different environments where the environment-assignment is unknown\n- They provide extensive experiments on simulated as well as real-world data\n- The paper is structured clearly, and easy to follow, but some of the mathematical notation can be streamlined (see points in weaknesses).\n\n**Weaknesses:**\n- The method rests on the strong assumption of linearity as recognized by the authors. The authors claim in the conclusion that the method can \"easily incorporate richer models\", I think it would have strengthened the contribution to explore this in some way, e.g. in the experiments.\n- There seems to be a strand of work missing in the related work section, namely methods that try to detect changing contexts/regimes, e.g. R-PCMCI [1].\n- The contribution of framing the problem in terms of a mixture model with multiple latent variables seems somewhat minor to me, also the proposed method seems straightforward for the problem and rests heavily on existing methods such as GES, EM, and TOPIC. This limits the novelty a bit, however I recognize that the application to the described causal discovery problem is novel.\n- In my opinion the introduction of the causal model in section 2 can be streamlined a bit. For example, I think it would help to express $\\beta$ as a function of the latent mixing variable (instead of allowing for general influence in the mechanism $f$ of $X_j$). This would show that this is the only point of influence for the latent. I also find it a bit confusing that you distinguish between the parent set of $X_j$ and the latent $La_j$: Isn't the mixing variable also a parent of some of the $X_j$s? Or do you define the parent set $Pa_j$ in line 88 as a subset of the observed variables?\n- Some of the concepts that not all readers are familiar with could be recapped/introduced in the main text, e.g. Adjusted Mutual Information, furthermore the notation $BIC_{\\hat{Z}}$ could be introduced more explicitly.\n- Typos:\n  - line 64 \"be extended the mixed setting\"\n  - line 80 resp.\n  - line 127 \"a frameworks\"\n  - line 180 distinguish instead of identify?", "was_revised": true, "final_justification": "The authors detailed various ways to improve the clarity of the paper during the rebuttal.", "rebuttal": "Thank you for your extensive response; we truly appreciate your time and constructive feedback.\n\nPlease consider our responses to your comments below.\n\n# Response to General Comments\n\n## Presentation\n\n* In the related work, we will include a reference to time series methods such as RPCMCI. These methods use EM to discover regimes along the temporal dimension and, therefore, additionally rely on the temporal ordering of the data.\n * We also value the suggestions for streamlining our formulation of the causal model; we agree with your viewpoint and will express $\\beta$ to be a function of \nthe respective mixing variable. We also note that you are correct that $Z$ are themselves causal parents, but we write $\\text{Pa}_i$ to distinguish the observed from the unobserved parents.\n* We appreciate the effort taken to point out our typos, we have corrected them in the current manuscript.\n \n\n## Nonlinear Models\n\nWhile we consider focusing our analysis on linear models useful on its own, we would like to justify our claim on the \"easy\" extensibility of our framework to nonlinear models. We have therefore we conducted a small proof-of-concept study to support this claim.\\\nPlease consider our results, below.\n\nIn the following, we replace the linear-regression mixtures with nonlinear variants, here a natural spline. We again use the EM algorithm to infer the mixture components and the BIC criterion to pick their number.\nThe following table shows how well we can reconstruct the mixture components  (Fig. 4 in the manuscript), measured by Adjusted Mutual Information (AMI) averaged over each node in 10-node graphs.\\\nWe show the CMM with the MLR in as our main presentation (left) and the one with a mixture of natural splines (right) under different true generating functions (rows). \n\n| _AMI_           | **CMM-linear**     | **CMM-natural-spline** | **GMM** (baseline)            |\n|-----------------|--------------------|-------------------------|---------------------|\n| *f*: linear     | **0.7624** ± 0.0378    | **0.7659** ± 0.0296         | 0.3598 ± 0.0431     |\n| *f*: quadratic  | 0.6465 ± 0.0935    | **0.7163** ± 0.0507         | 0.3116 ± 0.0452     |\n| *f*: cubic      | 0.5984 ± 0.0893    | **0.6876** ± 0.0589         | 0.2841 ± 0.0554     |\n| *f*: exp        | 0.6345 ± 0.0860    | **0.7025** ± 0.0544         | 0.3007 ± 0.0604     |\n| *f*: log        | 0.6628 ± 0.0849    | **0.7100** ± 0.0527         | 0.3100 ± 0.0615     |\n| *f*: sin        | 0.6452 ± 0.0884    | **0.7035** ± 0.0561         | 0.3144 ± 0.0603     |\n\nThe results match our expectations, where the models perform very closely in the linear case, but the MLR degrades in performance under non-linearity. Replacing it with the mixture of splines allows covering the non-linear cases reasonably well. \n\nThis proof-of-concept experiment supports exploring our algorithm with more flexible regression mixtures, and we can include it as an ablation study to encourage this. Indeed, our guarantees for the latent-aware oracle BIC can be transferred to a nonlinear setting. However, this would require access to oracle mixture parameters, and the connection to estimates obtained using EM is less clear under non-linearity. This point would require a deeper analysis in future work, but we hope that the reviewers agree that the parametric case is of sufficient contribution for the scope of this work. \n \n\n### Implementation Details\nAs we cannot provide source code here, we summarize the changes we made to the implementation. The source code included in the supplement requires the following small changes,\n\n1. using a different regression-mixture within our algorithm. This can be done in ``src/mixtures/mixing/mixing.py`` by changing the syntax of the regression formula passed to the R-script.\n2.  using a different data-generating function $f$ for each causal relationship in the randomly generated DAGs. Implementation support for this already exists under ``src/exp/gen/synthetic/data_gen_mixing.py`` where the linear function used in each cause-effect relationship can be replaced by other functional forms. To visualize these functional forms, one can refer to the notebook ``src/examples/demo_mixing.ipynb`` and modify the parameter ``F``, such as  ``params['F'] = FunType.CUB`` for cubic functions.\n\nThe experimental setting corresponds to the same base parameters that we used in main manuscript. \n \n# Response to Questions \n\n### Motivating Example\n\nThis is a good point. We can indeed extend our motivational example to motivate multiple latent mixing variables. For example, independent latent variables could correspond to (1) unknown batch effects due to different laboratories from which measurements come and (2) prior exposure related to demographic location. Following your suggestion, we improved the motivational premise in the updated manuscript as follows.\n\n   > Take for example a country-wide study of anti-microbial resistance in hospitalised patients focused on a resistant pathogen such as Methycillin-Resistant Staphylococcus Aureus (MRSA) [4]. For each patient, phenotypic susceptibility to multiple antibiotics is measured; for each antibiotic, each medical center may choose one of the collaborating laboratories to perform each test, thereby introducing a latent variable corresponding to a batch effect, each independent from the others.\\\n  Further, each patient's medical history includes prior exposure to pathogens with known cross-resistance to MRSA, such as Enteroccocus [5]. The plasmid profile of the latter in each geographical region largely affects its susceptibility, say to Vancomycin [2], which in turn affects the cross-resistance of MRSA [1]. Despite being well documented, this variable is not routinely measured by medical centers, and is hence an independent latent variable that defines the mechanism under which the presence of Enteroccocus affects the MRSA cross-resistance [3].\\\n   Such settings, where multiple observed variables are each affected by one independent latent factor, are common in practice. Yet, this premise is not currently addressed by standard causal inference methods, which often rely on assumptions like causal sufficiency or no unmeasured confounders. Both of these assumptions are violated in this setting, often leading to the discovery of spurious dependencies, or to missing true causal relations, altogether  [6, 7].\n\n    * [1] Arredondo-Alonso S, Top J, McNally A, Puranen S, Pesonen M, Pensar J, Marttinen P, Braat JC, Rogers MRC, van Schaik W, Kaski S, Willems RJL, Corander J, Schürch AC. Plasmids Shaped the Recent Emergence of the Major Nosocomial Pathogen Enterococcus faecium. mBio. 2020 Feb 11;11(1):e03284-19. doi: 10.1128/mBio.03284-19. PMID: 32047136; PMCID: PMC7018651.\n  \n    * [2] Boumasmoud M, Dengler Haunreiter V, Schweizer TA, Meyer L, Chakrakodi B, Schreiber PW, Seidl K, Kühnert D, Kouyos RD, Zinkernagel AS. Genomic Surveillance of Vancomycin-Resistant Enterococcus faecium Reveals Spread of a Linear Plasmid Conferring a Nutrient Utilization Advantage. mBio. 2022 Apr 26;13(2):e0377121. doi: 10.1128/mbio.03771-21. Epub 2022 Mar 28. PMID: 35343787; PMCID: PMC9040824.\n   \n    * [3] Cong Y, Yang S, Rao X. Vancomycin resistant Staphylococcus aureus infections: A review of case updating and clinical features. J Adv Res. 2019 Oct 12;21:169-176. doi: 10.1016/j.jare.2019.10.005. PMID: 32071785; PMCID: PMC7015472.\n \n    * [4] Hasanpour AH, Sepidarkish M, Mollalo A, Ardekani A, Almukhtar M, Mechaal A, Hosseini SR, Bayani M, Javanian M, Rostami A. The global prevalence of methicillin-resistant Staphylococcus aureus colonization in residents of elderly care centers: a systematic review and meta-analysis. Antimicrob Resist Infect Control. 2023 Jan 29;12(1):4. doi: 10.1186/s13756-023-01210-6. PMID: 36709300; PMCID: PMC9884412.\n  \n    * [5] Li G, Walker MJ, De Oliveira DMP. Vancomycin Resistance in Enterococcus and Staphylococcus aureus. Microorganisms. 2022 Dec 21;11(1):24. doi: 10.3390/microorganisms11010024. PMID: 36677316; PMCID: PMC9866002.\n\n    * [6] Huang B, Zhang K, Zhang J, Ramsey J, Sanchez‑Romero R, Glymour C, Scholkopf B. Causal Discovery from Heterogeneous/Nonstationary Data: Skeleton Estimation and Orientation Determination. Journal of Machine Learning Research. 2020 May;21(89):1–53.\n   \n    * [7]  Mazaheri B, Squires C, Uhler C. Synthetic Potential Outcomes and Causal Mixture Identifiability. In: Proceedings of the 28th International Conference on Artificial Intelligence and Statistics (AISTATS 2025); 2025 May; Mai Khao, Thailand. PMLR, vol. 258.\n\n### Algorithm Notation\n\nIn the description of the algorithm, we will refer to $\\text{Pa(Y)}$ instead of $\\text{Pa}_j$, thank you for pointing out this inconsistency. To clarify, we refer indeed to the edge $X \\rightarrow Y$ that we consider including in the current parent set $\\text{pa}(Y)$ for $Y$, for which we fit mixture models under these given parent sets.\n\n\nWe are happy to address any additional points that may arise."}]

Author rebuttal:

Meta-review: This work is about causal discovery using observational datasets where samples come from multiple unknown distinct environments. Two reviewers assigned clearly positive scores, highlighting the following strengths: good and convincing motivation of the problem, clear description of the approach, and convincing experiments.
In particular, one reviewer mentioned that the proposed extension of score-based causal discovery with the Bayesian Information Criterion has the potential to significantly enhance the practical applicability of statistical causal discovery methods.

There were also some critical concerns, mainly about (possibly too) strong assumptions, difficult interpretation of the inferred latent variables, and missing approximation guarantees. Most of these concerns, however, could be addressed by the authors in their rebuttal. I also found the rebuttal plausible and convincing, and for me, the positive aspects of this paper clearly outweigh the weaknesses. Therefore, I recommend acceptance.
Decision: Accept (poster)