Moment Matters: Mean and Variance Causal Graph Discovery from Heteroscedastic Observational Data

Yoichi Chikahara

stat.MLFeb 27, 2026

Moment Matters: Mean and Variance Causal Graph Discovery from Heteroscedastic Observational Data

Authors: Yoichi Chikahara

Organizations: Communication Science Laboratories, NTT, Inc., Kyoto, Japan

Abstract

Heteroscedasticity -- where the variance of a variable changes with other variables -- is pervasive in real data, and elucidating why it arises from the perspective of statistical moments is crucial in scientific knowledge discovery and decision-making. However, standard causal discovery does not reveal which causes act on the mean versus the variance, as it returns a single moment-agnostic graph, limiting interpretability and downstream intervention design. We propose a Bayesian, moment-driven causal discovery framework that infers separate mean and variance causal graphs from observational heteroscedastic data. We first derive the identification results by establishing sufficient conditions under which these two graphs are separately identifiable. Building on this theory, we develop a variational inference method that learns a posterior distribution over both graphs, enabling principled uncertainty quantification of structural features (e.g., edges, paths, and subgraphs). To address the challenges of parameter optimization in heteroscedastic models with two graph structures, we take a curvature-aware optimization approach and develop a prior incorporation technique that leverages domain knowledge on node orderings, improving sample efficiency. Experiments on synthetic, semi-synthetic, and real data show that our approach accurately recovers mean and variance structures and outperforms state-of-the-art baselines.

Figures & tables

Figure 1. Local protein regulatory network example represented by (a) Moment-agnostic causal graph and (b) mean and variance causal graphs (shown in red and blue edges)

Figure 2. Mean and variance causal graph inference performance on synthetic and SERGIO datasets with sample size $n=500$ . Achieving both lower SHD rate (left) and higher F1 score (right) is better.

Figure 3. Mean and variance causal graph inference performance on synthetic datasets ( $n=500$ ) under dense graphs

	$d=2$		$d=3$		$d=4$
	TV	KL	TV	KL	TV	KL
MC3	$\mathbf{0.01\pm 0.00}$	$\mathbf{0.01\pm 0.00}$	$\mathbf{0.10\pm 0.01}$	$\mathbf{0.09\pm 0.01}$	$0.42\pm 0.10$	$6.52\pm 0.91$
DDS	$0.31\pm 0.10$	$1.94\pm 0.28$	$0.32\pm 0.08$	$1.81\pm 0.21$	$0.45\pm 0.11$	$5.52\pm 0.73$
Proposed	$0.26\pm 0.06$	$1.01\pm 0.46$	$0.21\pm 0.09$	$0.34\pm 0.09$	$\mathbf{0.34\pm 0.08}$	$\mathbf{1.51\pm 0.28}$

Table 1. Total variation (TV) distance and KL divergence (KL) of posteriors inferred from linear Gaussian ANM datasets (

n=500

). Achieving both lower TV and KL values is better.

Figure 4. Moment-agnostic causal graph inference performance on non-Gaussian synthetic datasets (

n=500

) with (a) Laplace noise and (b) student-t noise. Achieving both lower SHD (left) and lower SID (right) is better.

	$n=100$		$n=200$		$n=853$
	SHD ( $\downarrow$ )	F1 ( $\uparrow$ )	SHD ( $\downarrow$ )	F1 ( $\uparrow$ )	SHD ( $\downarrow$ )	F1 ( $\uparrow$ )
MC3	22.6 $\pm$ 1.1	0.19 $\pm$ 0.04	20.3 $\pm$ 1.3	0.20 $\pm$ 0.07	18.9 $\pm$ 0.7	0.14 $\pm$ 0.05
DDS	17.6 $\pm$ 0.7	0.20 $\pm$ 0.03	16.6 $\pm$ 0.6	0.20 $\pm$ 0.02	15.0 $\pm$ 0.7	0.28 $\pm$ 0.02
ICDH	20.0 $\pm$ 1.0	0.21 $\pm$ 0.01	17.5 $\pm$ 0.5	0.22 $\pm$ 0.01	14.5 $\pm$ 0.5	0.27 $\pm$ 0.03
HOST	16.1 $\pm$ 0.5	0.19 $\pm$ 0.01	15.0 $\pm$ 0.3	0.33 $\pm$ 0.02	13.5 $\pm$ 0.8	0.38 $\pm$ 0.03
Proposed	16.0 $\pm$ 0.3	0.20 $\pm$ 0.02	14.9 $\pm$ 0.4	0.31 $\pm$ 0.02	13.7 $\pm$ 0.5	0.36 $\pm$ 0.02

Table 2. Moment-agnostic graph inference performance on Sachs dataset.

\downarrow

and

\uparrow

denote "lower is better" and "higher is better".

	Run time [sec]
DDS	367 $\pm$ 11
Proposed	733 $\pm$ 19
Proposed $+25\%$	852 $\pm$ 38
Proposed $+50\%$	1260 $\pm$ 51

Table 3. Run time on Sachs dataset (

n=853

). Our Proposed needs about twice the run time of DDS due to two-graph inference and can use prior knowledge with small overhead.

	$n=100$	$n=200$	$n=853$
Proposed	$0.585\pm 0.009$	$0.592\pm 0.008$	$0.620\pm 0.019$

Table 4. Estimated posterior probability of MEK

\rightarrow

ERK in inferred variance causal graph structure

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Mean ( $d=5$ )	Variance ( $d=5$ )	Mean ( $d=10$ )	Variance ( $d=10$ )
Proposed	$\bm{2.34\pm 0.20}$	$\bm{3.33\pm 0.30}$	$\bm{15.5\pm 1.2}$	$\bm{19.3\pm 1.1}$
ICDH	$5.05\pm 0.42$	$4.55\pm 0.40$	$20.1\pm 1.1$	$20.9\pm 1.2$
HOST	$4.84\pm 0.49$	$4.60\pm 0.45$	$19.6\pm 1.8$	$21.6\pm 1.4$
VarSort	$5.00\pm 1.00$	$6.00\pm 1.66$	$18.5\pm 6.2$	$21.9\pm 4.4$

Appendix

Table 5. SHD score comparison with VarSort on mean-variance HNM datasets (

n=500

) with number of nodes

d=5,10

Figure 5. Moment-agnostic causal graph inference performance on nonlinear Gaussian ANM datasets (

n=500

) with number of nodes

d=5,10,20,50

\downarrow

and

\uparrow

denote "lower is better" and "higher is better".

Figure 6. Moment-agnostic causal graph inference performance on nonlinear Gaussian HNM datasets (

n=500

) with number of nodes

d=5,10,20,50

\downarrow

and

\uparrow

denote "lower is better" and "higher is better".

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Causal discovery infers cause-and-effect structures from observational and interventional data, distinguishing causal relationships from correlation.

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SVI-DAG: A Structured Variational Inference Approach to Bayesian Causal Discovery

Shrenik Zinage

Massachusetts Institute of Technology · This work was conducted while the author was a PhD student at Purdue University, independently of the author’s dissertation research.

Causal Discovery Variational Inference

Open paper AskPDF arXiv

Bayesian causal discovery seeks to determine the posterior distribution of causal theories, which are interpreted as directed acyclic graphs (DAGs) that explain the observed data. The resulting posterior allows systematic reasoning regarding epistemic uncertainty within these theories. Nonetheless, finding such graphs is difficult due to identifiability problems and limited observational data. Furthermore, precisely approximating posterior over graphs is challenging given vast range of potential DAGs. Recent Bayesian approaches have addressed some of these challenges, yet they remain limited as they fail to encode dependencies between edges, and lack principled ways to incorporate domain knowledge as inductive biases during the search process. To overcome these limitations, we propose SVI-DAG, a structured variational inference approach to Bayesian causal discovery using observational data and prior beliefs that uses normalizing flows to model dependencies between edges, supporting expressive and multimodal posterior learning over DAGs. To mitigate mode seeking behaviour in evidence lower bound optimization and promote mode coverage, we use stein variational gradient descent to update the node potentials using a kernel in acyclicity space. We evaluate SVI-DAG against 5 state-of-the-art Bayesian DAG learning methods and demonstrate superior performance in uncertainty quantification while remaining competitive in terms of structural accuracy.

Abstract

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