Moment Matters: Mean and Variance Causal Graph Discovery from Heteroscedastic Observational Data
| Authors | Yoichi Chikahara |
| Year | 2026 |
| Field | Statistics / ML |
| arXiv | 2602.23602 |
| Download | |
| Categories | stat.ML, cs.LG |
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 \textit{mean} and \textit{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.
Engineering Breakdown
Plain English
This paper tackles a fundamental gap in causal discovery: standard methods reveal what causes what, but not how causes operate. When real-world data exhibits heteroscedasticity (variance that changes across observations), a single causal graph masks whether a variable influences another's mean, variance, or both. The authors propose a Bayesian framework that infers two separate causal graphs—one for mean effects and one for variance effects—from observational data. This dual-graph approach makes causal mechanisms more interpretable and enables better intervention design by revealing which causal pathways are purely about noise versus signal.
Core Technical Contribution
The core novelty is decomposing causal discovery into moment-specific graphs: a mean graph and a variance graph, inferred jointly from heteroscedastic data. The authors establish formal identification results (sufficient conditions under which these two graphs are uniquely recoverable from observational data), filling a theoretical gap that prior work ignored. Traditional causal discovery methods like constraint-based or score-based approaches return a single DAG agnostic to whether edges represent mean shifts, variance changes, or both—this work is the first to systematically separate these mechanisms in a principled Bayesian framework. The identification conditions likely involve assumptions about the functional form of variance models and the absence of confounding on variance-generating mechanisms.
How It Works
The framework models each variable's distribution conditioned on its parents using a generalized linear model or similar structure where both the mean and variance depend on parent variables. For a variable Y with parents Pa(Y), the method jointly estimates parameters that control how Pa(Y) affects E[Y|Pa(Y)] (mean) and Var(Y|Pa(Y)) (variance). The Bayesian approach places priors on the causal graph structure and moment coefficients, then uses posterior inference to identify which edges belong to the mean graph, variance graph, or both. The identification theory constrains which graph structures are consistent with the observed data distribution, reducing posterior uncertainty. Finally, the algorithm samples or optimizes over candidate graphs, scoring each by how well it explains the heteroscedastic patterns in the data.
Production Impact
In production ML systems, this addresses a concrete problem: when you intervene on a causal variable, knowing whether it affects mean or variance fundamentally changes your decision-making. For example, in A/B testing, an intervention might reduce average latency (mean) but increase variance (unpredictability), which standard causal discovery would conflate into a single edge. With separate graphs, you can design interventions that stabilize variance independently of mean shifts, improving system reliability. The computational cost is higher than single-graph methods (2x parameters plus graph structure search), and you need sufficient heteroscedastic signal in the data—sparse variance changes may be unrecoverable. Integration requires generalized linear modeling support and careful prior specification; misspecified variance models can corrupt mean graph estimates.
Limitations and When Not to Use This
The paper assumes that the heteroscedasticity itself has a causal structure (not arbitrary noise), which may not hold if variance changes stem from measurement error or unobserved confounding. Identification results likely require strong assumptions—e.g., no hidden confounders for variance edges, or specific functional forms for variance models—that are hard to verify in practice. The method may struggle with mixed discrete-continuous data or when variance is extremely low in some subgroups (near-singular distributions). Scalability to high-dimensional settings (hundreds or thousands of variables) is unclear; structure learning over graphs already faces exponential search, and doubling the graph space could be prohibitive without approximations like greedy search or constraint-based pruning.
Research Context
This work extends causal discovery theory beyond standard DAGs, building on recent advances in non-linear causal models (e.g., neural network-based discovery) and heteroscedastic regression. It connects to the causal machine learning literature (Pearl, Spirtes, Shimizu) and the growing interest in causal representation learning. The paper likely benchmarks against classical methods (PC, GES, LiNGAM) and baselines that ignore variance structure, showing improved recovery of true causal mechanisms on synthetic and semi-synthetic data. This opens a new research direction: generalizing causal discovery to other moments (skewness, kurtosis) and exploring interactions between mean and variance graphs under interventions.
:::tip Subscribe Get weekly breakdowns of papers like this in AI Letters - the newsletter for engineers building production AI systems. :::