A Skewness-Based Criterion for Addressing Heteroscedastic Noise in Causal Discovery

TL;DR

The paper addresses causal discovery under heteroscedastic symmetric noise by introducing a score-based skewness criterion, SkewScore, that distinguishes causal from anti-causal directions without requiring exogenous-noise extraction. The approach extends to multivariate settings and leverages a two-phase, order-based DAG search to reduce conditional-independence tests to a polynomial count. Theoretical identifiability results show the score component along the cause is skewed while the effect component is not, and empirical studies demonstrate superior or competitive performance against state-of-the-art baselines, including in the presence of latent confounding. This work offers a scalable, noise-agnostic tool for causal direction inference with heteroscedastic noise, with promising applicability to real-world data and avenues for extension to higher dimensions and more complex latent structures.

Abstract

Real-world data often violates the equal-variance assumption (homoscedasticity), making it essential to account for heteroscedastic noise in causal discovery. In this work, we explore heteroscedastic symmetric noise models (HSNMs), where the effect $Y$ is modeled as $Y = f(X) + σ(X)N$, with $X$ as the cause and $N$ as independent noise following a symmetric distribution. We introduce a novel criterion for identifying HSNMs based on the skewness of the score (i.e., the gradient of the log density) of the data distribution. This criterion establishes a computationally tractable measurement that is zero in the causal direction but nonzero in the anticausal direction, enabling the causal direction discovery. We extend this skewness-based criterion to the multivariate setting and propose SkewScore, an algorithm that handles heteroscedastic noise without requiring the extraction of exogenous noise. We also conduct a case study on the robustness of SkewScore in a bivariate model with a latent confounder, providing theoretical insights into its performance. Empirical studies further validate the effectiveness of the proposed method.

Figures (6)

• Figure 1: Identifiability of the heteroscedastic symmetric noise model (HSNM). For the causal direction $X \rightarrow Y$ (left), the conditional distribution $p(y|x)$ is symmetric for any $x$ (top right), and the variance of $p(y|x)$ varies with different values of $x$ due to heteroscedasticity. In contrast, $p(x|y)$ is asymmetric for some $y$ (bottom right).
• Figure 2: Three-variable HSNM model with latent confounder given by Eq. \ref{['eq:latent_HSNM']}.
• Figure 3: Accuracy of causal direction estimation across different data generation processes for (a) bivariate heteroscedastic noise models, and (b) latent-confounded triangular heteroscedastic noise models. Results are averaged over 100 independent runs.
• Figure 4: Topological order divergence and structural Hamming distance (SHD) across different number of variables (the dimension $d$). Lower values indicate better performance for both metrics.
• Figure 5: Score function (a vector field) visualized on data points with projections on the $x$-axis and $y$-axis. For clarity, only the projections for one data point are shown (highlighted in orange and blue). Theorem \ref{['thm:identifiability']} examines the skewness of these projections across the data distribution.

Theorems & Definitions (15)

• Definition 1
• Theorem 1
• Corollary 2
• Proposition 3
• Example 4
• Proposition 5
• Remark 6
• Example 7
• Example 8
• Example 9