Sample Complexity of Causal Identification with Temporal Heterogeneity

TL;DR

This paper investigates the conditions under which causal graphs can be identified from observational data with temporal and heterogeneity structures, analyzing the impact of noise distributions on the statistical limits of recovery.

Contribution

It unifies temporal and heterogeneity assumptions for causal identification, providing necessary conditions and analyzing sample complexity under different noise distributions.

Findings

Temporal structure can substitute for environmental heterogeneity in identifiability.

Heavy-tailed noise increases sample complexity for covariance-based methods.

Explicit bounds quantify the cost of robustness in non-Gaussian settings.

Abstract

Recovering a unique causal graph from observational data is an ill-posed problem because multiple generating mechanisms can lead to the same observational distribution. This problem becomes solvable only by exploiting specific structural or distributional assumptions. While recent work has separately utilized time-series dynamics or multi-environment heterogeneity to constrain this problem, we integrate both as complementary sources of heterogeneity. This integration yields unified necessary identifiability conditions and enables a rigorous analysis of the statistical limits of recovery under thin versus heavy-tailed noise. In particular, temporal structure is shown to effectively substitute for missing environmental diversity, possibly achieving identifiability even under insufficient heterogeneity. Extending this analysis to heavy-tailed (Student's t) distributions, we demonstrate that while geometric identifiability conditions remain invariant, the sample complexity diverges significantly from the Gaussian baseline. Explicit information-theoretic bounds quantify this cost of robustness, establishing the fundamental limits of covariance-based causal graph recovery methods in realistic non-stationary systems. This work shifts the focus from whether causal structure is identifiable to whether it is statistically recoverable in practice.