Partial Homoscedasticity in Causal Discovery with Linear Models

TL;DR

This paper introduces a framework for causal discovery in linear models with partial homoscedasticity, allowing variables to share error variances within blocks, and provides algorithms for identifying equivalent DAGs under this assumption.

Contribution

It extends existing models by characterizing when two DAGs are distributionally equivalent under partial homoscedasticity and offers an efficient algorithm to learn the associated CPDAG.

Findings

The framework generalizes previous all-equal or arbitrary variance assumptions.

The algorithm effectively constructs the CPDAG from data.

Simulation results show greedy search can exploit partial homoscedasticity.

Abstract

Recursive linear structural equation models and the associated directed acyclic graphs (DAGs) play an important role in causal discovery. The classic identifiability result for this class of models states that when only observational data is available, each DAG can be identified only up to a Markov equivalence class. In contrast, recent work has shown that the DAG can be uniquely identified if the errors in the model are homoscedastic, i.e., all have the same variance. This equal variance assumption yields methods that, if appropriate, are highly scalable and also sheds light on fundamental information-theoretic limits and optimality in causal discovery. In this paper, we fill the gap that exists between the two previously considered cases, which assume the error variances to be either arbitrary or all equal. Specifically, we formulate a framework of partial homoscedasticity, in which the variables are partitioned into blocks and each block shares the same error variance. For any such groupwise equal variances assumption, we characterize when two DAGs give rise to identical Gaussian linear structural equation models. Furthermore, we show how the resulting distributional equivalence classes may be represented using a completed partially directed acyclic graph (CPDAG), and we give an algorithm to efficiently construct this CPDAG. In a simulation study, we demonstrate that greedy search provides an effective way to learn the CPDAG and exploit partial knowledge about homoscedasticity of errors in structural equation models.