Identifiability of Gaussian Structural Equation Models with Homogeneous and Heterogeneous Error Variances

TL;DR

This paper proves the identifiability of Gaussian SEMs with both known and unknown error variances, introducing a milder assumption that leverages edge weights, and provides a consistent, feasible structure learning algorithm.

Contribution

It establishes new identifiability conditions for Gaussian SEMs with heterogeneous error variances and develops a novel, statistically consistent structure learning algorithm.

Findings

Theoretical proof of identifiability under new assumptions

Algorithm performs well in simulations and real data

Outperforms existing structure learning methods

Abstract

In this work, we consider the identifiability assumption of Gaussian linear structural equation models (SEMs) in which each variable is determined by a linear function of its parents plus normally distributed error. It has been shown that linear Gaussian structural equation models are fully identifiable if all error variances are the same or known. Hence, this work proves the identifiability of Gaussian SEMs with both homogeneous and heterogeneous unknown error variances. Our new identifiability assumption exploits not only error variances, but edge weights; hence, it is strictly milder than prior work on the identifiability result. We further provide a structure learning algorithm that is statistically consistent and computationally feasible, based on our new assumption. The proposed algorithm assumes that all relevant variables are observed, while it does not assume causal minimality and faithfulness. We verify our theoretical findings through simulations and real multivariate data, and compare our algorithm to state-of-the-art PC, GES and GDS algorithms.