Goodness-of-Fit Tests for Linear Non-Gaussian Structural Equation Models

TL;DR

This paper introduces new goodness-of-fit tests to verify the linear non-Gaussian assumption in causal discovery models, including extensions with latent confounders, using algebraic relations among moments.

Contribution

It develops novel statistical tests based on moment relations and rank constraints to assess the validity of linear non-Gaussian structural equation models in causal discovery.

Findings

Tests effectively identify violations of linear non-Gaussian assumptions.

Method performs well on benchmark cause-effect datasets.

Bootstrap approach provides reliable null distribution approximations.

Abstract

The field of causal discovery develops model selection methods to infer cause-effect relations among a set of random variables. For this purpose, different modelling assumptions have been proposed to render cause-effect relations identifiable. One prominent assumption is that the joint distribution of the observed variables follows a linear non-Gaussian structural equation model. In this paper, we develop novel goodness-of-fit tests that assess the validity of this assumption in the basic setting without latent confounders as well as in extension to linear models that incorporate latent confounders. Our approach involves testing algebraic relations among second and higher moments that hold as a consequence of the linearity of the structural equations. Specifically, we show that the linearity implies rank constraints on matrices and tensors derived from moments. For a practical implementation of our tests, we consider a multiplier bootstrap method that uses incomplete U-statistics to estimate subdeterminants, as well as asymptotic approximations to the null distribution of singular values. The methods are illustrated, in particular, for the T\"ubingen collection of benchmark data sets on cause-effect pairs.