On the causal interpretation of acyclic mixed graphs under multivariate normality

TL;DR

This paper investigates when acyclic mixed graphs under multivariate normality can be interpreted causally through hidden variables, establishing a key decomposability condition for equivalence with acyclic digraph models.

Contribution

It provides a necessary and sufficient condition, decomposability of the bidirected part, for mixed graphs to have a causally interpretable acyclic digraph model under multivariate normality.

Findings

Decomposability of the bidirected part is necessary and sufficient.

Characterizes when mixed graphs correspond to hidden variable models.

Clarifies the causal interpretation of acyclic mixed graphs.

Abstract

In multivariate statistics, acyclic mixed graphs with directed and bidirected edges are widely used for compact representation of dependence structures that can arise in the presence of hidden (i.e., latent or unobserved) variables. Indeed, under multivariate normality, every mixed graph corresponds to a set of covariance matrices that contains as a full-dimensional subset the covariance matrices associated with a causally interpretable acyclic digraph. This digraph generally has some of its nodes corresponding to hidden variables. We seek to clarify for which mixed graphs there exists an acyclic digraph whose hidden variable model coincides with the mixed graph model. Restricting to the tractable setting of chain graphs and multivariate normality, we show that decomposability of the bidirected part of the chain graph is necessary and sufficient for equality between the mixed graph model and some hidden variable model given by an acyclic digraph.