Reading Dependencies from Covariance Graphs

TL;DR

This paper introduces a graphical criterion to identify dependencies in covariance graphs, applicable to Gaussian distributions, and proves its soundness and completeness under certain assumptions.

Contribution

It presents a new graphical criterion for reading dependencies from covariance graphs, extending the understanding of dependency structures in probabilistic models.

Findings

The criterion is sound and complete under the specified assumptions.

All regular Gaussian distributions satisfy the assumptions.

The approach enhances dependency analysis in covariance graphs.

Abstract

The covariance graph (aka bi-directed graph) of a probability distribution $p$ is the undirected graph $G$ where two nodes are adjacent iff their corresponding random variables are marginally dependent in $p$. In this paper, we present a graphical criterion for reading dependencies from $G$, under the assumption that $p$ satisfies the graphoid properties as well as weak transitivity and composition. We prove that the graphical criterion is sound and complete in certain sense. We argue that our assumptions are not too restrictive. For instance, all the regular Gaussian probability distributions satisfy them.