Qualitative inequalities for squared partial correlations of a Gaussian random vector

TL;DR

This paper establishes qualitative inequalities for squared partial correlations in Gaussian vectors, providing tools for model comparison and selection in Gaussian graphical models, especially trees and polytrees.

Contribution

It introduces new sufficient conditions for comparing partial correlations and dependencies in Gaussian graphical models, including trees and polytrees, with applications to model selection.

Findings

Dependence in Gaussian trees characterized by path lengths.

Rules for comparing association degrees in Gaussian graphical models.

Application of inequalities to model selection and mutual information.

Abstract

We describe various sets of conditional independence relationships, sufficient for qualitatively comparing non-vanishing squared partial correlations of a Gaussian random vector. These sufficient conditions are satisfied by several graphical Markov models. Rules for comparing degree of association among the vertices of such Gaussian graphical models are also developed. We apply these rules to compare conditional dependencies on Gaussian trees. In particular for trees, we show that such dependence can be completely characterized by the length of the paths joining the dependent vertices to each other and to the vertices conditioned on. We also apply our results to postulate rules for model selection for polytree models. Our rules apply to mutual information of Gaussian random vectors as well.