Positivity for Gaussian graphical models

TL;DR

This paper extends the theoretical framework of Gaussian graphical models by providing explicit, cancellation-free formulas for subdeterminant expansions, enhancing understanding of dependence structures.

Contribution

It introduces explicit formulas for nonzero subdeterminants in Gaussian graphical models, building on previous trek separation results.

Findings

Derived explicit formulas for subdeterminants

Extended trek separation to cancellation-free formulas

Enhanced understanding of dependence in Gaussian models

Abstract

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of nonzero subdeterminants.