Trek separation for Gaussian graphical models

TL;DR

This paper introduces a graph-theoretic criterion called trek separation for Gaussian graphical models, generalizing d-separation, to determine when submatrices of covariance matrices have low rank, with proofs based on trek rules and algebraic combinatorics.

Contribution

It provides a precise characterization of low-rank submatrices in Gaussian graphical models using trek separation, extending existing criteria to a broader class of mixed graphs.

Findings

Generalizes d-separation to trek separation for mixed graphs

Provides a graph-theoretic criterion for low-rank submatrices

Uses algebraic combinatorics to prove the main results

Abstract

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases. Our new trek separation criterion generalizes the familiar $d$-separation criterion. Proofs are based on the trek rule, the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials.