TREK SEPARATION FOR GAUSSIAN GRAPHICAL MODELS
Sullivant, S, Talaska, K, Draisma, J Jan · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 2008
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 in directed and undirected graphical models. 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.