Trek separation for Gaussian graphical models
Seth Sullivant, Kelli Talaska, Jan Draisma
The Annals of Statistics · 2010 · Volume 38 · Issue 3
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.
Keywords
05A15, 62H99, 62J05, Bayesian network, Conditional independence, Gessel–Viennot–Lindström lemma, Graphical model, Linear regression, trek rule
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