The Annals of Statistics

Trek separation for Gaussian graphical models

Seth Sullivant, Kelli Talaska, and Jan Draisma

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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.

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Ann. Statist., Volume 38, Number 3 (2010), 1665-1685.

First available in Project Euclid: 24 March 2010

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Zentralblatt MATH identifier

Primary: 62H99: None of the above, but in this section 62J05: Linear regression
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

Graphical model Bayesian network Gessel–Viennot–Lindström lemma trek rule linear regression conditional independence


Sullivant, Seth; Talaska, Kelli; Draisma, Jan. Trek separation for Gaussian graphical models. Ann. Statist. 38 (2010), no. 3, 1665--1685. doi:10.1214/09-AOS760.

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