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.

Article information

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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