Trek separation for Gaussian graphical models

S. Sullivant, K. Talaska, J. Draisma

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43 Citations (Scopus)
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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.
Original languageEnglish
Pages (from-to)1665-1685
JournalThe Annals of Statistics
Issue number3
Publication statusPublished - 2010


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