Trek separation for Gaussian graphical models

Details Trek separation for Gaussian graphical models Trek separation for Gaussian graphical models

2008-12-10

arxiv.org/abs/0812.1938v3

Annals of Statistics 2010, Vol. 38, No. 3, 1665-1685

Statistics

Machine Learning

Published in at http://dx.doi.org/10.1214/09-AOS760 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/09-AOS760

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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