Exact Inference on Gaussian Graphical Models of Arbitrary Topology using Path-Sums

P.-L. Giscard; Z. Choo; S. J. Thwaite; D. Jaksch

We present the path-sum formulation for exact statistical inference of marginals on Gaussian graphical models of arbitrary topology. The path-sum formulation gives the covariance between each pair of variables as a branched continued fraction of finite depth and breadth. Our method originates from the closed- form resummation of infinite families of terms of the walk-sum representation of the covariance matrix. We prove that the path- sum formulation always exists for models whose covariance matrix is positive definite: i.e. it is valid for both walk-summable and non-walk-summable graphical models of arbitrary topology. We show that for graphical models on trees the path-sum formulation is equivalent to Gaussian belief propagation. We also recover, as a corollary, an existing result that uses determinants to calculate the covariance matrix. We show that the path-sum formulation formulation is valid for arbitrary partitions of the inverse covariance matrix. We give detailed examples demonstrating our results.

Exact Inference on Gaussian Graphical Models of Arbitrary Topology using Path-Sums

Abstract