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

TL;DR

This paper introduces the path-sum formulation for exact inference of covariances in Gaussian graphical models of any topology, unifying and extending existing methods.

Contribution

It develops a novel path-sum approach that provides a finite continued fraction representation for covariances, valid for all positive definite models, including non-walk-summable cases.

Findings

Path-sum formulation always exists for positive definite models.

Equivalent to belief propagation on trees.

Recovers existing determinant-based covariance calculations.

Abstract

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.