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

Giscard, P.-L.; Choo, Z.; Thwaite, S. J. and Jaksch, D. (2016): Exact Inference on Gaussian Graphical Models of Arbitrary Topology using Path-Sums. In: Journal of Machine Learning Research, Vol. 17, 1

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 in finite 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 de finite: 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.

• Item Type: Journal article
• Faculties: Physics
• Subjects: 500 Science > 530 Physics
• ISSN: 1532-4435
• Language: English