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Journal Articles Journal of Machine Learning Research Year : 2016

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

Z. Choo
  • Function : Author
S. J. Thwaite
  • Function : Author
D. Jaksch
  • Function : Author

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.

Dates and versions

hal-03597507 , version 1 (04-03-2022)

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P.-L Giscard, Z. Choo, S. J. Thwaite, D. Jaksch. Exact Inference on Gaussian Graphical Models of Arbitrary Topology using Path-Sums. Journal of Machine Learning Research, 2016, ⟨10.1137/15M1054535⟩. ⟨hal-03597507⟩

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