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

*P.-L. Giscard, Z. Choo, S. J. Thwaite, D. Jaksch*; 17(71):1−19, 2016.

### 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.

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