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We present the path-sum formulation for exact statistical inference ofmarginals on Gaussian graphical models of arbitrary topology. The path-sumformulation gives the covariance between each pair of variables as a branchedcontinued fraction of finite depth and breadth. Our method originates from theclosed-form resummation of infinite families of terms of the walk-sumrepresentation of the covariance matrix. We prove that the path-sum formulationalways exists for models whose covariance matrix is positive definite: i.e.~itis valid for both walk-summable and non-walk-summable graphical models ofarbitrary topology. We show that for graphical models on trees the path-sumformulation is equivalent to Gaussian belief propagation. We also recover, as acorollary, an existing result that uses determinants to calculate thecovariance matrix. We show that the path-sum formulation formulation is validfor arbitrary partitions of the inverse covariance matrix. We give detailedexamples demonstrating our results.

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