Loop series expansion with propagation diagrams

TL;DR

This paper introduces a novel loop series expansion for binary pairwise Markov random fields using propagation diagrams, linking exact marginals with Bethe approximations through a polynomial formulation.

Contribution

It derives a new loop series expansion with propagation diagrams, providing a polynomial expression and a relation between exact and approximate marginals.

Findings

Loop series expressed as a polynomial with positive integer coefficients.

Propagation diagrams describe message propagation rules.

New formula relates exact marginals to Bethe approximations.

Abstract

The Bethe approximation is a successful method for approximating partition functions of probabilistic models associated with a graph. Recently, Chertkov and Chernyak derived an interesting formula called Loop Series Expansion, which is an expansion of the partition function. The main term of the series is the Bethe approximation while other terms are labelled by subgraphs called generalized loops. In this paper, we derive a loop series expansion of binary pairwise Markov random fields with propagation diagrams, which describe rules how first messages and secondary messages propagate. Our approach allows to express the Loop Series in the form of a polynomial with coefficients positive integers. Using the propagation diagrams, we establish a new formula that shows a relation between the exact marginal probabilities and their Bethe approximations.