Graph polynomials and approximation of partition functions with Loopy Belief Propagation

TL;DR

This paper explores the mathematical properties of graph polynomials related to the Bethe approximation and loop series expansion, providing clearer derivations and extending the framework to marginals in probabilistic graphical models.

Contribution

It introduces a polynomial formulation of the Loop Series Expansion with positive integer coefficients and extends the analysis to marginals, clarifying previous results.

Findings

Derived a polynomial form of the Loop Series Expansion

Extended the expansion to include marginals

Provided clearer derivations and discussion of polynomial properties

Abstract

The Bethe approximation, or loopy belief propagation algorithm is a successful method for approximating partition functions of probabilistic models associated with a graph. 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 labeled by subgraphs called generalized loops. In our recent paper, we derive the loop series expansion in form of a polynomial with coefficients positive integers, and extend the result to the expansion of marginals. In this paper, we give more clear derivation for the results and discuss the properties of the polynomial which is introduced in the paper.