Graph polynomials and approximation of partition functions with Loopy Belief Propagation

The Bethe approximation, or loopy belief propagation algorithm, is a successful method for approximating partition functions of probabilistic models associated with graphs. 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 the 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 of the results and discuss the properties of newly introduced polynomials.