Partition function loop series for a general graphical model: free energy corrections and message-passing equations

TL;DR

This paper develops a loop series expansion for the partition function of general graphical models, linking it to belief propagation fixed points and mean-field spin glass theory, and extends it to replica-symmetry-breaking levels.

Contribution

It generalizes previous work by deriving a loop series expansion that relates to Bethe-Peierls free energy and message-passing equations at various RSB levels, without relying on the Bethe-Peierls approximation.

Findings

First term recovers Bethe-Peierls free energy at RS level

Corrections come from subgraphs without dangling edges

Extension to RSB levels and higher partition functions

Abstract

A loop series expansion for the partition function of a general statistical model on a graph is carried out. If the auxiliary probability distributions of the expansion are chosen to be a fixed point of the belief-propagation equation, the first term of the loop series gives the Bethe-Peierls free energy functional at the replica-symmetric level of the mean-field spin glass theory, and corrections are contributed only by subgraphs that are free of dangling edges. This result generalize the early work of Chertkov and Chernyak on binary statistical models. If the belief-propagation equation has multiple fixed points, a loop series expansion is performed for the grand partition function. The first term of this series gives the Bethe-Peierls free energy functional at the first-step replica-symmetry-breaking (RSB) level of the mean-field spin-glass theory, and corrections again come only from subgraphs that are free of dangling edges, provided that the auxiliary probability distributions of the expansion are chosen to be a fixed point of the survey-propagation equation. The same loop series expansion can be performed for higher-level partition functions, obtaining the higher-level RSB Bethe-Peierls free energy functionals (and the correction terms) and message-passing equations without using the Bethe-Peierls approximation.