Loop series for discrete statistical models on graphs

TL;DR

This paper elaborates on the loop calculus for discrete statistical models on graphs, expressing partition functions as series with BP and loop contributions, and discusses derivations, gauge invariance, and applications.

Contribution

It introduces detailed derivations and insights into the loop series expansion, connecting BP solutions with loop corrections and gauge invariance in statistical models.

Findings

Loop series expressed as finite sums with BP and loop terms

Two alternative derivations of the loop series are provided

Gauge symmetry clarifies invariance of the partition function

Abstract

In this paper we present derivation details, logic, and motivation for the loop calculus introduced in \cite{06CCa}. Generating functions for three inter-related discrete statistical models are each expressed in terms of a finite series. The first term in the series corresponds to the Bethe-Peierls (Belief Propagation)-BP contribution, the other terms are labeled by loops on the factor graph. All loop contributions are simple rational functions of spin correlation functions calculated within the BP approach. We discuss two alternative derivations of the loop series. One approach implements a set of local auxiliary integrations over continuous fields with the BP contribution corresponding to an integrand saddle-point value. The integrals are replaced by sums in the complimentary approach, briefly explained in \cite{06CCa}. A local gauge symmetry transformation that clarifies an important invariant feature of the BP solution, is revealed in both approaches. The partition function remains invariant while individual terms change under the gauge transformation. The requirement for all individual terms to be non-zero only for closed loops in the factor graph (as opposed to paths with loose ends) is equivalent to fixing the first term in the series to be exactly equal to the BP contribution. Further applications of the loop calculus to problems in statistical physics, computer and information sciences are discussed.