Loop series for discrete statistical models on graphs

Abstract

In this paper we present the derivation details, logic, and motivation for the three loopcalculus introduced in Chertkov and Chernyak (2006 Phys. Rev. E 73 065102(R)). Generatingfunctions for each of the three interrelated discrete statistical models are expressed in termsof a finite series. The first term in the series corresponds to the Bethe–Peierlsbelief–propagation (BP) contribution; the other terms are labelled by loops on the factorgraph. All loop contributions are simple rational functions of spin correlation functionscalculated within the BP approach. We discuss two alternative derivations of the loopseries. One approach implements a set of local auxiliary integrations over continuous fieldswith the BP contribution corresponding to an integrand saddle-point value. Theintegrals are replaced by sums in the complementary approach, briefly explained inChertkov and Chernyak (2006 Phys. Rev. E 73 065102(R)). Local gauge symmetrytransformations that clarify an important invariant feature of the BP solutionare revealed in both approaches. The individual terms change under the gaugetransformation while the partition function remains invariant. The requirement for allindividual terms to be nonzero only for closed loops in the factor graph (as opposed topaths with loose ends) is equivalent to fixing the first term in the series to beexactly equal to the BP contribution. Further applications of the loop calculus toproblems in statistical physics, computer and information sciences are discussed.