Belief Propagation and Loop Series on Planar Graphs

TL;DR

This paper presents a method to efficiently evaluate the partition function of planar graph models in Bayesian inference by reducing the problem to a dimer matching model using loop series truncation and Pfaffian formulas.

Contribution

It introduces a novel approach to compute the partition function on planar graphs by connecting loop series truncation to dimer models and Pfaffian calculations, enabling tractable inference.

Findings

Truncating the loop series reduces to evaluating a dimer model.

The Pfaffian formula allows efficient computation of the partition function.

The method extends to the full loop series with Pfaffian contributions.

Abstract

We discuss a generic model of Bayesian inference with binary variables defined on edges of a planar graph. The Loop Calculus approach of [1, 2] is used to evaluate the resulting series expansion for the partition function. We show that, for planar graphs, truncating the series at single-connected loops reduces, via a map reminiscent of the Fisher transformation [3], to evaluating the partition function of the dimer matching model on an auxiliary planar graph. Thus, the truncated series can be easily re-summed, using the Pfaffian formula of Kasteleyn [4]. This allows to identify a big class of computationally tractable planar models reducible to a dimer model via the Belief Propagation (gauge) transformation. The Pfaffian representation can also be extended to the full Loop Series, in which case the expansion becomes a sum of Pfaffian contributions, each associated with dimer matchings on an extension to a subgraph of the original graph. Algorithmic consequences of the Pfaffian representation, as well as relations to quantum and non-planar models, are discussed.