Loop Calculus in Statistical Physics and Information Science

TL;DR

This paper introduces a loop calculus framework that expresses the partition function of discrete statistical models as a finite series, enhancing understanding of the Bethe-Peierls approximation's accuracy and applicability.

Contribution

It provides an exact series expansion for the partition function, linking BP solutions to loop contributions, which was not previously formalized.

Findings

The series converges rapidly in many cases, explaining BP's success.

Loop contributions can be computed directly from BP solutions.

Applications demonstrate improved accuracy in statistical physics and information science.

Abstract

Considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series. The leading term in the series is the Bethe-Peierls (Belief Propagation)-BP contribution, the rest are expressed as loop-contributions on the factor graph and calculated directly using the BP solution. The series unveils a small parameter that often makes the BP approximation so successful. Applications of the loop calculus in statistical physics and information science are discussed.