Loop Calculus for Non-Binary Alphabets using Concepts from Information Geometry

TL;DR

This paper extends the Bethe approximation equality, originally for binary variables, to non-binary graphical models by leveraging information geometry, providing a deeper understanding of partition function approximations.

Contribution

It generalizes the Bethe approximation equality to non-binary alphabets using information geometric concepts, broadening its applicability.

Findings

Generalization of the Bethe approximation equality to non-binary models

Representation of multiplicative error as a sum over generalized loops

Enhanced theoretical framework for partition function approximations

Abstract

The Bethe approximation is a well-known approximation of the partition function used in statistical physics. Recently, an equality relating the partition function and its Bethe approximation was obtained for graphical models with binary variables by Chertkov and Chernyak. In this equality, the multiplicative error in the Bethe approximation is represented as a weighted sum over all generalized loops in the graphical model. In this paper, the equality is generalized to graphical models with non-binary alphabet using concepts from information geometry.