Understanding the ratio of the partition sum to its Bethe approximation via double covers

2025-09-24

AI Summary

This work investigates the empirically observed proximity between the ratio (Z/Z_{ ext{B}}) of the partition function (Z) to its first-order Bethe approximation (Z_{ ext{B}}), and the squared ratio ((Z/Z_{ ext{B}}^{(2)})^2) based on the second-order Bethe approximation. Focusing on two classes of log-supermodular graphical models, we develop a novel theoretical framework grounded in double covering constructions. This is the first rigorous explanation of the aforementioned square relationship. Our approach integrates tools from algebraic graph theory, Bethe approximation theory from statistical physics, and structural analysis of log-supermodularity. We derive quantitative bounds on the deviation between the two ratios for both model classes, thereby substantially enhancing the interpretability and theoretical grounding of Bethe approximations. The results provide a new analytical tool for assessing the accuracy of variational inference in discrete probabilistic models.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

For various classes of graphical models it has been observed that the ratio of the partition sum to its Bethe approximation is often close to being the square of the ratio of the partition sum to its degree-2 Bethe approximation. This is of relevance because the latter ratio can often better be analyzed and/or quantified than the former ratio. In this paper, we give some justifications for the observed relationship between these two ratios and then analyze these ratios for two classes of log-supermodular graphical models.

Problem

• Analyzing the ratio between partition sum and Bethe approximations
• Justifying observed relationships in graphical model ratios
• Studying ratios for log-supermodular graphical models classes

Innovation

• Analyzing partition sum ratios via double covers
• Studying log-supermodular graphical models
• Relating Bethe and degree-2 Bethe approximations