# A Homological Approach to Belief Propagation and Bethe Approximations

## TL;DR

This paper introduces a homological framework for understanding belief propagation and Bethe approximations, revealing conserved quantities and new insights into their equilibria through a differential complex of local observables.

## Contribution

It presents a novel homological approach that connects belief propagation with Bethe free energy critical points, providing deeper theoretical understanding.

## Key findings

- Identifies conserved quantities in belief propagation
- Links equilibria of belief propagation to Bethe free energy critical points
- Provides a differential complex framework for local observables

## Abstract

We introduce a differential complex of local observables given a decomposition of a global set of random variables into subsets. Its boundary operator allows us to define a transport equation equivalent to Belief Propagation. This definition reveals a set of conserved quantities under Belief Propagation and gives new insight on the relationship of its equilibria with the critical points of Bethe free energy.