# Belief Propagation Converges to Gaussian Distributions in Sparsely-Connected Factor Graphs

arXiv: https://arxiv.org/abs/2601.21935 · HTML: https://mlsanity.com/paper/2601.21935/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2026.

## Built on by (most cited first, 1 of 1)

- [Hierarchy-GBP: Accelerating Factor Graph Inference via Abstraction and Recovery](https://mlsanity.com/paper/2610.06978/) (2026)

## Abstract (arXiv)

Belief Propagation (BP) is a powerful algorithm for distributed inference in probabilistic graphical models, however it quickly becomes infeasible for practical compute and memory budgets. Many efficient, non-parametric forms of BP have been developed, but the most popular is Gaussian Belief Propagation (GBP), a variant that assumes all distributions are locally Gaussian. GBP is widely used due to its efficiency and empirically strong performance in applications like computer vision or sensor networks - even when modelling non-Gaussian problems. In this paper, we seek to provide a theoretical guarantee for when Gaussian approximations are valid in highly non-Gaussian, sparsely-connected factor graphs performing BP (common in spatial AI). We leverage the Central Limit Theorem (CLT) to prove mathematically that variables' beliefs under BP converge to a Gaussian distribution in complex, loopy factor graphs obeying our 4 key assumptions. We then confirm experimentally that variable beliefs become increasingly Gaussian after just a few BP iterations in a stereo depth estimation task.