Belief Propagation Converges to Gaussian Distributions in Sparsely-Connected Factor Graphs
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arXiv
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| Format: | Preprint |
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2026
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| author | Yates, Tom Cheng, Yuzhou Alzugaray, Ignacio Akarca, Danyal Mediano, Pedro A. M. Davison, Andrew J. |
| author_facet | Yates, Tom Cheng, Yuzhou Alzugaray, Ignacio Akarca, Danyal Mediano, Pedro A. M. Davison, Andrew J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21935 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Belief Propagation Converges to Gaussian Distributions in Sparsely-Connected Factor Graphs Yates, Tom Cheng, Yuzhou Alzugaray, Ignacio Akarca, Danyal Mediano, Pedro A. M. Davison, Andrew J. Distributed, Parallel, and Cluster Computing 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. |
| title | Belief Propagation Converges to Gaussian Distributions in Sparsely-Connected Factor Graphs |
| topic | Distributed, Parallel, and Cluster Computing |
| url | https://arxiv.org/abs/2601.21935 |