The Betti Mixture Model for Bayesian Estimation of Topological Features from Point Clouds

Abstract

We develop the Betti mixture model (BMM), a Bayesian estimator for the Betti numbers of a point cloud from its persistence diagram. BMM partitions the sorted persistence lifetimes into a signal block and a noise block, marginalises the per-block component parameters analytically under conjugate normal-inverse-gamma priors, and returns a finite categorical posterior over each Betti number---closed in form, requiring no Markov chain Monte Carlo and no tuning constants. We establish posterior consistency under an explicit signal--noise separation condition, an inconsistency converse when the condition fails, and a finite-sample lower bound on the coverage of the highest-posterior-density credible set. On simulation benchmarks BMM outperforms the established baselines (Martínez, BayesTDA, and the bottleneck bootstrap) on overall point-estimation accuracy. 

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