Bayesian Inference for Persistent Homology.

Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains absent. This paper, relying on the theory of point processes, lays the foundation for Bayesian inference with persistence diagrams. We model persistence diagrams as Poisson point processes with prior intensities and compute posterior intensities by adopting techniques from the theory of marked point processes. We then propose a family of conjugate prior intensities via Gaussian mixtures and proceed with a classification application in materials science using Bayes factors.