A Bayesian Framework for Persistent Homology

Abstract

Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that use persistence diagrams in statistical inference, a full Bayesian treatment remains absent. This paper, relying on the theory of point processes, presents a generalized Bayesian framework for inference with persistence diagrams relying on a substitution likelihood argument. In essence, 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 to obtain a closed form of the posterior intensity. Finally, we demonstrate the utility of this generalized Bayesian framework with a classification problem in materials science using Bayes factors.