# A Bayesian Framework for Persistent Homology

## TL;DR

This paper introduces a Bayesian approach to analyze persistence diagrams in topological data analysis, modeling them as Poisson point processes and enabling statistical inference with conjugate priors.

## Contribution

It develops a novel Bayesian framework for persistence diagrams using point process theory, providing closed-form posterior intensities with Gaussian mixture priors.

## Key findings

- Effective classification in materials science using Bayes factors
- Closed-form posterior intensities derived from Gaussian mixture priors
- Framework extends statistical inference capabilities for topological data analysis

## Abstract

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, presents a 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 Bayesian framework with a classification problem in materials science using Bayes factors.