  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.
