The research areas of statistical machine learning and Bayesian inference in the context of topological data analysis will be explained. We begin with an explanation of two foundational frameworks for inference: statistical machine leaning and Bayesian inference. We use a geometric problem, dimension reduction
as the setting to provide a basis for these two frameworks.
We then describe the work in topological data analysis that falls under these settings. This includes learning the topology of a manifold or stratified space and probabilistic modeling with persistence diagrams.
We will also discuss the problem of manifold learning and the centrality of the Laplace-Beltrami operator. We will discuss generalizations of this problem that utilize the combinatorial Laplacian (Hodge operator) and the relation of this to clustering and random walks.