Probability measures on the space of persistence diagrams

This paper shows that the space of persistence diagrams has properties that allow for the definition of probability measures which support expectations, variances, percentiles and conditional probabilities. This provides a theoretical basis for a statistical treatment of persistence diagrams, for example computing sample averages and sample variances of persistence diagrams. We first prove that the space of persistence diagrams with the Wasserstein metric is complete and separable. We then prove a simple criterion for compactness in this space. These facts allow us to show the existence of the standard statistical objects needed to extend the theory of topological persistence to a much larger set of applications.