  We show that an embedding in Euclidean space based on tropical geometry
generates stable sufficient statistics for barcodes. In topological data
analysis, barcodes are multiscale summaries of algebraic topological
characteristics that capture the `shape' of data; however, in practice, they
have complex structures that make them difficult to use in statistical
settings. The sufficiency result presented in this work allows for classical
probability distributions to be assumed on the tropical geometric
representation of barcodes. This makes a variety of parametric statistical
inference methods amenable to barcodes, all while maintaining their initial
interpretations. More specifically, we show that exponential family
distributions may be assumed, and that likelihood functions for persistent
homology may be constructed. We conceptually demonstrate sufficiency and
illustrate its utility in persistent homology dimensions 0 and 1 with concrete
parametric applications to human immunodeficiency virus and avian influenza
data.
