One Step Entropy Variation in Sequential Sampling of Species for the Poisson-Dirichlet Process

Abstract

We consider the sequential sampling of species, where observed samples are classified into the species they belong to. We are particularly interested in studying some quantities describing the sampling process when there is a new species discovery. We assume that the observations and species are organized as a two-parameter Poisson-Dirichlet Process, which is commonly used as a Bayesian prior in the context of entropy estimation, and we use the computation of the mean posterior entropy given a sample developed in Archer et al. (J. Mach. Learn. Res. 15(1):2833–2868, 2014). Our main result shows the existence of a monotone functional, constructed from the difference between the maximal entropy and the mean entropy throughout the sampling process. We show that this functional remains constant only when a new species discovery occurs.