Bayesian Estimators of Diversity Indexes on Exchangeable Random Partitions
Servet Martı́nez · Markov Processes And Related Fields · 2026
We study indexes of diversity of the abundance of species when their proportions are organized as an exchangeable random partition and we take a sample from them. Firstly, we prove a general result: the sequence of Bayesian estimators of any integrable function defined on countable partitions of the unit interval is an integrable martingale that converges a.s. and in L^1 to the function, when the sample size diverges to infinity. Hence, the Bayesian estimator fluc- tuates as an integrable martingale. For the Poisson-Dirichlet Process, we study the estimators of the entropy and the Gini indexes in more detail. A series of results are devoted to revealing that the behavior of the Bayesian estimators share a number of similarities with the plug-in estimators. These include the a.s. limit behavior, but we also consider behaviors expressing local relations between these estimators. This is the case for the one-step difference of the conditional plug-in entropy of the individuals given that their species is known. We prove that it can be rephrased for the Bayesian entropy estimator and this gives a one-step difference between processes that does not jump only when a new species is found. Similar behavior is established for the Gini index.