Bayesian Estimators of Diversity Indexes on Exchangeable Random Partitions

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.