Bayesian Estimators of Diversity Indexes on Exchangeable Random Partitions

S. Martinez

2026, v.32, Issue 1, 123-148

ABSTRACT

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.

doi:10.61102/1024-2953-mprf.2026.32.1.006

Keywords: Bayesian estimators, exchangeable random partitions, Integrable martingales, Poisson-Dirichlet Process, Shannon entropy and Gini index

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