Bayesian nonparametric estimation of Simpson's evenness index under $\alpha-$Gibbs priors

TL;DR

This paper develops a Bayesian nonparametric method to estimate Simpson's evenness index, a measure of species diversity, using $oldsymbol{ extit{ extalpha}}$-Gibbs priors, providing explicit estimators and variance calculations.

Contribution

It introduces a Bayesian estimator for Simpson's evenness index under $ extalpha$-Gibbs priors, extending previous work on species richness estimation.

Findings

Derived explicit Bayesian estimator for Simpson's index

Calculated variance of the estimator under specific priors

Enhanced understanding of species evenness estimation in Bayesian nonparametrics

Abstract

A Bayesian nonparametric approach to the study of species diversity based on choosing a random discrete distribution as a prior model for the unknown relative abundances of species has been recently introduced in Lijoi et al. (2007, 2008). Explicit posterior predictive estimation of {\it species richness} has been obtained under priors belonging to the $\alpha$-Gibbs class (Gnedin & Pitman, 2006). Here we focus on posterior estimation of {\it species evenness} which accounts for diversity in terms of the proximity to the situation of uniform distribution of the population into different species. We focus on Simpson's index and provide a Bayesian estimator under quadratic loss function, with its variance, under some specific $\alpha-$Gibbs priors.