  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.
