A note on nonparametric inference for species variety with Gibbs-type priors

A Bayesian nonparametric methodology has been recently introduced for estimating, given an initial observed sample, the species variety featured by an additional unobserved sample of size m. Although this methodology led to explicit posterior distributions under the general framework of Gibbs-type priors, there are situations of practical interest where m is required to be very large and the computational burden for evaluating these posterior distributions makes impossible their concrete implementation. In this paper we present a solution to this problem for a large class of Gibbs-type priors which encompasses the two parameter Poisson-Dirichlet prior and, among others, the normalized generalized Gamma prior. Our solution relies on the study of the large m asymptotic behaviour of the posterior distribution of the number of new species in the additional sample. In particular we introduce a simple characterization of the limiting posterior distribution in terms of a scale mixture with respect to a suitable latent random variable; this characterization, combined with the adaptive rejection sampling, leads to derive a large m approximation of any feature of interest from the exact posterior distribution. We show how to implement our results through a simulation study and the analysis of a dataset in linguistics.