Bayesian nonparametric inference for discovery probabilities: credible intervals and large sample asymptotics

TL;DR

This paper develops Bayesian nonparametric methods to estimate discovery probabilities in species sampling, deriving credible intervals and analyzing asymptotic behavior for large samples under Gibbs-type priors.

Contribution

It introduces a methodology for credible interval estimation of discovery probabilities and compares two popular Gibbs-type priors in this context.

Findings

Credible intervals for discovery probabilities are derived.

Asymptotic behavior of estimators is characterized for large samples.

Comparison between Poisson-Dirichlet and normalized generalized Gamma priors is provided.

Abstract

Given a sample of size $n$ from a population of individuals belonging to different species with unknown proportions, a popular problem of practical interest consists in making inference on the probability $D_{n}(l)$ that the $(n+1)$-th draw coincides with a species with frequency $l$ in the sample, for any $l=0,1,\ldots,n$. This paper contributes to the methodology of Bayesian nonparametric inference for $D_{n}(l)$. Specifically, under the general framework of Gibbs-type priors we show how to derive credible intervals for a Bayesian nonparametric estimation of $D_{n}(l)$, and we investigate the large $n$ asymptotic behaviour of such an estimator. Of particular interest are special cases of our results obtained under the specification of the two parameter Poisson--Dirichlet prior and the normalized generalized Gamma prior, which are two of the most commonly used Gibbs-type priors. With respect to these two prior specifications, the proposed results are illustrated through a simulation study and a benchmark Expressed Sequence Tags dataset. To the best our knowledge, this illustration provides the first comparative study between the two parameter Poisson--Dirichlet prior and the normalized generalized Gamma prior in the context of Bayesian nonparemetric inference for $D_{n}(l)$.