A note on a Bayesian nonparametric estimator of the discovery probability

TL;DR

This paper corrects previous formulas for Bayesian nonparametric estimators of species discovery probabilities, providing accurate formulas using a new technique and verifying their correctness through explicit counterproofs.

Contribution

It offers corrected formulas for Bayesian nonparametric estimators under Gibbs and Poisson-Dirichlet priors, improving previous results with a novel derivation method.

Findings

Corrected formulas for Gibbs priors

Explicit formulas for Poisson-Dirichlet priors

Verification through counterproofs

Abstract

Favaro, Lijoi, and Pruenster (2012, Biometrics, 68, 1188--1196) derive a novel Bayesian nonparametric estimator of the probability of detecting at the $(n+m+1)$th observation a species already observed with any given frequency in an enlarged sample of size $n+m$, conditionally on a basic sample of size $n$. Unfortunately the general result under Gibbs priors (Theorem 2), and consequently the explicit result under $(\alpha, \theta)$ Poisson-Dirichlet priors (Proposition 3), appear to be wrong. Here we provide the correct formulas for both the results, obtained by means of a new technique devised in Cerquetti (2013). We verify the correctness of our derivation by an explicit counterproof for the two-parameter Poisson-Dirichlet case.