Statistica Sinica 27 (2017), 839-858

BAYESIAN NONPARAMETRIC INFERENCE
FOR DISCOVERY PROBABILITIES: CREDIBLE
INTERVALS AND LARGE SAMPLE ASYMPTOTICS
Julyan Arbel1,2, Stefano Favaro1,3, Bernardo Nipoti4 and Yee Whye Teh5
1Collegio Carlo Alberto, 2Bocconi University, 3University of Torino,
4Trinity College Dublin and 5University of Oxford

Abstract: Given a sample of size $n$ from a population of individuals belonging to different species with unknown proportions, a problem of practical interest consists in making inference on the probability that the $\left(n+1\right)$ -th draw coincides with a species with frequency in the sample, for any . This paper contributes to the methodology of Bayesian nonparametric inference for Specifically, under the general framework of Gibbs-type priors we show how to derive credible intervals for a Bayesian nonparametric estimation of , and we investigate the large 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. With respect for these prior specifications, the proposed results are illustrated through a simulation study and a benchmark Expressed Sequence Tags dataset. To the best our knowledge, this provides the first comparative study between the twoparameter Poisson–Dirichlet prior and the normalized generalized Gamma prior in the context of Bayesian nonparemetric inference for .

Key words and phrases: Asymptotics, Bayesian nonparametrics, credible intervals, discovery probability, Gibbs-type priors, Good–Turing estimator, normalized generalized Gamma prior, smoothing technique, two-parameter Poisson–Dirichlet.