  The problem of estimating discovery probabilities originated in the context
of statistical ecology, and in recent years it has become popular due to its
frequent appearance in challenging applications arising in genetics,
bioinformatics, linguistics, designs of experiments, machine learning, etc. A
full range of statistical approaches, parametric and nonparametric as well as
frequentist and Bayesian, has been proposed for estimating discovery
probabilities. In this paper we investigate the relationships between the
celebrated Good-Turing approach, which is a frequentist nonparametric approach
developed in the 1940s, and a Bayesian nonparametric approach recently
introduced in the literature. Specifically, under the assumption of a two
parameter Poisson-Dirichlet prior, we show that Bayesian nonparametric
estimators of discovery probabilities are asymptotically equivalent, for a
large sample size, to suitably smoothed Good-Turing estimators. As a by-product
of this result, we introduce and investigate a methodology for deriving exact
and asymptotic credible intervals to be associated with the Bayesian
nonparametric estimators of discovery probabilities. The proposed methodology
is illustrated through a comprehensive simulation study and the analysis of
Expressed Sequence Tags data generated by sequencing a benchmark complementary
DNA library.
