Bayesian Non-Parametric Inference for Species Variety with a Two-Parameter Poisson–Dirichlet Process Prior

Abstract

A Bayesian non-parametric methodology has been recently proposed to deal with the issue of prediction within species sampling problems. Such problems concern the evaluation, conditional on a sample of size "n", of the species variety featured by an additional sample of size "m". Genomic applications pose the additional challenge of having to deal with large values of both "n" and "m". In such a case the computation of the Bayesian non-parametric estimators is cumbersome and prevents their implementation. We focus on the two-parameter Poisson-Dirichlet model and provide completely explicit expressions for the corresponding estimators, which can be easily evaluated for any sizes of "n" and "m". We also study the asymptotic behaviour of the number of new species conditionally on the observed sample: such an asymptotic result, combined with a suitable simulation scheme, allows us to derive asymptotic highest posterior density intervals for the estimates of interest. Finally, we illustrate the implementation of the proposed methodology by the analysis of five expressed sequence tags data sets. Copyright (c) 2009 Royal Statistical Society.

Posterior analysis of rare variants in Gibbs-type species sampling models

“…See Favaro et al [13] for a similar approach in Bayesian nonparametric inference for the number of new species generated by the additional sample. In order to obtain asymptotic credible intervals forR…”

“…In order to determine the quantiles s 1 and s 2 , we devised an algorithm for sampling the limiting random variable S (n,j) α,θ . To this end, we combine the algorithm proposed in Favaro et al [13] with the fast rejection algorithm for sampling from an exponentially tilted positive α-stable random variable. See Hofert [17] for details.…”

Species sampling models: consistency for the number of species

“…The idea we present here is motivated by recent work appearing in Lijoi et al (2007Lijoi et al ( , 2008 and Favaro et al (2009). The problem is to estimate the number of species in a population, early work on which can be found in many papers.…”

“…Lijoi et al (2007) are predominantly concerned with estimating the number of new species in a further sample of size m having previously observed a sample of size n. For this, Bayesian nonparametric models are employed and, specifically, discrete random probability measures are used, such as the Dirichlet process and the two parameter Poisson-Dirichlet process. More generally, two classes used are the class of normalized random measures, which are driven by nondecreasing Lévy processes, and Gibbs-type priors (Lijoi et al, 2008, Favaro et al, 2009). These models assume that the number of species is infinite, claiming that if the number of species in the population is large, then it is reasonable to assume that it is infinite (Favaro et al, 2009, Lijoi et al, 2007.…”

“…More generally, two classes used are the class of normalized random measures, which are driven by nondecreasing Lévy processes, and Gibbs-type priors (Lijoi et al, 2008, Favaro et al, 2009). These models assume that the number of species is infinite, claiming that if the number of species in the population is large, then it is reasonable to assume that it is infinite (Favaro et al, 2009, Lijoi et al, 2007. Probably this was done because the mathematics is more attractive for such models.…”