Bayesian nonparametric estimators derived from conditional Gibbs structures

Abstract

We consider discrete nonparametric priors which induce Gibbstype exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predicting various features of additional samples. The results provide useful tools for genomic applications where prediction of future outcomes is required.

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

“…A particularly important example is represented by the estimation of the number of new species that will be observed in the additional sample. See Lijoi et al [29], Favaro et al [12], Favaro et al [11] and Bacallado et al [1] for estimators of other features related to species richness under Gibbs-type priors. This class of priors stands out for both mathematical tractability and flexibility.…”

“…These species will be referred to as old species. See Lijoi et al [29] and Favaro et al [12] for a description of the random variables (14)- (16) by means of conditional partition probability functions when data are generated by Gibbs-type priors.…”

“…. , τ , then (29) characterizes the posterior distribution of the rare variants. Similarly to Corollaries 2 and 3, in the next corollary we present a collection of results which can be derived by Theorem 4 under the assumption of a two parameter Poisson-Dirichlet prior.…”

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.…”