Bayesian Nonparametric Estimation of the Probability of Discovering New Species

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

“…Under the framework (1), withP being in the class of the Gibbs-type random probability measures by Pitman [35], Lijoi et al [25] introduced a novel Bayesian nonparametric methodology for making inferences on quantities related to an additional unobserved sample (X n+1 , . .…”

“…In this paper we derive the prior distribution of the rare variants under the more general assumption ofP being a Gibbs-type random probability measure. Furthermore, following ideas set forth in Lijoi et al [25], we derive and investigate the posterior distribution of the rare variants. Such a posterior distribution corresponds to the conditional joint distribution of the frequency counts of the rare variants induced by an additional sample (X n+1 , .…”

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.…”

“…See, for instance, Efron & Thisted (1976), Hill (1979), Boender & Rinnooy Kan (1987), Chao & Lee (1992), Chao & Bunge (2002), Chao et al (2009), Zhang & Stern (2005), Wang & Lindsay (2005), Wang (2010) and Barger & Bunge (2010). 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).…”

“…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.…”