A Bayesian Nonparametric Approach to Species Sampling Problems with Ordering

Abstract

Species-sampling problems (SSPs) refer to a vast class of statistical problems that, given an observable sample from an unknown population of individuals belonging to some species, call for estimating (discrete) functionals of the unknown species composition of additional unobservable samples. A common feature of SSPs is the invariance with respect to species labelling, i.e. species' labels are immaterial in defining the functional of interest, which is at the core of the development of the Bayesian nonparametric (BNP) approach to SSPs under the popular Pitman-Yor process (PYP) prior. In this paper, we consider SSPs that are not "invariant" to species labelling, in the sense that an ordering or ranking is assigned to species' labels, and we develop a BNP approach to such problems. In particular, inspired by the population genetics literature on age-ordered alleles' compositions, with a renowned interest in the frequency of the oldest allele, we study the following SSP with ordering: given an observable sample from unknown population of individuals belonging to some species (alleles), with species' labels being ordered according to weights (ages), estimate the frequencies of the first r order species' labels in an enlarged sample obtained by including additional unobservable samples. Our BNP approach relies on an ordered version of the PYP prior, which leads to an explicit posterior distribution of the first r order frequencies, with corresponding estimates being simple and computationally efficient. We apply our approach to the analysis of genetic variation, showing its effectiveness in the estimation of the frequency of the oldest allele, and then discuss other applications in the contexts of citations to academic articles and online purchases of items.