A Bayesian Nonparametric Approach to Species Sampling Problems with Ordering

TL;DR

This paper introduces a Bayesian nonparametric method using an ordered Pitman-Yor process to estimate species' frequencies in populations where species are ranked by age, with applications in genetics.

Contribution

It develops a novel BNP approach for ordered species sampling problems, providing explicit, computationally efficient posterior estimates for species' frequencies based on their age order.

Findings

Effective estimation of the oldest allele's frequency in genetic data.

Explicit posterior distribution for ordered species frequencies.

Method demonstrates computational efficiency and practical applicability.

Abstract

Species-sampling problems (SSPs) refer to a vast class of statistical problems calling for the estimation of (discrete) functionals of the unknown species composition of an unobservable population. A common feature of SSPs is their invariance with respect to species labeling, which is at the core of the Bayesian nonparametric (BNP) approach to SSPs under the popular Pitman-Yor process (PYP) prior. In this paper, we develop a BNP approach to SSPs that are not "invariant" to species labeling, in the sense that an ordering or ranking is assigned to species' labels. Inspired by the population genetics literature on age-ordered alleles' compositions, we study the following SSP with ordering: given an observable sample from an unknown population of individuals belonging to 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. By relying on an ordered PYP prior, we obtain an explicit posterior distribution of the first $r$ order frequencies, with estimates being of easy implementation and computationally efficient. We apply our approach to the analysis of genetic variation, showing its effectiveness in estimating the frequency of the oldest allele, and then we discuss other potential applications.