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Species-sampling problems (SSPs) refer to a vast class of statisticalproblems calling for the estimation of (discrete) functionals of the unknownspecies composition of an unobservable population. A common feature of SSPs istheir invariance with respect to species labelling, which is at the core of theBayesian nonparametric (BNP) approach to SSPs under the popular Pitman-Yorprocess (PYP) prior. In this paper, we develop a BNP approach to SSPs that arenot "invariant" to species labelling, in the sense that an ordering or rankingis assigned to species' labels. Inspired by the population genetics literatureon age-ordered alleles' compositions, we study the following SSP with ordering:given an observable sample from an unknown population of individuals belongingto species (alleles), with species' labels being ordered according to weights(ages), estimate the frequencies of the first $r$ order species' labels in anenlarged sample obtained by including additional unobservable samples. Byrelying on an ordered PYP prior, we obtain an explicit posterior distributionof the first $r$ order frequencies, with estimates being of easy implementationand computationally efficient. We apply our approach to the analysis of geneticvariation, showing its effectiveness in estimating the frequency of the oldestallele, and then we discuss other potential applications.

A Bayesian Nonparametric Approach to Species Sampling Problems with Ordering