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.
Funding Statement
The first and third authors gratefully acknowledge funding from the European Research Council (ERC), under the European Union’s Horizon 2020 research and innovation programme, Grant agreement No. 817257. The second and third authors gratefully acknowledge the support from the Italian Ministry of University and Research (MUR), “Dipartimenti di Eccellenza” grant 2023-2027. The second author was supported by the European Union – Next Generation EU funds, component M4C2, investment 1.1., PRIN-PNRR 2022 (P2022H5WZ9).
Acknowledgments
The authors are grateful to the Associate Editor and two anonymous Referees for their comments and corrections that allow them to improve remarkably the paper. The authors wish to thank Paul Jenkins for useful discussions on the use of age-ordered random partitions in population genetics. Federico Camerlenghi is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Stefano Favaro is also affiliated to IMATI-CNR “Enrico Magenes” (Milan, Italy).
Citation
Cecilia Balocchi. Federico Camerlenghi. Stefano Favaro. "A Bayesian Nonparametric Approach to Species Sampling Problems with Ordering." Bayesian Anal. Advance Publication 1 - 26, 2024. https://doi.org/10.1214/24-BA1418
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