Looking-backward probabilities for Gibbs-type exchangeable random partitions

TL;DR

This paper investigates the conditional distributions of old species in Gibbs-type exchangeable random partitions, enhancing understanding of species re-observation probabilities in Bayesian nonparametrics.

Contribution

It provides new results on the conditional properties of Gibbs-type partitions related to previously observed species, extending prior focus mainly on new species.

Findings

Derived explicit formulas for the distribution of re-observed species

Enhanced understanding of species sampling in Bayesian models

Extended analysis to include old species in Gibbs-type partitions

Abstract

Gibbs-type random probability measures and the exchangeable random partitions they induce represent the subject of a rich and active literature. They provide a probabilistic framework for a wide range of theoretical and applied problems that are typically referred to as species sampling problems. In this paper, we consider the class of looking-backward species sampling problems introduced in Lijoi et al. (Ann. Appl. Probab. 18 (2008) 1519-1547) in Bayesian nonparametrics. Specifically, given some information on the random partition induced by an initial sample from a Gibbs-type random probability measure, we study the conditional distributions of statistics related to the old species, namely those species detected in the initial sample and possibly re-observed in an additional sample. The proposed results contribute to the analysis of conditional properties of Gibbs-type exchangeable random partitions, so far focused mainly on statistics related to those species generated by the additional sample and not already detected in the initial sample.