Conditional formulae for Gibbs-type exchangeable random partitions

TL;DR

This paper studies the properties of Gibbs-type exchangeable random partitions, deriving explicit formulas for their distributions and moments, with applications to species sampling and genomic data analysis.

Contribution

It provides new explicit expressions for the distributions and moments of Gibbs-type partitions, enhancing Bayesian nonparametric methods for species estimation.

Findings

Explicit formulas for distributions and moments of Gibbs-type partitions

Asymptotic behaviors of block counts derived

Application to genomic data for species estimation

Abstract

Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs-type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviors. Such expressions can be interpreted as Bayesian nonparametric estimators of the rare species variety and their performance is tested on some real genomic data.