Clustering structure for species sampling sequences with general base measure

TL;DR

This paper studies the clustering behavior of species sampling sequences with general base measures, providing new stochastic representations and explicit formulas for their partition structures, with implications for Bayesian nonparametrics.

Contribution

It introduces a stochastic representation for species sampling sequences with general base measures and derives explicit EPPF expressions, advancing understanding of their clustering properties.

Findings

Derived a stochastic representation in terms of latent exchangeable partitions.

Provided explicit formulas for the EPPF of the generated partitions.

Analyzed the asymptotic behavior of the number of clusters and their sizes.

Abstract

We investigate the clustering structure of species sampling sequences $(\xi_n)_n$, with general base measure. Such sequences are exchangeable with a species sampling random probability as directing measure. The clustering properties of these sequences are interesting for Bayesian nonparametrics applications, where mixed base measures are used, for example, to accommodate sharp hypotheses in regression problems and provide sparsity. In this paper, we prove a stochastic representation for $(\xi_n)_n$ in terms of a latent exchangeable random partition. We provide explicit expression of the EPPF of the partition generated by $(\xi_n)_n$ in terms of the EPPF of the latent partition. We investigate the asymptotic behaviour of the total number of blocks and of the number of blocks with fixed cardinality in the partition generated by $(\xi_n)_n$.