Defining Predictive Probability Functions for Species Sampling Models

TL;DR

This paper explores the relationship between predictive probability functions and exchangeable partition probability functions in species sampling models, providing new conditions for their correspondence and methods for inference.

Contribution

It introduces novel conditions for when a putative PPF can define an EPPF and discusses inference techniques for complex PPFs not linear in cluster size.

Findings

All PPFs in a certain class must define probabilities linear in cluster size.

Provides a new necessary and sufficient condition for PPFs to define an EPPF.

Discusses a numerical method for deriving PPFs for complex models.

Abstract

We review the class of species sampling models (SSM). In particular, we investigate the relation between the exchangeable partition probability function (EPPF) and the predictive probability function (PPF). It is straightforward to define a PPF from an EPPF, but the converse is not necessarily true. In this paper we introduce the notion of putative PPFs and show novel conditions for a putative PPF to define an EPPF. We show that all possible PPFs in a certain class have to define (unnormalized) probabilities for cluster membership that are linear in cluster size. We give a new necessary and sufficient condition for arbitrary putative PPFs to define an EPPF. Finally, we show posterior inference for a large class of SSMs with a PPF that is not linear in cluster size and discuss a numerical method to derive its PPF.