Random Discrete Probability Measures Based on Negative Binomial Process

TL;DR

This paper introduces a new class of random discrete probability measures based on the negative binomial process, enhancing flexibility in nonparametric Bayesian modeling and relating to existing priors like Dirichlet and Poisson-Dirichlet processes.

Contribution

It develops a generalized Poisson-Kingman distribution using the negative binomial process, providing new priors and series representations that connect with established Bayesian nonparametric models.

Findings

Introduces a new flexible class of priors for Bayesian nonparametrics.

Derives a novel series representation for the Poisson-Dirichlet process.

Establishes relationships between the new measures and existing Bayesian priors.

Abstract

An important functional of Poisson random measure is the negative binomial process (NBP). We use NBP to introduce a generalized Poisson-Kingman distribution and its corresponding random discrete probability measure. This random discrete probability measure provides a new set of priors with more flexibility in nonparametric Bayesian models. It is shown how this random discrete probability measure relates to the non-parametric Bayesian priors such as Dirichlet process, normalized positive {\alpha}-stable process, Poisson-Dirichlet process (PDP), and others. An extension of the DP with its almost sure approximation is presented. Using our representation for NBP, we derive a new series representation for the PDP.