Random Discrete Probability Measures Based on Negative Binomial Process

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.