Random discrete probability measures based on a negative binomial process

Abstract A distinctive functional of the Poisson point process is the negative binomial process for which the increments are not independent but are independent conditional on an underlying gamma variable. Using a new point process representation for the negative binomial process, we generalize the Poisson–Kingman distribution and its corresponding random discrete probability measure. This new proposed family of discrete random probability measures, which is defined by normalizing the points of the negative binomial process, provides a new set of useful priors for Bayesian nonparametric models with more flexibility than the random discrete probability measure which are obtained by normalizing the points of a Poisson point process. We illustrate how this family of random discrete probability measures contains the nonparametric Bayesian priors such as the Dirichlet process, the normalized positive ‐stable process, the Poisson–Dirichlet process (PDP), and others. With the same gamma Lévy measure, we derive an extension of the Dirichlet process and its almost sure approximation. Using our representation for the negative binomial process, we develop a new series representation for the PDP. We demonstrate through simulations how using priors from this family can enhance the accuracy of Bayesian nonparametric hierarchical models.