  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.
