  One of the main research areas in Bayesian Nonparametrics is the proposal and
study of priors which generalize the Dirichlet process. Here we exploit
theoretical properties of Poisson random measures in order to provide a
comprehensive Bayesian analysis of random probabilities which are obtained by
an appropriate normalization. Specifically we achieve explicit and tractable
forms of the posterior and the marginal distributions, including an explicit
and easily used description of generalizations of the important
Blackwell-MacQueen P\'olya urn distribution. Such simplifications are achieved
by the use of a latent variable which admits quite interesting interpretations
which allow to gain a better understanding of the behaviour of these random
probability measures. It is noteworthy that these models are generalizations of
models considered by Kingman (1975) in a non-Bayesian context. Such models are
known to play a significant role in a variety of applications including
genetics, physics, and work involving random mappings and assemblies. Hence our
analysis is of utility in those contexts as well. We also show how our results
may be applied to Bayesian mixture models and describe computational schemes
which are generalizations of known efficient methods for the case of the
Dirichlet process. We illustrate new examples of processes which can play the
role of priors for Bayesian nonparametric inference and finally point out some
interesting connections with the theory of generalized gamma convolutions
initiated by Thorin and further developed by Bondesson.
