  Discrete random probability measures and the exchangeable random partitions
they induce are key tools for addressing a variety of estimation and prediction
problems in Bayesian inference. Indeed, many popular nonparametric priors, such
as the Dirichlet and the Pitman-Yor process priors, select discrete probability
distributions almost surely and, therefore, automatically induce exchangeable
random partitions. Here we focus on the family of Gibbs-type priors, a recent
and elegant generalization of the Dirichlet and the Pitman-Yor process priors.
These random probability measures share properties that are appealing both from
a theoretical and an applied point of view: (i) they admit an intuitive
characterization in terms of their predictive structure justifying their use in
terms of a precise assumption on the learning mechanism; (ii) they stand out in
terms of mathematical tractability; (iii) they include several interesting
special cases besides the Dirichlet and the Pitman-Yor processes. The goal of
our paper is to provide a systematic and unified treatment of Gibbs-type priors
and highlight their implications for Bayesian nonparametric inference. We will
deal with their distributional properties, the resulting estimators,
frequentist asymptotic validation and the construction of time-dependent
versions. Applications, mainly concerning hierarchical mixture models and
species sampling, will serve to convey the main ideas. The intuition inherent
to this class of priors and the neat results that can be deduced for it lead
one to wonder whether it actually represents the most natural generalization of
the Dirichlet process.
