Sufficientness postulates for Gibbs-type priors and hierarchical generalizations

A fundamental problem in Bayesian nonparametrics consists of selecting a prior distribution by assuming that the corresponding predictive probabilities obey certain properties. An early discussion of such a problem, although in a parametric framework, dates back to the seminal work by English philosopher W. E. Johnson, who introduced a noteworthy characterization for the predictive probabilities of the symmetric Dirichlet prior distribution. This is typically referred to as Johnson's "sufficientness" postulate. In this paper we review some nonparametric generalizations of Johnson's postulate for a class of nonparametric priors known as species sampling models. In particular we revisit and discuss the "sufficientness" postulate for the two parameter Poisson-Dirichlet prior within the more general framework of Gibbstype priors and their hierarchical generalizations.