Sufficientness postulates for Gibbs-type priors and hierarchial 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 Gibbs-type priors and their hierarchical generalizations