Posterior Consistency of Species Sampling Priors

Recently there has been increasing interest in species sampling priors, the nonparametric priors defined as the directing random probability measures of the species sampling sequences. In this paper, we show that not all of the species sampling priors produce consistent posteriors. In particular, in the class of Pitman-Yor process priors, the only priors rendering posterior consistency are essentially the Dirichlet process priors. Under certain conditions, we also give a set of necessary and sufficient conditions for the posterior consistency for the general species sampling prior. Considered examples include the normalized inverse-Gaussian process, the Poisson-Kingman partition, and the Gibbs partition.