Posterior consistency of species sampling priors

Recently there have been increasing interests on 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. For the general species sampling priors, we also give a necessary and sufficient condition for the posterior consistency.