Posterior distributions of Gibbs-type priors

Gibbs type priors have been shown to be natural generalizations of Dirichlet process (DP) priors used for intricate applications of Bayesian nonparametric methods. This includes applications to mixture models and to species sampling models arising in populations genetics. Notably these latter applications, and also applications where power law behavior such as that arising in natural language models are exhibited, provide instances where the DP model is wholly inadequate. Gibbs type priors include the DP, the also popular Pitman-Yor process and closely related normalized generalized gamma process as special cases. However, there is in fact a richer infinite class of such priors, where, despite knowledge about the exchangeable marginal structures produced by sampling $n$ observations, descriptions of the corresponding posterior distribution, a crucial component in Bayesian analysis, remain unknown. This paper presents descriptions of the posterior distributions for the general class, utilizing a novel proof that leverages the exclusive Gibbs properties of these models. The results are applied to several specific cases for further illustration.