Bayesian nonparametric inference beyond the Gibbs-type framework

The definition and investigation of general classes of nonparametric priors has recently been an active research line in Bayesian statistics. Among the various proposals, the Gibbs-type family, which includes the Dirichlet process as a special case, stands out as the most tractable class of nonparametric priors for exchangeable sequences of observations. This is the consequence of a key simplifying assumption on the learning mechanism, which, however, has justification except that of ensuring mathematical tractability. In this paper, we remove such an assumption and investigate a general class of random probability measures going beyond the Gibbs-type framework. More specifically, we present a nonparametric hierarchical structure based on transformations of completely random measures, which extends the popular hierarchical Dirichlet process. This class of priors preserves a good degree of tractability, given that we are able to determine the fundamental quantities for Bayesian inference. In particular, we derive the induced partition structure and the prediction rules and characterize the posterior distribution. These theoretical results are also crucial to devise both a marginal and a conditional algorithm for posterior inference. An illustration concerning prediction in genomic sequencing is also provided.