Gibbs sampling methods for stick-breaking priors

A rich and flexible class of random probability measures, which we call sti ck-breaking pliers, can be constructed using a sequence of independent beta random variables. Examples of random measures that have this characterizat ion include the Dirichlet process, its two-parameter extension, the two-par ameter Poisson-Dirichlet process, finite dimensional Dirichlet priors, and beta two-parameter processes. The rich nature of stick-breaking priors offe rs Bayesians a useful class of priors for nonparametric problems, while the similar construction used in each prior can be exploited to develop a gene ral computational procedure for fitting them. In this article we present tw o general types of Gibbs samplers that can be used to lit posteriors of Bay esian hierarchical models based on stick-breaking priors. The first type of Gibbs sampler, referred to as a Polya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling method currently employed for Diric hlet process computing. This method applies to stick-breaking priors with a known Polya urn characterization, that is, priors with an explicit and sim ple prediction rule. Our second method, the blocked Gibbs sampler, is based on an entirely different approach that works by directly sampling values f rom the posterior of the random measure. The blocked Gibbs sampler can be v iewed as a more general approach because it works without requiring an expl icit prediction rule. We find that the blocked Gibbs avoids some of the lim itations seen with the Polya urn approach and should be simpler for nonexpe rts to use.