A rich and flexible class of random probability measures, which we call stick-breaking priors, can be constructed using a sequence of independent beta random variables. Examples of random measures that have this characterization include the Dirichlet process, its two-parameter extension, the two-parameter Poisson–Dirichlet process, finite dimensional Dirichlet priors, and beta two-parameter processes. The rich nature of stick-breaking priors offers Bayesians a useful class of priors for nonparametric problems, while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them. In this article we present two general types of Gibbs samplers that can be used to fit posteriors of Bayesian hierarchical models based on stick-breaking priors. The first type of Gibbs sampler, referred to as a Pólya urn Gibbs sampler, is a generalized version of a widely used Gibbs sampling method currently employed for Dirichlet process computing. This method applies t...

A rich nonparametric analysis of the finite normal mixture model is obtained by working with a precise truncation approximation of the Dirichlet process. Model fitting is carried out by a simple Gi...

Probability Surveys publishes survey articles in theoretical and applied probability. The style of articles may range from reviews of recent research to graduate textbook exposition. Articles may be broad or narrow in scope. The essential requirements are a well specified topic and target audience, together with clear exposition.

We consider the problem of estimating the number of components d and the unknown mixing distribution in a finite mixture model, in which d is bounded by some fixed finite number N. Our approach relies on the use of a prior over the space of mixing distributions with at most N components. By decomposing the resulting marginal density under this prior, we discover a weighted Bayes factor method for consistently estimating d that can be implemented by an iid generalized weighted Chinese restaurant (GWCR) Monte Carlo algorithm. We also discuss a Gibbs sampling method (the blocked Gibbs sampler) for estimating d and also the mixing distribution. We show that our resulting posterior is consistent and achieves the frequentist optimal Op (n−1/4) rate of estimation. We compare the performance of the new GWCR model selection procedure with that of the Akaike information criterion and the Bayes information criterion implemented through an EM algorithm. Applications of our methods to five real datasets and simulation...

This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailor-made to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,…,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12 (1984) 351–357] and [Ann. Inst. Statist. Math. 41 (1989) 227–245]. In order to illustrate the flexibility of the approach, large classes of random probability measures and random hazards or intensities which can be expressed as functionals of Poisson random measures are described. We describe a unified posterior analysis of classes of discrete random probability which identifies and exploits features common to all these models. The analysis circumvents many of the difficult issues involved in Bayesian nonparametric calculus, including a combinatorial component. This allows one to focus on the unique features of each process which are characterized via real valued functions h. The applicability of the technique is further illustrated by obtaining explicit posterior expressions for Lévy–Cox moving average processes within the general setting of multiplicative intensity models. In addition, novel computational procedures, similar to efficient procedures developed for the Dirichlet process, are briefly discussed for these models.