Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models

We present some easy-to-construct random probability measures which approximate the Dirichlet process and an extension which we will call the beta two-parameter process. The nature of these constructions makes it simple to implement Markov chain Monte Carlo algorithms for fitting nonparametric hierarchical models and mixtures of nonparametric hierarchical models. For the Dirichlet process,we consider a truncation approximation as well as a weak limit approximation based on a mixture of Dirichlet processes. The same type of truncation approximation can also be applied to the beta two-parameter process. Both methods lead to posteriors which can be fitted using Markov chain Monte Carlo algorithms that take advantage of blocked coordinate updates. These algorithms promote rapid mixing of the Markov chain and can be readily applied to normal mean mixture models and to density estimation problems. We prefer the truncation approximations, since a simple device for monitoring the adequacy of the approximation can be easily computed from the output of the Gibbs sampler. Furthermore, for the Dirichlet process, the truncation approximation offers an exponentially higher degree of accuracy over the weak limit approximation for the same computational effort. We also find that a certain beta two-parameter process may be suitable for finite mixture modelling because the distinct number of sampled values from this process tends to match closely the number of components of the underlying mixture distribution.