Fast Markov chain Monte Carlo in approximate Dirichlet and beta-Stacy process hierarchical models

Abstract


I will present some easy to construct random probability measures which approximate the Dirichlet and beta-Stacy processes. The simplicity of these constructions makes it possible to implement fast Markov chain Monte Carlo (MCMC) algorithms for fitting nonparametric hierarchical models (NPHM) and mixtures of nonparametric hierarchical models (MNPHM). For the Dirichlet process, I will 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 versions of the beta-Stacy process. Both methods lead to posteriors which can be fit using MCMC algorithms that take advantage of multivariate 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. The truncation approximations appear to be preferable, 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 amount of computation.

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