A note on convergence rates of Gibbs sampling for nonparametric mixtures.
Sonia Petrone, Gareth O. Roberts, Jeffrey S. Rosenthal · 1999
We consider a mixture model where the mixing distribution is random and is given a Dirichlet process prior. We describe the general structure of two Gibbs sampling algorithms that are useful for approximating Bayesian inferences in this problem. When the kernel f(x j `) of the mixture is bounded, we show that the Markov chains resulting from the Gibbs sampling are uniformly ergodic, and we provide an explicit rate bound. Unfortunately, the bound is not sharp in general; improving sensibly the bound seems however quite difficult. 1 Introduction. In many statistical problems it seems appropriate to specify the distribution of the data as a mixture of parametric densities, i.e. to assume that the data X i are independent and identically distributed, conditionally to a distribution function G, with density f(x j G) = R f(x j `) dG(`). The mixing distribution G is unknown and, in a Bayesian nonparametric analysis, it is considered as a random distribution function which is usually given ...