Calculating posterior distributions and modal estimates in Markov mixture models

This paper is concerned with finite mixture models in which the populations from one observation to the next are selected according to an unobserved Markov process. A new, full Bayesian approach based on the method of Gibbs sampling is developed. Calculations are simplified by data augmentation, achieved by introducing a population index variable into the list of unknown parameters. It is shown that the latent variables, one for each observation, can be simulated from their joint distribution given the data and the remaining parameters. This result serves to accelerate the convergence of the Gibbs sample, Modal estimates are also computed by stochastic versions of the EM algorithm. These provide an alternative to a full Bayesian approach and to existing methods of locating the maximum likelihood estimate. The ideas are applied in detail to Poisson data, mixtures of multivariate normal distributions, and autoregressive time series.