Markov chain Monte Carlo (MCMC) sampling strategies can be used to simulate hidden Markov model (HMM) parameters from their posterior distribution given observed data. Some MCMC methods (for computing likelihood, conditional probabilities of hidden states, and the most likely sequence of states) used in practice can be improved by incorporating established recursive algorithms. The most important is a set of forward-backward recursions calculating conditional distributions of the hidden states given observed data and model parameters. We show how to use the recursive algorithms in an MCMC context and demonstrate mathematical and empirical results showing a Gibbs sampler using the forward-backward recursions mixes more rapidly than another sampler often used for HMM's. We introduce an augmented variables technique for obtaining unique state labels in HMM's and finite mixture models. We show how recursive computing allows statistically efficient use of MCMC output when estimating the hidden states. We directly calculate the posterior distribution of the hidden chain's state space size by MCMC, circumventing asymptotic arguments underlying the Bayesian information criterion, which is shown to be inappropriate for a frequently analyzed data set in the HMM literature. The use of log-likelihood for assessing MCMC convergence is illustrated, and posterior predictive checks are used to investigate application specific questions of model adequacy.

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The last ten years have witnessed the development of sampling frameworks that permit the construction of Markov chains which simultaneously traverse both parameter and model space. In this time substantial methodological progress has been made. In this article we present a survey of the current state of the art and evaluate some of the most recent advances in this field. We also discuss future research perspectives in the context of the drive to develop sampling mechanisms with high degrees of both efficiency and automation. 1

We present full Bayesian analysis of finite mixtures of multivariate normals with unknown number of components. We adopt reversible jump Markov chain Monte Carlo and we construct, in a manner similar to that of Richardson and Green (1997), split and merge moves that produce good mixing of the Markov chains. The split moves are constructed on the space of eigenvectors and eigenvalues of the current covariance matrix so that the proposed covariance matrices are positive definite. Our proposed methodology has applications in classification and discrimination as well as heterogeneity modelling. We test our algorithm with real and simulated data.

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Abstract. In the past ten years there has been a dramatic increase of interest in the Bayesian analysis of finite mixture models. This is primarily because of the emergence of Markov chain Monte Carlo (MCMC) methods. Whilst MCMC provides a convenient way to draw inference from complicated statistical models, there are many, perhaps under appreciated, problems associated with the MCMC analysis of mixtures. The problems are mainly caused by the nonidentifiability of the components under symmetric priors, which leads to so called label switching in the MCMC output. This will mean that ergodic averages of component specific quantities will be identical and thus useless for inference. We review the solutions to the label switching problem, such as artificial identifiability constraints (e.g. Diebolt & Robert (1994)), relabelling algorithms (Stephens 1997a) and label invariant loss functions (Celeux, Hurn & Robert 2000). We also review various MCMC sampling schemes that have been suggested for mixture models and discuss posterior sensitivity to prior specification.

Mixture models have very wide application. However, the problem of determining the number of components, K say, is one of the challenging problems in this area. There are several approaches in the literature for testing K for different values. A recent alternative approach is to treat K as unknown, and to model K and the mix- ture component parameters jointly; see for example Richardson and Green (1997). In this paper, we propose an approach based on a birth-death process. In contrast with the approach in Stephens(2000), we make use of the latent indicators so that the approach can be used to solve the problems with missing data when calculation of the likelihood requires knowledge of the unobservable latent variables. Specifically, we use the method to analyse hidden Markov models with unknown numbers of states. The model and the algorithm are illustrated by some real examples.

This paper provides a brief review of the more popular methods for comparing models in a Bayesian framework. Personal experience in implementing these methods in problems requiring mixture models is also referenced. 1