This paper deals with both exploration and interpretation problems related to posterior distributions for mixture models. The specification of mixture posterior distributions means that the presence of k! modes is known immediately. Standard Markov chain Monte Carlo techniques usually have difficulties with well-separated modes such as occur here; the Markov chain Monte Carlo sampler stays within a neighbourhood of a local mode and fails to visit other equally important modes. We show that exploration of these modes can be imposed on the Markov chain Monte Carlo sampler using tempered transitions based on Langevin algorithms. However, as the prior distribution does not distinguish between the different components, the posterior mixture distribution is symmetric and thus standard estimators such as posterior means cannot be used. Since this is also true for most non-symmetric priors, we propose alternatives for Bayesian inference for permutation invariant posteriors, including a cluster...

In this paper, we show how Bayesian inference for switching regression models and their generalisations can be achieved by the specification of loss functions which overcome the label switching problem common to all mixture models. We also derive an extension to models where the number of components in the mixture is unknown, based on the birthand -death technique developed in Stephens (2000a). The methods are illustrated on various real datasets.

this paper. The reason for the paradoxical complexity of the mixture model is due to the product structure of the likelihood function, L(` 1 ; : : : ; ` k jx 1 ; : : : ; xn ) =

There is increasing need for efficient estimation of mixture distributions, especially following the explosion in the use of these as modelling tools in many applied fields. We propose in this paper a Bayesian noninformative approach for the estimation of normal mixtures which relies on a reparameterisation of the secondary components of the mixture in terms of divergence from the main component. As well as providing an intuitively appealing representation at the modelling stage, this reparameterisation has important bearing on both the prior distribution and the performance of MCMC algorithms. We compare two possible reparameterisations extending Mengersen and Robert (1996) and show that the reparameterisation which does not link the secondary components together is associated with poor convergence properties of MCMC algorithms. Keywords: Bayesian inference; Convergence of Markov chains; Gibbs sampling; Identifiability; Noninformative prior; Normal distribution. 1. Introduction The ...