Markov Chain Monte Carlo Methods and the Label Switching Problem in Bayesian Mixture Modelling

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.