Reversible Jump MCMC in mixtures of normal distributions with the same component means

The Bayesian estimation of a special case of mixtures of normal distributions with an unknown number of components is considered. More specifically, the case where some components may have identical means is studied. The standard Reversible Jump MCMC algorithm for the estimation of a normal mixture model consisting of components with distinct parameters naturally fails to give precise results in the case where (at least) two of the mixture components have equal means. In particular, this algorithm either tends to combine such components resulting in a posterior distribution for their number having mode at a model with fewer components than those of the true one, or overestimates the number of components. This problem is overcome by defining-for every number of components-models with different number of parameters and introducing a new move type that bridges these competing models. The proposed method is applied in conjunction with suitable modifications of the standard split-combine and birth-death moves for updating the number of components. The method is illustrated by using two simulated datasets and the well-known galaxy dataset.