Reversible Jump MCMC in mixtures of normal distributions with the same
component means
P. Papastamoulis and G. Iliopoulos
The
purpose of this paper is the Bayesian estimation of a special case of mixtures
of normal distributions with unknown number of components. More specifically,
we consider the case where some components may have identical means. The
Reversible Jump MCMC algorithm introduced by Richardson and Green (1997) 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. We overcome
this problem 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 Richardson and Green's split-combine and birth-death moves for
updating the number of components. We illustrate the method using two simulated
datasets and the well-known galaxy dataset.
Key words and phrases: Mixtures
of normal distributions; Bayesian model selection; Reversible Jump MCMC; galaxy
dataset.