  Estimating the model evidence - or mariginal likelihood of the data - is a
notoriously difficult task for finite and infinite mixture models and we
reexamine here different Monte Carlo techniques advocated in the recent
literature, as well as novel approaches based on Geyer (1994) reverse logistic
regression technique, Chib (1995) algorithm, and Sequential Monte Carlo (SMC).
Applications are numerous. In particular, testing for the number of components
in a finite mixture model or against the fit of a finite mixture model for a
given dataset has long been and still is an issue of much interest, albeit yet
missing a fully satisfactory resolution. Using a Bayes factor to find the right
number of components K in a finite mixture model is known to provide a
consistent procedure. We furthermore establish the consistence of the Bayes
factor when comparing a parametric family of finite mixtures against the
nonparametric 'strongly identifiable' Dirichlet Process Mixture (DPM) model.
