Model Likelihoods and Bayes Factors for Switching and Mixture Models

In the present paper we explore various approaches of computing model likelihoods from the MCMC output for mixture and switching models, among them the candidate's formula, importance sampling, reciprocal importance sampling and bridge sampling. We demonstrate that the candidate's formula is sensitive to label switching. It turns out that the best method to estimate the model likelihood is the bridge sampling technique, where the MCMC sample is combined with an iid sample from an importance density. The importance density is constructed in an unsupervised manner from the MCMC output using a mixture of complete data posteriors. Whereas the importance sampling estimator as well as the reciprocal importance sampling estimator are sensitive to the tail behaviour of the importance density, we demonstrate that the bridge sampling estimator is far more robust in this concern. Our case studies range from from selecting the number of classes in a mixture of multivariate normal distributions, testing for the inhomogeneity of a discrete time Poisson process, to testing for the presence of Markov switching and order selection in the MSAR model. (author's abstract)