Parallel hierarchical sampling: a practical multiple-chains sampler for Bayesian model selection

This paper introduces the parallel hierarchical sampler (PHS), a Markov chain Monte Carlo algorithm using several chains simultaneously. The connections between PHS and the parallel tempering (PT) algorithm are illustrated, convergence of PHS joint transition kernel is proved and and its practical advantages are emphasized. We illustrate the inferences obtained using PHS, parallel tempering and the Metropolis-Hastings algorithm for three Bayesian model selection problems, namely Gaussian clustering, the selection of covariates for a linear regression model and the selection of the structure of a treed survival model.