Transitional Markov chain Monte Carlo: Observations and improvements

The Transitional Markov chain Monte Carlo (TMCMC) method is a widely used method for Bayesian updating and Bayesian model class selection. The method is based on successively sampling from a sequence of distributions that gradually approach the posterior target distribution. The samples of the intermediate distributions are used to obtain an estimate of the evidence, which is needed in the context of Bayesian model class selection. The properties of the TMCMC method are discussed and the following three modifications to the TMCMC method are proposed: (1) The sample weights should be adjusted after each MCMC step; (2) a burn-in period in the MCMC sampling step can improve the posterior approximation; and (3) the scale of the proposal distribution of the MCMC algorithm can be selected adaptively to achieve a near-optimal acceptance rate. The performance of the proposed modifications is compared with the original TMCMC method by means of three example problems. The proposed modifications reduce the bias in the estimate of the evidence, and improve the convergence behavior of posterior estimates.