Accept–reject Metropolis–Hastings sampling and marginal likelihood estimation

We describe a method for estimating the marginal likelihood, based on Chib (1995) and C hib and Jeliazkov (2001) , when simulation from the posterior distribution of the model parameters is by the accept–reject Metropolis–Hastings (ARMH) algorithm. The method is developed for one-block and multiple-block ARMH algorithms and does not require the (typically) unknown normalizing constant of the proposal density. The problem of calculating the numerical standard error of the estimates is also considered and a procedure based on batch means is developed. Two examples, dealing with a multinomial logit model and a Gaussian regression model with non-conjugate priors, are provided to illustrate the efficiency and applicability of the method.