Weak Consistency of Markov Chain Monte Carlo Methods

We study poor behavior of Markov chain Monte Carlo methods in large sample framework. We define weak consistency to measure the convergence rate of Monte Carlo procedure. This property is studied by convergence of step Markov process to a diffusion process. We apply weak consistency to a popular data augmentation for simple mixture model. The Monte Carlo method is known to work poorly when one of a mixture proportion is close to 0. We show that it is not (local) consistent but (local) weak consistent. As an alternative, we propose a Metropolis-Hastings algorithm which is local consistent for the same model. These results come from a weak convergence property of Monte Carlo procedures which is difficult to obtain from Harris recurrence approach.