Exact Transition Probabilities for the Independence Metropolis Sampler

A recent result of Jun Liu's has shown how to compute explicitly the eigen-values and eigenvectors for the Markov chain derived from a special case of the Hastings sampling algorithm, known as the indepdendence Metropolis sampler. In this note, we show how to extend the result to obtain exact n-step transition probabilities for any n. This is done rst for a chain on a nite state space, and then extended to a general (discrete or continuous) state space. The paper concludes with some implications for diagnostic tests of convergence of Markov chain samplers.