Metropolized independent sampling with comparisons to rejection sampling and importance sampling

Although Markov chain Monte Carlo methods have been widely used in many disciplines, exact eigen analysis for such generated chains has been rare. In this paper, a special Metropolis-Hastings algorithm, Metropolized independent sampling, proposed first in Hastings (1970), is studied in full detail. The eigenvalues and eigenvectors of the corresponding Markov chain, as well as a sharp bound for the total variation distance between the nth updated distribution and the target distribution, are provided. Furthermore, the relationship between this scheme, rejection sampling, and importance sampling are studied with emphasis on their relative efficiencies. It is shown that Metropolized independent sampling is superior to rejection sampling in two respects: asymptotic efficiency and ease of computation.