Importance is Important: Generalized Markov Chain Importance Sampling Methods

We show that for any multiple-try Metropolis algorithm, one can always accept the proposal and evaluate the importance weight that is needed to correct for the bias without extra computational cost. This results in a general, convenient, and rejection-free Markov chain Monte Carlo (MCMC) sampling scheme. By further leveraging the importance sampling perspective on Metropolis–Hastings algorithms, we propose an alternative MCMC sampler on discrete spaces that is also outside the Metropolis–Hastings framework, along with a general theory on its complexity. Numerical examples suggest that the proposed algorithms are consistently more efficient than the original Metropolis–Hastings versions.