Importance is important: generalized Markov chain importance sampling methods

We show that for any correlated 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 class of convenient and rejection-free Markov chain Monte Carlo (MCMC) sampling schemes. By further leveraging the importance sampling perspective on Metropolis–Hastings algorithms, we propose an alternative MCMC sampler on discrete spaces along with a general theory on its complexity. Both proposed methods can be viewed as using importance weighting to overcome the inefficiency of the Metropolis–Hastings framework caused by repeated proposal rejections. Numerical examples suggest that the proposed algorithms are consistently more efficient than their Metropolis–Hastings counterparts.