From Minimax Optimal Importance Sampling to Uniformly Ergodic Importance-tempered MCMC

We make two closely related theoretical contributions to the use of importance sampling schemes. First, for independent sampling, we prove that the minimax optimal trial distribution coincides with the target if and only if the target distribution has no atom with probability greater than $1/2$, where "minimax" means that the worst-case asymptotic variance of the self-normalized importance sampling estimator is minimized. When a large atom exists, it should be downweighted by the trial distribution. A similar phenomenon holds for a continuous target distribution concentrated on a small set. Second, we argue that it is often advantageous to run the Metropolis--Hastings algorithm with a tempered stationary distribution, $\pi(x)^\beta$, and correct for the bias by importance weighting. The dynamics of this "importance-tempered" sampling scheme can be described by a continuous-time Markov chain. We prove that for one-dimensional targets with polynomial tails, $\pi(x) \propto (1 + |x|)^{-\gamma}$, this chain is uniformly ergodic if and only if $1/\gamma < \beta < (\gamma - 2)/\gamma$. These results suggest that for target distributions with light or polynomial tails of order $\gamma > 3$, importance tempering can improve the precision of time-average estimators and essentially eliminate the need for burn-in.