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

TL;DR

This paper provides theoretical insights into importance sampling and importance-tempered MCMC, showing conditions for optimal trial distributions and ergodicity, which can enhance sampling efficiency for certain target distributions.

Contribution

It characterizes when the minimax optimal trial distribution matches the target and analyzes the ergodicity of importance-tempered MCMC for polynomial tail distributions.

Findings

Minimax optimal trial distribution equals the target only if the target has no large atom.

Importance tempering can improve estimator precision for polynomial tail targets.

Uniform ergodicity of importance-tempered chains depends on tail decay and tempering parameter.

Abstract

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.