On a Metropolis-Hastings importance sampling estimator

TL;DR

This paper introduces a new Metropolis-Hastings importance sampling estimator that leverages all proposed states, providing theoretical guarantees and potentially outperforming traditional methods in moderate dimensions.

Contribution

It proposes a novel MH importance sampling estimator that incorporates rejected proposals, with proven convergence properties and explicit error bounds, differing from classical MH estimators.

Findings

Estimator satisfies strong law of large numbers and CLT.

Asymptotic variance is free of correlation terms.

Numerical experiments show improved performance in moderate dimensions.

Abstract

A classical approach for approximating expectations of functions w.r.t. partially known distributions is to compute the average of function values along a trajectory of a Metropolis-Hastings (MH) Markov chain. A key part in the MH algorithm is a suitable acceptance/rejection of a proposed state, which ensures the correct stationary distribution of the resulting Markov chain. However, the rejection of proposals causes highly correlated samples. In particular, when a state is rejected it is not taken any further into account. In contrast to that we consider a MH importance sampling estimator which explicitly incorporates all proposed states generated by the MH algorithm. The estimator satisfies a strong law of large numbers as well as a central limit theorem, and, in addition to that, we provide an explicit mean squared error bound. Remarkably, the asymptotic variance of the MH importance sampling estimator does not involve any correlation term in contrast to its classical counterpart. Moreover, although the analyzed estimator uses the same amount of information as the classical MH estimator, it can outperform the latter in scenarios of moderate dimensions as indicated by numerical experiments.