Markov Interacting Importance Samplers

TL;DR

The paper introduces the Markov Interacting Importance Sampler (MIIS), a novel MCMC method that uses importance sampling and control variates to improve efficiency and variance reduction, especially when Gibbs sampling is infeasible.

Contribution

It presents the MIIS algorithm, combining importance sampling with Markov chains, and demonstrates its advantages over traditional MCMC methods in variance reduction and efficiency.

Findings

Significantly reduces variance of Monte Carlo estimates.

Outperforms standard MCMC in Bayesian analysis.

Enhances convergence speed with MIIS random walk.

Abstract

We introduce a new Markov chain Monte Carlo (MCMC) sampler called the Markov Interacting Importance Sampler (MIIS). The MIIS sampler uses conditional importance sampling (IS) approximations to jointly sample the current state of the Markov Chain and estimate conditional expectations, possibly by incorporating a full range of variance reduction techniques. We compute Rao-Blackwellized estimates based on the conditional expectations to construct control variates for estimating expectations under the target distribution. The control variates are particularly efficient when there are substantial correlations between the variables in the target distribution, a challenging setting for MCMC. An important motivating application of MIIS occurs when the exact Gibbs sampler is not available because it is infeasible to directly simulate from the conditional distributions. In this case the MIIS method can be more efficient than a Metropolis-within-Gibbs approach. We also introduce the MIIS random walk algorithm, designed to accelerate convergence and improve upon the computational efficiency of standard random walk samplers. Simulated and empirical illustrations for Bayesian analysis show that the method significantly reduces the variance of Monte Carlo estimates compared to standard MCMC approaches, at equivalent implementation and computational effort.