Honest Importance Sampling with Multiple Markov Chains

Area of Science:

• Computational Statistics
• Monte Carlo Methods
• Markov Chain Monte Carlo (MCMC)

Background:

• Importance sampling is a Monte Carlo technique for estimating expectations using samples from a different probability density.
• While standard importance sampling has established theory (consistency, central limit theorem), its application within Markov chain Monte Carlo (MCMC) introduces complexities in standard error estimation.
• Existing MCMC importance sampling methods face challenges with central limit theorem conditions and consistent estimation of asymptotic variance.

Purpose of the Study:

• To develop a method for overcoming the complexities of standard error estimation in MCMC importance sampling.
• To establish a central limit theorem (CLT) for multiple-chain importance sampling estimators based on regenerative simulation.
• To provide a simple, consistent estimator for the asymptotic variance in the MCMC importance sampling context.

Main Methods:

• Utilized regenerative simulation to address challenges in MCMC importance sampling.
• Developed multiple-chain importance sampling estimators assuming availability of Markov chain samples from several probability densities.
• Established a CLT for these estimators under geometric convergence rates of Markov chains to their target distributions.

Main Results:

• The proposed multiple-chain importance sampling estimators obey a CLT based on regeneration.
• Under geometric convergence, moment conditions similar to the independent and identically distributed (i.i.d.) case are sufficient for the CLT.
• A simple, consistent estimator for the asymptotic variance is available due to the regenerative nature of the CLT.

Conclusions:

• Regenerative simulation effectively resolves standard error estimation issues in MCMC importance sampling.
• The method provides a robust theoretical framework (CLT) and practical tools for variance estimation.
• Applications in Bayesian sensitivity analysis, including random effects models and variable selection, demonstrate the utility of multiple-chain importance sampling.