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Honest Importance Sampling with Multiple Markov Chains.

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Summary
This summary is machine-generated.

Importance sampling, a Monte Carlo method, estimates expectations using samples from different probability densities. This study introduces regenerative simulation for more reliable standard error estimation in Markov chain Monte Carlo (MCMC) importance sampling.

Keywords:
Geometric ergodicityMarkov chain Monte Carloimportance samplingregenerative simulationstandard errors

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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.