Monte Carlo Integration with Markov Chain
There are two conceptually distinct parts in Markov chain Monte Carlo (MCMC): a sampler is designed for simulating a Markov chain and then an estimator is constructed on the Markov chain for computing integrals of interest. We aim to improve the second part in this article. While taking the likelihood approach of Kong et al., we basically treat the Markov chain scheme as a random design and define a stratified estimator of the baseline measure. We propose useful techniques including subsampling, regulation, and amplification for achieving overall computational efficiency. We also introduce approximate variance estimators for the point estimators. The method can yield substantially improved accuracy compared with Chib's estimator or the crude Monte Carlo estimator, as illustrated with three examples.