Abstract
Regeneration is a useful tool in Markov chain Monte Carlo simulation because it can be used to side-step the burn-in problem and to construct better estimates of the variance of parameter estimates themselves. It also provides a simple way to introduce adaptive behavior into a Markov chain, and to use parallel processors to build a single chain. Regeneration is often difficult to take advantage of because, for most chains, no recurrent proper atom exists, and it is not always easy to use Nummelin's splitting method to identify regeneration times. This article describes a constructive method for generating a Markov chain with a specified target distribution and identifying regeneration times. As a special case of the method, an algorithm which can be “wrapped” around an existing Markov transition kernel is given. In addition, a specific rule for adapting the transition kernel at regeneration times is introduced, which gradually replaces the original transition kernel with an independence-sampling Metropolis-Hastings kernel using a mixture normal approximation to the target density as its proposal density. Computational gains for the regenerative adaptive algorithm are demonstrated in examples.
Highlights
Markov chain Monte Carlo (MCMC) methods have become popular in the last decade as a tool for exploring properties of distributions which are known only up to a constant of proportionality
We present an alternative way of identifying regeneration times, which relies on constructing a Markov chain on an enlarged state-space
In addition to presenting this “constructive” method for identification of regeneration times, we describe how an existing transition kernel can be embedded into a new kernel for which identification of regeneration times is trivial
Summary
Markov chain Monte Carlo (MCMC) methods have become popular in the last decade as a tool for exploring properties of distributions which are known only up to a constant of proportionality. There are several important problems associated with standard methods of constructing Markov chains for Monte Carlo simulation. One does not generally know how long it takes before the chain is (by some measure) sufficiently close to its limiting distribution. A second problem is the inherent correlation between successive elements of the chain, which makes it difficult to estimate the variance of the Monte Carlo estimates. Errors in integrals approximated by MCMC simulation are approximately proportional to the inverse of the square root of the length of the chain, Markov chains often need to be relatively long to give accurate estimates of integrals
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