Estimates and Standard Errors for Ratios of Normalizing Constants from Multiple Markov Chains via Regeneration

In the classical biased sampling problem, we have k densities π₁(·), …, π (·), each known up to a normalizing constant, i.e. for l = 1, …, k, π (·) = ν (·)/m , where ν (·) is a known function and m is an unknown constant. For each l, we have an iid sample from π ,·and the problem is to estimate the ratios m for all l and all s. This problem arises frequently in several situations in both frequentist and Bayesian inference. An estimate of the ratios was developed and studied by Vardi and his co-workers over two decades ago, and there has been much subsequent work on this problem from many different perspectives. In spite of this, there are no rigorous results in the literature on how to estimate the standard error of the estimate. We present a class of estimates of the ratios of normalizing constants that are appropriate for the case where the samples from the π 's are not necessarily iid sequences, but are Markov chains. We also develop an approach based on regenerative simulation for obtaining standard errors for the estimates of ratios of normalizing constants. These standard error estimates are valid for both the iid case and the Markov chain case.