Abstract: The naive importance sampling estimator, based on samples from a single importance density, can be numerically unstable. We consider generalized importance sampling estimators where samples from more than one probability distribution are combined. We study this problem in the Markov chain Monte Carlo context, where independent samples are replaced with Markov chain samples. If the chains converge to their respective target distributions at a polynomial rate, then under two finite moment conditions, we show a central limit theorem holds for the generalized estimators. We develop an easy-to-implement method to calculate valid asymptotic standard errors based on batch means. We provide a batch means estimator for calculating asymptotically valid standard errors of Geyer's (1994) reverse logistic estimator. We illustrate the method via three examples. In particular, the generalized importance sampling estimator is used for Bayesian spatial modeling of binary data and to perform empirical Bayes variable selection where the batch means estimator enables standard error calculations in high-dimensional settings.

Key words and phrases: Bayes factors, Markov chain Monte Carlo, polynomial ergodicity, ratios of normalizing constants, reverse logistic estimator.