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The naive importance sampling estimator, based on samples from a singleimportance density, can be numerically unstable. Instead, we considergeneralized importance sampling estimators where samples from more than oneprobability distribution are combined. We study this problem in the Markovchain Monte Carlo context, where independent samples are replaced with Markovchain samples. If the chains converge to their respective target distributionsat a polynomial rate, then under two finite moment conditions, we show acentral limit theorem holds for the generalized estimators. Further, we developan easy to implement method to calculate valid asymptotic standard errors basedon batch means. We also provide a batch means estimator for calculatingasymptotically valid standard errors of Geyer(1994) reverse logistic estimator.We illustrate the method using a Bayesian variable selection procedure inlinear regression. In particular, the generalized importance sampling estimatoris used to perform empirical Bayes variable selection and the batch meansestimator is used to obtain standard errors in a high-dimensional setting wherecurrent methods are not applicable.

Estimating standard errors for importance sampling estimators with multiple Markov chains