  The naive importance sampling estimator, based on samples from a single
importance density, can be numerically unstable. Instead, 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. Further, we develop
an easy to implement method to calculate valid asymptotic standard errors based
on batch means. We also provide a batch means estimator for calculating
asymptotically valid standard errors of Geyer(1994) reverse logistic estimator.
We illustrate the method using a Bayesian variable selection procedure in
linear regression. In particular, the generalized importance sampling estimator
is used to perform empirical Bayes variable selection and the batch means
estimator is used to obtain standard errors in a high-dimensional setting where
current methods are not applicable.
