Two-Stage Importance Sampling With Mixture Proposals

Abstract

For importance sampling (IS), multiple proposals can be combined to address different aspects of a target distribution. There are various methods for IS with multiple proposals, including Hesterberg's stratified IS estimator, Owen and Zhou's regression estimator, and Tan's maximum likelihood estimator. For the problem of efficiently allocating samples to different proposals, it is natural to use a pilot sample to select the mixture proportions before the actual sampling and estimation. However, most current discussions are in an empirical sense for such a two-stage procedure. In this article, we establish a theoretical framework of applying the two-stage procedure for various methods, including the asymptotic properties and the choice of the pilot sample size. By our simulation studies, these two-stage estimators can outperform estimators with naive choices of mixture proportions. Furthermore, while Owen and Zhou's and Tan's estimators are designed for estimating normalizing constants, we extend their usage and the two-stage procedure to estimating expectations and show that the improvement is still preserved in this extension.

Keywords:

• Control variates
• Normalizing constant
• Pilot samples

Acknowledgments

Chen's research is sponsored in part by NSF grants DMS 0800183, DMS 0905763, and DMS0915139. Tan's research is sponsored in part by NSF grant DMS 0749418. The authors thank the two editors, an associate editor, and two anonymous referees for their helpful comments.