Adaptive importance sampling in monte carlo integration

Abstract

An Adaptive Importance Sampling (AIS) scheme is introduced to compute integrals of the form as a mechanical, yet flexible, way of dealing with the selection of parameters of the importance function. AIS starts with a rough estimate for the parameters λ of the importance function g, and runs importance sampling in an iterative way to continually update λ using only linear accumulation. Consistency of AIS is established. The efficiency of the algorithm is studied in three examples and found to be substantially superior to ordinary importance sampling.

Keywords:

• Monte Carlo integration
• basic importance sampling
• adaptive importance sampling
• importance function
• linear accumulation
• martingale limit theorem
• approximate normality

^*Research was supported by the National Science Foundation, grants DMS-8702620 and DMS-8717799, and by a David Ross Fellowship from Purdue University, as part of the first author's Ph.D.thesis. Parts of the work were also done while the authors were visiting at Duke University. The authorsthank Arup Bose for his help with the convergence proofs.

^*Research was supported by the National Science Foundation, grants DMS-8702620 and DMS-8717799, and by a David Ross Fellowship from Purdue University, as part of the first author's Ph.D.thesis. Parts of the work were also done while the authors were visiting at Duke University. The authorsthank Arup Bose for his help with the convergence proofs.

Notes

^*Research was supported by the National Science Foundation, grants DMS-8702620 and DMS-8717799, and by a David Ross Fellowship from Purdue University, as part of the first author's Ph.D.thesis. Parts of the work were also done while the authors were visiting at Duke University. The authorsthank Arup Bose for his help with the convergence proofs.