Adaptive importance sampling in monte carlo integration

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.