Adaptive Mixture Importance Sampling

Importance sampling involves approximation of functionals (such as expectations) of a target distribution by sampling from a design distribution. In many applications, it is natural or convenient to use a design distribution which is a mixture of given distributions. One typically has wide latitude in selecting the mixing probabilities of the design distribution. Furthermore, one can reduce variance by drawing fixed sized samples from the components of the design distribution, rather than drawing a random sample from the mixture. Here, we investigate the optimal allocation of sample sizes to each component and the optimal mixing probabilities. As the optimal choices involve typically unknown quantities, a two stage procedure is proposed which makes use of a pilot sample for estimating the allocation of sample sizes. After the final samples are available, optimal mixing probabilities can be obtained for each functional of the target distribution which is of interest. Applications are gi..