Safe and effective importance sampling

We present two improvements on the technique of importance sampling. First, we show that importance sampling from a mixture of densities, using those densities as control variates, results in a useful upper bound on the asymp totic variance. That bound is a small multiple of the asymptotic variance o f importance sampling from the best single component density. This allows o ne to benefit from the great variance reductions obtainable by importance s ampling, while protecting against the equally great variance increases that might take the practitioner by surprise. The second improvement is to show how importance sampling from two Dr more densities can be used to approach a zero sampling variance even for integrands that take both positive acid negative values.