On sample generation and weight calculation in multiple importance sampling

Importance sampling is a Monte Carlo technique that approximates moments of target densities by means of weighted samples. These samples are traditionally drawn from a single proposal density. In multiple importance sampling (MIS) a set of different proposal densities is available. In this paper, we propose a formal framework that allows different ways of drawing samples from a set of proposals and different proper weighting functions that can be applied. In particular, we describe three sampling methods and five generic weighting functions. As proper sampling/weighting combinations, six unique MIS schemes (three of them are novel) are discussed throughout the paper. All the methods are analyzed in terms of the variance of the associated estimators, establishing a ranking regarding their performance.