Importance Sampling methods are broadly used to approximate posterior
distributions or some of their moments. In its standard approach, samples are
drawn from a single proposal distribution and weighted properly. However, since
the performance depends on the mismatch between the targeted and the proposal
distributions, several proposal densities are often employed for the generation
of samples. Under this Multiple Importance Sampling (MIS) scenario, many works
have addressed the selection or adaptation of the proposal distributions,
interpreting the sampling and the weighting steps in different ways. In this
paper, we establish a general framework for sampling and weighing procedures
when more than one proposal are available. The most relevant MIS schemes in the
literature are encompassed within the new framework, and, moreover novel valid
schemes appear naturally. All the MIS schemes are compared and ranked in terms
of the variance of the associated estimators. Finally, we provide illustrative
examples which reveal that, even with a good choice of the proposal densities,
a careful interpretation of the sampling and weighting procedures can make a
significant difference in the performance of the method.

本研究在多个 proposal 分布下建立了一个新的 Importance Sampling 方法，支持多个 proposal 下的权重分配并可以通过优化 proposal 分布的选择来提高算法效率。