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。

广义多重重要性抽样

Generalized Multiple Importance Sampling

In multiple importance sampling we combine samples from a finite list of
proposal distributions. When those proposal distributions are used to create
control variates, it is possible (Owen and Zhou, 2000) to bound the ratio of
the resulting variance to that of the unknown best proposal distribution in our
list. The minimax regret arises by taking a uniform mixture of proposals, but
that is conservative when there are many components. In this paper we optimize
the mixture component sampling rates to gain further efficiency. We show that
the sampling variance of mixture importance sampling with control variates is
jointly convex in the mixture probabilities and control variate regression
coefficients. We also give a sequential importance sampling algorithm to
estimate the optimal mixture from the sample data.

本研究旨在通过优化混合成分采样率来提高多元重要性采样的效率。我们证明在控制变量回归系数和混合概率的联合情况下，多元重要性采样的采样方差是凸的，并给出了一个可用于估计样本数据中最优混合的顺序重要性采样算法。

多重重要性采样中的最优混合权重

Optimal mixture weights in multiple importance sampling

We consider various versions of adaptive Gibbs and Metropolis-within-Gibbs
samplers, which update their selection probabilities (and perhaps also their
proposal distributions) on the fly during a run by learning as they go in an
attempt to optimize the algorithm. We present a cautionary example of how even
a simple-seeming adaptive Gibbs sampler may fail to converge. We then present
various positive results guaranteeing convergence of adaptive Gibbs samplers
under certain conditions.

研究了自适应 Gibbs 和 Metropolis-within-Gibbs 采样器的各个版本，在运行过程中通过学习来动态更新其选择概率（和可能是其提议分布），并试图优化算法。我们提供了一个谨慎的例子，说明即使是看似简单的自适应 Gibbs 采样器也可能失败收敛。然后，我们提出了各种正面结果，保证了在某些条件下自适应 Gibbs 采样器的收敛性。