The effective sample size (ESS) is widely used in sample-based simulation
methods for assessing the quality of a Monte Carlo approximation of a given
distribution and of related integrals. In this paper, we revisit the
approximation of the ESS in the specific context of importance sampling (IS).
The derivation of this approximation, that we will denote as
$\widehat{\text{ESS}}$, is partially available in Kong (1992). This
approximation has been widely used in the last 25 years due to its simplicity
as a practical rule of thumb in a wide variety of importance sampling methods.
However, we show that the multiple assumptions and approximations in the
derivation of $\widehat{\text{ESS}}$, makes it difficult to be considered even
as a reasonable approximation of the ESS. We extend the discussion of the
$\widehat{\text{ESS}}$ in the multiple importance sampling (MIS) setting, we
display numerical examples, and we discuss several avenues for developing
alternative metrics. This paper does not cover the use of ESS for MCMC
algorithms.

本文重访了有效样本量（ESS）在重要性采样（IS）中的近似，得出该近似难以被视为 ESS 的合理近似，并提出了替代指标的若干方法。

重新思考有效样本数

Rethinking the Effective Sample Size

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

Multiple importance sampling (MIS) methods use a set of proposal
distributions from which samples are drawn. Each sample is then assigned an
importance weight that can be obtained according to different strategies. This
work is motivated by the trade-off between variance reduction and computational
complexity of the different approaches (classical vs. deterministic mixture)
available for the weight calculation. A new method that achieves an efficient
compromise between both factors is introduced in this paper. It is based on
forming a partition of the set of proposal distributions and computing the
weights accordingly. Computer simulations show the excellent performance of the
associated \mbox{\emph{partial deterministic mixture} MIS estimator.

本文介绍了一种基于集合划分的新的多重重要性采样方法，该方法通过权重计算的高效折衷来减少方差和计算复杂度，并提供了相关的计算机模拟结果。

高效的多重重要性采样估计器

Efficient Multiple Importance Sampling Estimators

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。