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. The derivation of this approximation, that we will denote as $\widehat{\text{ESS}}$, is partially available in a 1992 foundational technical report of Augustine Kong. 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 $\widehat{\text{ESS}}$ make 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 setting, we display numerical examples and we discuss several avenues for developing alternative metrics. This paper does not cover the use of ESS for Markov chain Monte Carlo algorithms.

中文翻译：

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重新思考有效样本量

有效样本量 (ESS) 广泛用于基于样本的模拟方法中，用于评估给定分布和相关积分的蒙特卡罗近似质量。在本文中，我们在重要性抽样的特定背景下重新审视了 ESS 的近似。这个近似的推导，我们将表示为$\widehat{\text{ESS}}$，在 Augustine Kong 的 1992 年基础技术报告中部分可用。这种近似值在过去 25 年中被广泛使用，因为它作为各种重要性抽样方法中的实用经验法则很简单。然而，我们证明了推导中的多个假设和近似$\widehat{\text{ESS}}$使其难以被视为 ESS 的合理近似值。我们扩展讨论$\widehat{\text{ESS}}$在多重重要性抽样设置中，我们展示了数值示例，并讨论了开发替代指标的几种途径。本文不涉及将 ESS 用于马尔可夫链蒙特卡罗算法。