用于嵌套积分的中位数随机拟蒙特卡洛方法
Median randomized quasi-Monte Carlo for nested integration
AI总结：
该研究将中位数随机拟蒙特卡洛（mRQMC）扩展至嵌套积分，提出双循环mRQMC估计量，针对EIG应用设计重要性采样正则化方案，在多项式示例和EIG估计中均实现收敛性能提升。
摘要：
嵌套积分∫f(∫g(y,x)dx)dy（其中y∈[0,1]^d₁，x∈[0,1]^d₂）常出现在贝叶斯实验设计、医疗决策和计算金融领域。嵌套积分的统计估计量比标准积分的计算要求更高。近期研究表明，中位数随机拟蒙特卡洛（mRQMC）方法对解析被积函数可达到超多项式收敛速率。我们将mRQMC方法扩展至嵌套积分，提出一种新型双循环mRQMC估计量，若内外被积函数均为解析函数，则该估计量可达到超多项式收敛速率。相比之下，现有嵌套积分估计量（如双循环或多层蒙特卡洛）仅能达到代数收敛速率。然而，实验设计中的期望信息增益（EIG）应用通常不满足上述条件，原因在于外被积函数缺乏正则性。我们通过一种基于理论的重要性采样正则化方案解决该问题。尽管内积分仍可达到超多项式收敛速率，但对于EIG中的外积分，理论上仅能保证代数速率α+3/2−ε（其中ε>0，α∈N∪{0}表示正则化后外被积函数的混合导数阶数）。在中等高维场景下，针对简单多项式示例，我们的方法相比基于样本均值的双循环随机拟蒙特卡洛方法有显著提升；此外，我们证明了该中位数估计量在更具挑战性的EIG估计场景中收敛性能更优。
英文摘要：
Nested integrals $\int f\left(\int g(\boldsymbol{y},\boldsymbol{x})\mathrm{d}{}\boldsymbol{x}\right)\mathrm{d}{}\boldsymbol{y}$ where $\boldsymbol{y}\in[0,1]^{d_1}$ and $\boldsymbol{x}\in[0,1]^{d_2}$ frequently arise in Bayesian experimental design, medical decision making, and computational finance. Statistical estimators for nested integrals are more computationally demanding than for standard integrals. Recent studies demonstrate that the median randomized quasi-Monte Carlo (mRQMC) method achieves super-polynomial convergence rates for analytic integrands. We extend mRQMC methods to nested integrations and propose a novel double-loop mRQMC estimator with super-polynomial convergence rate if both the inner and outer integrands are analytic. Established estimators for nested integrals, such as double-loop or multilevel Monte Carlo, are by contrast limited to algebraic convergence rates. However, expected information gain (EIG) applications from experimental design typically do not fulfill this condition due to a lack of regularity in the outer integrand. We address this through a theoretically-grounded importance sampling regularization scheme. Although super-polynomial convergence rates remain achievable for the inner integral, only the algebraic rate $α+3/2-ε$ for any $ε>0$ can be theoretically guaranteed for the outer integral in the EIG, where $α\in\mathbb{N}\cup\{0\}$ denotes the order of the mixed derivatives of the regularized outer integrand. Our approach shows substantial improvements over double-loop randomized quasi-Monte Carlo methods based on the sample mean in moderately high-dimensional settings for a simple polynomial example. Moreover, we demonstrate improved convergence behavior of the median-based estimator for the more challenging setting of EIG estimation.
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