Evaluating the Expected Information Gain (EIG) is a critical task in many areas of computational science and statistics, necessitating the approximation of nested integrals. Available techniques for this problem based on Quasi-Monte Carlo (QMC) methods have primarily focused on enhancing the efficiency of the inner integral approximation. In this work, we introduce a novel approach that extends the scope of these efforts to address inner and outer expectations simultaneously. Leveraging the principles of Owen's scrambling, we develop a randomized quasi-Monte Carlo (RQMC) method that improves the approximation of nested integrals. We also indicate how to combine this methodology with Importance Sampling to address a measure concentration arising in the inner integral. Our RQMC method capitalizes on the unique structure of nested expectations to offer a more efficient approximation mechanism. By incorporating Owen's scrambling techniques, we handle integrands exhibiting infinite variation in the Hardy-Krause (HK) sense, paving the way for theoretically sound error estimates. We derive asymptotic error bounds for the bias and variance of our estimator. In addition, we provide nearly optimal sample sizes for the inner and outer RQMC approximations, which are helpful for the actual numerical implementations. We verify the quality of our estimator through numerical experiments in the context of Bayesian optimal experimental design. Specifically, we compare the computational efficiency of our RQMC method against standard nested Monte Carlo integration across two case studies: one in thermo-mechanics and the other in pharmacokinetics. These examples highlight our approach's computational savings and enhanced applicability, showcasing the advantages of estimating the Expected Information Gain with greater efficiency and reduced computational cost.

翻译：在许多计算科学和统计学领域中，评估期望信息增益（EIG）是一项关键任务，这需要近似计算嵌套积分。现有基于准蒙特卡洛（QMC）方法的技术主要集中于提高内层积分近似的效率。本文提出了一种新颖方法，将这一工作的范围拓展至同时处理内外层期望。利用Owen的加扰原理，我们开发了一种随机化准蒙特卡洛（RQMC）方法，改进了嵌套积分的近似。我们还指出如何将该方法与重要性采样相结合，以解决内层积分中出现的测度集中问题。我们的RQMC方法利用嵌套期望的独特结构，提供了更高效的近似机制。通过引入Owen的加扰技术，我们处理了在Hardy-Krause（HK）意义下具有无限变差的被积函数，为理论上的误差估计奠定了基础。我们推导了估计量偏差和方差的渐近误差界。此外，我们为内外层RQMC近似提供了近乎最优的样本量，这有助于实际数值实现。通过贝叶斯最优实验设计背景下的数值实验，我们验证了所提估计量的质量。具体而言，我们在两个案例研究中对比了RQMC方法与标准嵌套蒙特卡洛积分的计算效率：一个涉及热力学问题，另一个涉及药代动力学问题。这些例子凸显了该方法在计算量节省和适用性增强方面的优势，展示了以更高效率和更低计算成本估计期望信息增益的优越性。