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 focused on enhancing the efficiency of either the inner or outer 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 of digital nets, we develop a randomized QMC (rQMC) method that improves the convergence behavior of 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 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 sense, paving the way for theoretically sound error estimates. As the main contribution of this work, we derive asymptotic error bounds for the bias and variance of our estimator, along with regularity conditions under which these error bounds can be attained. In addition, we provide nearly optimal sample sizes for the rQMC approximations, which are helpful for the actual numerical implementations. Moreover, we verify the quality of our estimator through numerical experiments in the context of EIG estimation. Specifically, we compare the computational efficiency of our rQMC method against standard nested MC 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.

翻译：评估期望信息增益（EIG）是计算科学与统计领域中的关键任务，这需要近似计算嵌套积分。当前基于拟蒙特卡洛（QMC）方法的求解技术主要聚焦于提升内层或外层积分近似的效率。本研究提出了一种创新框架，将上述研究范畴拓展至同时处理内外层期望。基于Owen数字网格乱序技术原理，我们开发了一种随机化拟蒙特卡洛（rQMC）方法，显著改善了嵌套积分近似的收敛特性。同时，本文展示了如何将本方法与重要性采样相结合，以应对内层积分中出现的测度集中现象。我们的方法充分利用了嵌套期望的独特结构，构建出更高效的逼近机制。通过融入Owen乱序技术，我们成功处理了在Hardy-Krause意义下具有无限变差的被积函数，为建立理论严谨的误差估计奠定了基础。作为本文核心贡献，我们推导了估计量偏差与方差的新近误差界，并给出了实现这些误差界所需的正则性条件。此外，我们提供了rQMC近似的近最优样本量方案，这对实际数值实现具有指导意义。在EIG估计的数值实验中，我们验证了所提估计量的有效性：通过热力学与药物动力学两个典型案例，将所提rQMC方法与标准嵌套MC积分方法进行计算效率对比。这些实例充分展示了本方法在计算成本节约与适用性提升方面的显著优势。