Conditional Monte Carlo or pre-integration is a powerful tool for reducing variance and improving the regularity of integrands when using Monte Carlo and quasi-Monte Carlo (QMC) methods. To select the variable to pre-integrate, one must consider both the variable's importance and the tractability of the conditional expectation. For integrals over a Gaussian distribution, any linear combination of variables can potentially be pre-integrated. Liu and Owen (2022) propose to select the linear combination based on an active subspace decomposition of the integrand. However, pre-integrating the selected direction might be intractable. In this work, we address this issue by finding the active subspace subject to constraints such that pre-integration can be easily carried out. The proposed algorithm also provides a computationally-efficient alternative to dimension reduction for pre-integrated functions. The method is applied to examples from computational finance, density estimation, and computational chemistry, and is shown to achieve smaller errors than previous methods.

翻译：条件蒙特卡罗或预积分是一种强大的工具，可用于在使用蒙特卡罗和拟蒙特卡罗（QMC）方法时降低方差并改善被积函数的正则性。选择预积分变量时，必须同时考虑该变量的重要性以及条件期望的可处理性。对于高斯分布上的积分，变量的任何线性组合均可进行预积分。Liu与Owen（2022）提出基于被积函数的活动子空间分解来选择线性组合。然而，对所选方向进行预积分可能难以实现。本研究通过寻找满足约束条件（使得预积分易于执行）的活动子空间来解决此问题。所提出的算法还为预积分函数提供了一种计算高效的降维替代方案。该方法被应用于计算金融、密度估计和计算化学等领域的示例中，并证明相较于先前方法能够实现更小的误差。