We propose a novel approach for estimating conditional or parametric expectations in the setting where obtaining samples or evaluating integrands is costly. Through the framework of probabilistic numerical methods (such as Bayesian quadrature), our novel approach allows to incorporates prior information about the integrands especially the prior smoothness knowledge about the integrands and the conditional expectation. As a result, our approach provides a way of quantifying uncertainty and leads to a fast convergence rate, which is confirmed both theoretically and empirically on challenging tasks in Bayesian sensitivity analysis, computational finance and decision making under uncertainty.

翻译：本文提出了一种新颖的方法，用于在获取样本或计算被积函数成本高昂的场景下估计条件期望或参数化期望。基于概率数值方法（如贝叶斯求积法）的框架，我们的新方法能够纳入关于被积函数的先验信息，特别是关于被积函数及条件期望的先验平滑性知识。因此，该方法提供了一种量化不确定性的途径，并实现了快速的收敛速率。这一结论在贝叶斯敏感性分析、计算金融和不确定性决策等挑战性任务中均得到了理论与实证的双重验证。