Abstract The expected information gain is an important quality criterion of Bayesian experimental designs, which measures how much the information entropy about uncertain quantity of interest θ is reduced on average by collecting relevant data Y. However, estimating the expected information gain has been considered computationally challenging since it is defined as a nested expectation with an outer expectation with respect to Y and an inner expectation with respect to θ. In fact, the standard, nested Monte Carlo method requires a total computational cost of to achieve a root-mean-square accuracy of ε. In this paper we develop an efficient algorithm to estimate the expected information gain by applying a multilevel Monte Carlo (MLMC) method. To be precise, we introduce an antithetic MLMC estimator for the expected information gain and provide a sufficient condition on the data model under which the antithetic property of the MLMC estimator is well exploited such that optimal complexity of is achieved. Furthermore, we discuss how to incorporate importance sampling techniques within the MLMC estimator to avoid arithmetic underflow. Numerical experiments show the considerable computational cost savings compared to the nested Monte Carlo method for a simple test case and a more realistic pharmacokinetic model.

中文翻译：

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预期信息增益的多级蒙特卡罗估计

摘要 期望信息增益是贝叶斯实验设计的一个重要质量标准，它通过收集相关数据 Y 来衡量关于不确定感兴趣量 θ 的信息熵平均减少了多少。 然而，估计期望信息增益一直被认为在计算上具有挑战性因为它被定义为一个嵌套期望，其外部期望是关于 Y 的，而内部期望是关于 θ 的。事实上，标准的嵌套蒙特卡罗方法需要 的总计算成本来实现 ε 的均方根精度。在本文中，我们开发了一种有效的算法，通过应用多级蒙特卡罗 (MLMC) 方法来估计预期信息增益。准确地说，我们为预期的信息增益引入了一个对立的 MLMC 估计器，并为数据模型提供了充分条件，在该条件下，MLMC 估计器的对立特性得到了很好的利用，从而实现了 的最佳复杂度。此外，我们讨论了如何在 MLMC 估计器中结合重要性采样技术来避免算术下溢。数值实验表明，对于简单的测试案例和更现实的药代动力学模型，与嵌套蒙特卡罗方法相比，可节省大量计算成本。