带保证的条件期望的计算
Computation of conditional expectations with guarantees
摘要：
在理论上，对于给定的$d$维随机向量$X$，平方可积随机变量$Y$的条件期望可以通过在所有Borel可测函数$f \colon \mathbb{R}^d \to \mathbb{R}$上最小化$Y$和$f(X)$之间的均方距离得到.然而，在许多应用中，这个最小化问题不能被精确地解决，相反，必须使用一种数值方法来计算一个合适的波莱尔函数的子族的近似最小值。结果的质量取决于子族的充分性和数值方法的性能。在本文中，我们推导出一个期望值表示的最小均方距离，在许多应用中，可以有效地近似与标准的蒙特卡罗平均。这使我们能够保证给定条件期望的任何数值近似的准确性。我们说明了该方法通过评估质量的近似条件期望得到的线性，多项式和神经网络回归在不同的具体例子。
英文摘要：
Theoretically, the conditional expectation of a square-integrable random variable $Y$ given a $d$-dimensional random vector $X$ can be obtained by minimizing the mean squared distance between $Y$ and $f(X)$ over all Borel measurable functions $f \colon \mathbb{R}^d \to \mathbb{R}$. However, in many applications this minimization problem cannot be solved exactly, and instead, a numerical method which computes an approximate minimum over a suitable subfamily of Borel functions has to be used. The quality of the result depends on the adequacy of the subfamily and the performance of the numerical method. In this paper, we derive an expected value representation of the minimal mean squared distance which in many applications can efficiently be approximated with a standard Monte Carlo average. This enables us to provide guarantees for the accuracy of any numerical approximation of a given conditional expectation. We illustrate the method by assessing the quality of approximate conditional expectations obtained by linear, polynomial and neural network regression in different concrete examples.
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