论文标题
希尔伯特空间中的线性条件期望
The linear conditional expectation in Hilbert space
论文作者
论文摘要
线性条件期望（LCE）提供了有条件期望的最佳线性（或更确切地说是仿射）估计，因此在近似贝叶斯的推论中起着重要的作用，尤其是贝叶斯线性方法。本文在无限的希尔伯特空间环境中建立了LCE的分析特性。此外，在仿生的希尔伯特（Hilbert-Schmidt Operators）的空间中工作，我们为此LCE建立了正则化程序。作为一个重要的应用，我们获得了条件均值嵌入公式的简单替代推导和直观的理由，这是一个广泛用于机器学习的概念，可以通过将它们嵌入核心kernel hilbert Space来执行随机变量的条件。
The linear conditional expectation (LCE) provides a best linear (or rather, affine) estimate of the conditional expectation and hence plays an important rôle in approximate Bayesian inference, especially the Bayes linear approach. This article establishes the analytical properties of the LCE in an infinite-dimensional Hilbert space context. In addition, working in the space of affine Hilbert--Schmidt operators, we establish a regularisation procedure for this LCE. As an important application, we obtain a simple alternative derivation and intuitive justification of the conditional mean embedding formula, a concept widely used in machine learning to perform the conditioning of random variables by embedding them into reproducing kernel Hilbert spaces.
