### Linear conditional expectation in Hilbert space in Bernoulli

The article “The linear conditional expectation in Hilbert space” by Ilja Klebanov, Björn Sprungk, and myself has just appeared in its final form in the journal *Bernoulli*.
In this paper, we study the best approximation \(\mathbb{E}^{\mathrm{A}}[U|V]\) of the conditional expectation \(\mathbb{E}[U|V]\) of an \(\mathcal{G}\)-valued random variable \(U\) conditional upon a \(\mathcal{H}\)-valued random variable \(V\), where “best” means \(L^{2}\)-optimality within the class \(\mathrm{A}(\mathcal{H}; \mathcal{G})\) of affine functions of the conditioning variable \(V\).
This approximation is a powerful one and lies at the heart of the Bayes linear approach to statistical inference, but its analytical properties, especially for \(U\) and \(V\) taking values in infinite-dimensional spaces \(\mathcal{G}\) and \(\mathcal{H}\), are only partially understood — which this article aims to rectify.

I. Klebanov, B. Sprungk, and T. J. Sullivan. “The linear conditional expectation in Hilbert space.” *Bernoulli* 27(4):2267–2299, 2021.

**Abstract.**
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.

Published on Wednesday 25 August 2021 at 08:00 UTC #publication #tru2 #bayesian #rkhs #mean-embedding #klebanov #sprungk