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November 2021 The linear conditional expectation in Hilbert space
Ilja Klebanov, Björn Sprungk, T.J. Sullivan
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Bernoulli 27(4): 2267-2299 (November 2021). DOI: 10.3150/20-BEJ1308


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.

Funding Statement

IK and TJS are supported in part by the Deutsche Forschungsgemeinschaft (DFG) through project TrU-2 “Demand modelling and control for e-commerce using RKHS transfer operator approaches” of the Excellence Cluster “MATH+ The Berlin Mathematics Research Centre” (EXC-2046/1, project 390685689). TJS is further supported by the DFG project 415980428. BS has been supported by the DFG project 389483880.


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Ilja Klebanov. Björn Sprungk. T.J. Sullivan. "The linear conditional expectation in Hilbert space." Bernoulli 27 (4) 2267 - 2299, November 2021.


Received: 1 September 2020; Revised: 1 December 2020; Published: November 2021
First available in Project Euclid: 24 August 2021

Digital Object Identifier: 10.3150/20-BEJ1308

Rights: Copyright © 2021 ISI/BS


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Vol.27 • No. 4 • November 2021
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