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The linear conditional expectation in Hilbert space
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Klebanov, Ilja, Sprungk, Björn and Sullivan, Tim J. (2021) The linear conditional expectation in Hilbert space. Bernoulli, 27 (4). pp. 2267-2299. doi:10.3150/20-BEJ1308 ISSN 1350-7265.
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Official URL: http://dx.doi.org/10.3150/20-BEJ1308
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.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > Q Science (General) Q Science > QA Mathematics |
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Divisions: | Faculty of Science, Engineering and Medicine > Engineering > Engineering Faculty of Science, Engineering and Medicine > Science > Mathematics |
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Library of Congress Subject Headings (LCSH): | Bayesian statistical decision theory, Linear systems, Conditional expectations (Mathematics), Hilbert space, Kernel functions, Machine learning | ||||||||||||
Journal or Publication Title: | Bernoulli | ||||||||||||
Publisher: | Int Statistical Institute | ||||||||||||
ISSN: | 1350-7265 | ||||||||||||
Official Date: | November 2021 | ||||||||||||
Dates: |
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Volume: | 27 | ||||||||||||
Number: | 4 | ||||||||||||
Page Range: | pp. 2267-2299 | ||||||||||||
DOI: | 10.3150/20-BEJ1308 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Date of first compliant deposit: | 25 August 2021 | ||||||||||||
Date of first compliant Open Access: | 27 August 2021 | ||||||||||||
RIOXX Funder/Project Grant: |
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