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arXiv · 2008.12070

The linear conditional expectation in Hilbert space

Ilja Klebanov·Björn Sprungk·T. J. Sullivan
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Abstract

The linear conditional expectation (LCE) provides a best linear (or rather, affine) estimate of the conditional expectation and hence plays an important r\^ole 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.

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Ilja Klebanov, Björn Sprungk, T. J. Sullivan. 2020-08-27. The linear conditional expectation in Hilbert space. https://doi.org/10.3150/20-bej1308

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