The linear conditional expectation in Hilbert space
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 infinitedimensional Hilbert space context. In addition, working in the space of affine HilbertSchmidt 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 arXiv:
 arXiv:2008.12070
 Bibcode:
 2020arXiv200812070K
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Functional Analysis;
 Statistics  Machine Learning;
 46E22;
 28C20;
 62C10;
 62J05;
 62G05
 EPrint:
 32 pages