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# Computation of conditional expectations with guarantees

Theoretically, the conditional expectation of a square-integrable randomvariable $Y$ given a $d$-dimensional random vector $X$ can be obtained byminimizing the mean squared distance between $Y$ and $f(X)$ over all Borelmeasurable functions $f \colon \mathbb{R}^d \to \mathbb{R}$. However, in manyapplications this minimization problem cannot be solved exactly, and instead, anumerical method that computes an approximate minimum over a suitable subfamilyof Borel functions has to be used. The quality of the result depends on theadequacy of the subfamily and the performance of the numerical method. In thispaper, we derive an expected value representation of the minimal mean squaredistance which in many applications can efficiently be approximated with astandard Monte Carlo average. This enables us to provide guarantees for theaccuracy of any numerical approximation of a given conditional expectation. Weillustrate the method by assessing the quality of approximate conditionalexpectations obtained by linear, polynomial as well as neural networkregression in different concrete examples.

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