  Theoretically, the conditional expectation of a square-integrable random
variable $Y$ given a $d$-dimensional random vector $X$ can be obtained by
minimizing the mean squared distance between $Y$ and $f(X)$ over all Borel
measurable functions $f \colon \mathbb{R}^d \to \mathbb{R}$. However, in many
applications this minimization problem cannot be solved exactly, and instead, a
numerical method which computes an approximate minimum over a suitable
subfamily of Borel functions has to be used. The quality of the result depends
on the adequacy of the subfamily and the performance of the numerical method.
In this paper, we derive an expected value representation of the minimal mean
squared distance which in many applications can efficiently be approximated
with a standard Monte Carlo average. This enables us to provide guarantees for
the accuracy of any numerical approximation of a given conditional expectation.
We illustrate the method by assessing the quality of approximate conditional
expectations obtained by linear, polynomial and neural network regression in
different concrete examples.
