  Conditional mean embeddings (CMEs) have proven themselves to be a powerful
tool in many machine learning applications. They allow the efficient
conditioning of probability distributions within the corresponding reproducing
kernel Hilbert spaces (RKHSs) by providing a linear-algebraic relation for the
kernel mean embeddings of the respective joint and conditional probability
distributions. Both centred and uncentred covariance operators have been used
to define CMEs in the existing literature. In this paper, we develop a
mathematically rigorous theory for both variants, discuss the merits and
problems of each, and significantly weaken the conditions for applicability of
CMEs. In the course of this, we demonstrate a beautiful connection to Gaussian
conditioning in Hilbert spaces.
