  On the basis of a suggestive definition of a classical extension of quantum
mechanics in terms of statistical models, we prove that every such classical
extension is essentially given by the so-called Misra-Bugajski reduction map.
We consider how this map enables one to understand quantum mechanics as a
reduced classical statistical theory on the projective Hilbert space as phase
space and discuss features of the induced hidden-variables model. Moreover,
some relevant technical results on the topology and Borel structure of the
projective Hilbert space are reviewed.
