  We present three statistical descriptions for systems of classical particles
and consider their extension to hybrid quantum-classical systems. The classical
descriptions are ensembles on configuration space, ensembles on phase space,
and a Hilbert space approach using van Hove operators which provides an
alternative to the Koopman-von Neumann formulation. In all cases, there is a
natural way to define classical observables and a corresponding Lie algebra
that is isomorphic to the usual Poisson algebra in phase space. We show that in
the case of classical particles, the three descriptions are equivalent and
indicate how they are related. We then modify and extend these descriptions to
introduce hybrid models where a classical particle interacts with a quantum
particle. The approach of ensembles on phase space and the Hilbert space
approach, which are novel, lead to equivalent hybrid models, while they are not
equivalent to the hybrid model of the approach of ensembles on configuration
space. Thus, we end up identifying two inequivalent types of hybrid systems,
making different predictions, especially when it comes to entanglement. These
results are of interest regarding ``no-go'' theorems about quantum systems
interacting via a classical mediator which address the issue of whether gravity
must be quantized. Such theorems typically require assumptions that make them
model dependent. The hybrid systems that we discuss provide concrete examples
of inequivalent models that can be used to compute simple examples to test the
assumptions of the ``no-go'' theorems and their applicability.
