  Ontological models are attempts to quantitatively describe the results of a
probabilistic theory, such as Quantum Mechanics, in a framework exhibiting an
explicit realism-based underpinning. Unlike either the well known
quasi-probability representations, or the "r-p" vector formalism, these models
are contextual and by definition only involve positive probability
distributions (and indicator functions). In this article we study how the
ontological model formalism can be used to describe arbitrary statistics of a
system subjected to a finite set of preparations and measurements. We present
three models which can describe any such empirical data and then discuss how to
turn an indeterministic model into a deterministic one. This raises the issue
of how such models manifest contextuality, and we provide an explicit example
to demonstrate this. In the second half of the paper we consider the issue of
finding ontological models with as few ontic states as possible.
