  We consider a system with a discrete configuration space. We show that the
geometrical structures associated with such a system provide the tools
necessary for a reconstruction of discrete quantum mechanics once dynamics is
brought into the picture. We do this in three steps. Our starting point is
information geometry, the natural geometry of the space of probability
distributions. Dynamics requires additional structure. To evolve the
probabilities $P^k$, we introduce coordinates $S^k$ canonically conjugate to
the $P^k$ and a symplectic structure. We then seek to extend the metric
structure of information geometry, to define a geometry over the full space of
the $P^k$ and $S^k$. Consistency between the metric tensor and the symplectic
form forces us to introduce a K\"ahler geometry. The construction has notable
features. A complex structure is obtained in a natural way. The canonical
coordinates of the K\"ahler space are precisely the wave functions of quantum
mechanics. The full group of unitary transformations is obtained. Finally, one
may associate a Hilbert space with the K\"ahler space, which leads to the
standard version of quantum theory. We also show that the metric that we derive
here using purely geometrical arguments is precisely the one that leads to
Wootters' expression for the statistical distance for quantum systems.
