  We study a natural Wasserstein gradient flow on manifolds of probability
distributions with discrete sample spaces. We derive the Riemannian structure
for the probability simplex from the dynamical formulation of the Wasserstein
distance on a weighted graph. We pull back the geometric structure to the
parameter space of any given probability model, which allows us to define a
natural gradient flow there. In contrast to the natural Fisher-Rao gradient,
the natural Wasserstein gradient incorporates a ground metric on sample space.
We illustrate the analysis of elementary exponential family examples and
demonstrate an application of the Wasserstein natural gradient to maximum
likelihood estimation.
