  A discrete statistical model is a subset of a probability simplex. Its
maximum likelihood estimator (MLE) is a retraction from that simplex onto the
model. We characterize all models for which this retraction is a rational
function. This is a contribution via real algebraic geometry which rests on
results due to Huh and Kapranov on Horn uniformization. We present an algorithm
for constructing models with rational MLE, and we demonstrate it on a range of
instances. Our focus lies on models familiar to statisticians, like Bayesian
networks, decomposable graphical models, and staged trees.
