  We study multivariate Gaussian statistical models whose maximum likelihood
estimator (MLE) is a rational function of the observed data. We establish a
one-to-one correspondence between such models and the solutions to a nonlinear
first-order partial differential equation (PDE). Using our correspondence, we
reinterpret familiar classes of models with rational MLE, such as directed (and
decomposable undirected) Gaussian graphical models. We also find new models
with rational MLE. For linear concentration models with rational MLE, we show
that homaloidal polynomials from birational geometry lead to solutions to the
PDE. We thus shed light on the problem of classifying Gaussian models with
rational MLE by relating it to the open problem in birational geometry of
classifying homaloidal polynomials.
