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.

翻译：我们研究极大似然估计为观测数据有理函数的多变量高斯统计模型。我们在这类模型与非线性一阶偏微分方程的解之间建立了一一对应关系。借助该对应关系，我们重新诠释了具有有理极大似然估计的经典模型族，例如有向（及可分解无向）高斯图模型。我们还发现了具有有理极大似然估计的新模型。对于具有有理极大似然估计的线性浓度模型，我们证明了双有理几何中的同调多项式可导出该偏微分方程的解。由此，我们将高斯模型有理极大似然估计的分类问题与双有理几何中同调多项式的分类这一开放问题相关联，从而揭示了该问题的本质。