Differential Equations for Gaussian Statistical Models with Rational Maximum Likelihood Estimator

Abstract

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.

29 pages, 1 table

References in corpus (8)

• Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian
• The maximum likelihood degree of a very affine variety
• The Maximum Likelihood Degree of Toric Varieties
• Exponential Varieties
• Milnor numbers of projective hypersurfaces with isolated singularities
• Linear precision for toric surface patches
• On Huh's conjectures for the polar degree
• Computing symmetric determinantal representations

Cited by in corpus (2)

• Homaloidal Polynomials and Gaussian Models of Maximum Likelihood Degree One
• The $F$-adjoined Gauss Map and Gaussian Likelihood Geometry