Lukas Gustafsson: Gaussian likelihood geometry and rational MLE

We study the maximum likelihood degree (MLD) of centered multivariate Gaussian statistical models with homogeneous vanishing ideal. The MLD of a statistical model M counts the number of complex critical points of the log-likelihood function constrained to M, and this number is independent of the data. The maximum likelihood estimator (MLE) sends every data to the maximizer of the corresponding likelihood function on M and it is commonly evaluated by numerically solving the constrained optimization problem. It is exactly when MLD = 1 that the MLE is a rational function and easier to evaluate. In a joint paper with C. Améndola, K. Kohn, O. Marigliano, A. Seigal we establish a one-to-one correspondence between models with rational MLE (same as MLD =1) and the solutions to a nonlinear firstorder partial differential equation. Current work in progress, soon on arxiv, combines this correspondence with the "F-adjoined Gauss map" to classify projective curves with rational MLE and give a formula for the expected MLD of a model as a weighted sum of its polar classes.