Connecting Discrete and Gaussian Maximum Likelihood Degrees

In statistics, discrete probability distributions and gaussians centered at 0 are fundamental. The collection of discrete and centered gaussian distributions on n-variables can be modeled by the probability simplex and the positive definite cone of matrices respectively. A problem in statistics is to maximize the log-likelihood function restricted to a semi-algebraic subset of these models, given some statistical data. Transitioning to the complex numbers we may instead count the number of critical points, which we define to be the Maximum Likelihood Degree (ML-degree) of the corresponding subvariety. This concept is similar to the Euclidean Distance Degree (EDD) and yields an algebraic optimization problem. In the discrete case there are many nice results, such as the ML-degree being an Euler characteristic or the classification of all ML-degree 1 models. In my talk I will discuss ideas for proving analogues of these results in the Gaussian case.