Torus Actions and Maximum Likelihood Estimation

We describe connections between invariant theory and maximum likelihood estimation, in the context of log-linear models. Finding a maximum likelihood estimate (MLE) is an optimisation problem over a statistical model, to obtain the point that best fits observed data. We show that this is equivalent to a capacity problem - finding the point of minimal norm in an orbit under a corresponding torus action. The existence of the MLE can then be characterized by stability under the action. Moreover, algorithms from statistics can be used in invariant theory, and vice versa. Based on joint work with Carlos Améndola, Kathlén Kohn and Philipp Reichenbach. This is part one of a two part talk: in the second part, Philipp Reichenbach will discuss our results for multivariate Gaussian models.