  Gaussian mixture models are central to classical statistics, widely used in
the information sciences, and have a rich mathematical structure. We examine
their maximum likelihood estimates through the lens of algebraic statistics.
The MLE is not an algebraic function of the data, so there is no notion of ML
degree for these models. The critical points of the likelihood function are
transcendental, and there is no bound on their number, even for mixtures of two
univariate Gaussians.
