Maximum Likelihood Estimates for Gaussian Mixtures Are Transcendental
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.
Algebraic statistics, Expectation maximization, Maximum likelihood, Mixture model, Normal distribution, Transcendence theory
Dewey Decimal Classification:
Mathematical Aspects of Computer and Information Sciences
6th International Conference, MACIS 2015, Berlin, Germany, November 11-13, 2015, Revised Selected Papers