For Gaussian linear covariance models, the paper computes maximum likelihood degrees, provides a Julia package that finds all local maxima, and proves that the dual maximum likelihood estimator for binary Brownian motion tree models is an explicit rational function of the data.
Discrete Statistical Models with Rational Maximum Likelihood Estimator
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abstract
A discrete statistical model is a subset of a probability simplex. Its maximum likelihood estimator (MLE) is a retraction from that simplex onto the model. We characterize all models for which this retraction is a rational function. This is a contribution via real algebraic geometry which rests on results due to Huh and Kapranov on Horn uniformization. We present an algorithm for constructing models with rational MLE, and we demonstrate it on a range of instances. Our focus lies on models familiar to statisticians, like Bayesian networks, decomposable graphical models, and staged trees.
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Estimating linear covariance models with numerical nonlinear algebra
For Gaussian linear covariance models, the paper computes maximum likelihood degrees, provides a Julia package that finds all local maxima, and proves that the dual maximum likelihood estimator for binary Brownian motion tree models is an explicit rational function of the data.