Counting and locating the solutions of polynomial systems of maximum likelihood equations, I

In statistics, mixture models consisting of several component subpopulations are used widely to model data drawn from heterogeneous sources. In this paper, we consider maximum likelihood estimation for mixture models in which the only unknown parameters are the component proportions. By applying the theory of multivariable polynomial equations, we derive bounds for the number of isolated roots of the corresponding system of likelihood equations. If the component densities belong to certain familiar continuous exponential families, including the multivariate normal or gamma distributions, then our upper bound is, almost surely, the exact number of solutions. Key words: Bernstein's theorem, carrier sets, EM algorithm, facial resultant, finite mixture model, genetic algorithms, homotopy continuation methods, maximum likelihood estimation, mixed volume, numerical continuation algorithms.

• Type: Journal
• Year: 2006
• Where: JSC
• Authors: Max-Louis G. Buot, Donald St. P. Richards
• Keywords: JSC 2006, Maximum Likelihood Estimation, Mixture Model, Multivariable Polynomial Equations