The Geometry of Mixture Likelihoods: A General Theory

Abstract

In this paper certain fundamental properties of the maximum likelihoodestimator of a mixing distribution are shown to be geometric propertiesof the likelihood set. The existence, support size, likelihood equations,and uniqueness of the estimator are revealed to be directly relatedto the properties of the convex hull of the likelihood set and thesupport hyperplanes of that hull. It is shown using geometric techniquesthat the estimator exists under quite general conditions, with asupport size no larger than the number of distinct observations.Analysis of the convex dual of the likelihood set leads to a dualmaximization problem. A convergent algorithm is described. The definingequations for the estimator are compared with the usual parametriclikelihood equations for finite mixtures. Sufficient conditions foruniqueness are given. Part II will deal with a special theory forexponential family mixtures.