ECM and MM algorithms for mixtures with constrained parameters
Résumé
EM algorithms for obtaining maximum likelihood estimates of parameters in finite mixture models are well-known, and normal mixtures are the most commonly used in this category. In fact, certain types of constraints on the parameter space, such as the equality of variance assumption, lead to well-known EM algorithms. After briefly summarizing these well-known results, we then consider the problem of more general constraints on the parameter space for finite mixtures of normal components. Surprisingly, this simple extension has not been explored in the literature. Here, we show how the MLE problem succumbs to an EM generalization known as an ECM algorithm. With certain types of variance constraints, yet another generalization of EM, known as MM algorithms is required. After a brief explanation of these algorithmic ideas, we demonstrate how they may be applied to the problem of parameter estimation in finite mixtures of normal components in the presence of equality or linear constraints on the parameters. We provide software that implements these algorithms.
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