Mixture Models

Introduction

Mixture distributions are convex combinations of “component” distributions. In statistics, these are standard tools for modeling heterogeneity in the sense that different elements of a sample may belong to different components. However, they may also be used simply as flexible instruments for achieving a good fit to data when standard distributions fail. As good software for fitting mixtures is available, these play an increasingly important role in nearly every field of statistics.

It is convenient to explain finite mixtures (i.e., finite convex combinations) as theoretical models for cluster analysis (see Cluster Analysis: An Introduction), but of course the range of applicability is not at all restricted to the clustering context. Suppose that a feature vector X is observed in a heterogeneous population, which consists of k homogeneous subpopulations, the “components.” It is assumed that for $i = 1,\ldots ,k$, Xis distributed in the i-th component according to a...