Merging the Components of a Finite Mixture Using Posterior Probabilities

Methods in parametric cluster analysis commonly assume data can be modelled by means of a finite mixture of distributions. However, associating each mixture component to one cluster is frequently misleading because different mixture components can overlap, and then, associated clusters can overlap too suggesting a unique cluster. A number of approaches have already been proposed to construct the clusters by merging components using the posterior probabilities. This article presents a generic approach for building a hierarchy of mixture components that integrates and generalizes some techniques proposed earlier in the literature. Using this proposal, two new techniques based on the log-ratio of posterior probabilities are introduced. Moreover, to decide the final number of clusters, two new methods are presented. Simulated and real datasets are used to illustrate this methodology.