  Mixture model-based clustering has become an increasingly popular data
analysis technique since its introduction over fifty years ago, and is now
commonly utilized within a family setting. Families of mixture models arise
when the component parameters, usually the component covariance (or scale)
matrices, are decomposed and a number of constraints are imposed. Within the
family setting, model selection involves choosing the member of the family,
i.e., the appropriate covariance structure, in addition to the number of
mixture components. To date, the Bayesian information criterion (BIC) has
proved most effective for model selection, and the expectation-maximization
(EM) algorithm is usually used for parameter estimation. In fact, this EM-BIC
rubric has virtually monopolized the literature on families of mixture models.
Deviating from this rubric, variational Bayes approximations are developed for
parameter estimation and the deviance information criterion for model
selection. The variational Bayes approach provides an alternate framework for
parameter estimation by constructing a tight lower bound on the complex
marginal likelihood and maximizing this lower bound by minimizing the
associated Kullback-Leibler divergence. This approach is taken on the most
commonly used family of Gaussian mixture models, and real and simulated data
are used to compare the new approach to the EM-BIC rubric.
