A Variational Approximations-DIC Rubric for Parameter Estimation and Mixture Model Selection Within a Family Setting

TL;DR

This paper introduces a variational Bayes approach for parameter estimation and model selection in Gaussian mixture models within a family setting, offering an alternative to the traditional EM-BIC method.

Contribution

It develops variational Bayes approximations for both parameter estimation and deviance information criterion-based model selection in mixture models, diverging from the EM-BIC standard.

Findings

Variational Bayes provides a tight lower bound on the marginal likelihood.

The new approach compares favorably to EM-BIC in real and simulated data.

It offers an alternative framework for mixture model analysis.

Abstract

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.