Identifying Mixtures of Mixtures Using Bayesian Estimation

The use of a finite mixture of normal distributions in model-based clustering allows us to capture non-Gaussian data clusters. However, identifying the clusters from the normal components is challenging and in general either achieved by imposing constraints on the model or by using post-processing procedures. Within the Bayesian framework, we propose a different approach based on sparse finite mixtures to achieve identifiability. We specify a hierarchical prior, where the hyperparameters are carefully selected such that they are reflective of the cluster structure aimed at. In addition, this prior allows us to estimate the model using standard MCMC sampling methods. In combination with a post-processing approach which resolves the label switching issue and results in an identified model, our approach allows us to simultaneously (1) determine the number of clusters, (2) flexibly approximate the cluster distributions in a semiparametric way using finite mixtures of normals and (3) identify cluster-specific parameters and classify observations. The proposed approach is illustrated in two simulation studies and on benchmark datasets. Supplementary materials for this article are available online.