  In the framework of Bayesian model-based clustering based on a finite mixture
of Gaussian distributions, we present a joint approach to estimate the number
of mixture components and identify cluster-relevant variables simultaneously as
well as to obtain an identified model. Our approach consists in specifying
sparse hierarchical priors on the mixture weights and component means. In a
deliberately overfitting mixture model the sparse prior on the weights empties
superfluous components during MCMC. A straightforward estimator for the true
number of components is given by the most frequent number of non-empty
components visited during MCMC sampling. Specifying a shrinkage prior, namely
the normal gamma prior, on the component means leads to improved parameter
estimates as well as identification of cluster-relevant variables. After
estimating the mixture model using MCMC methods based on data augmentation and
Gibbs sampling, an identified model is obtained by relabeling the MCMC output
in the point process representation of the draws. This is performed using
$K$-centroids cluster analysis based on the Mahalanobis distance. We evaluate
our proposed strategy in a simulation setup with artificial data and by
applying it to benchmark data sets.
