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In the framework of Bayesian model-based clustering based on a finite mixtureof Gaussian distributions, we present a joint approach to estimate the numberof mixture components and identify cluster-relevant variables simultaneously aswell as to obtain an identified model. Our approach consists in specifyingsparse hierarchical priors on the mixture weights and component means. In adeliberately overfitting mixture model the sparse prior on the weights emptiessuperfluous components during MCMC. A straightforward estimator for the truenumber of components is given by the most frequent number of non-emptycomponents visited during MCMC sampling. Specifying a shrinkage prior, namelythe normal gamma prior, on the component means leads to improved parameterestimates as well as identification of cluster-relevant variables. Afterestimating the mixture model using MCMC methods based on data augmentation andGibbs sampling, an identified model is obtained by relabeling the MCMC outputin the point process representation of the draws. This is performed using$K$-centroids cluster analysis based on the Mahalanobis distance. We evaluateour proposed strategy in a simulation setup with artificial data and byapplying it to benchmark data sets.

Model-based clustering based on sparse finite Gaussian mixtures