  This paper is about variable selection, clustering and estimation in an
unsupervised high-dimensional setting. Our approach is based on fitting
constrained Gaussian mixture models, where we learn the number of clusters $K$
and the set of relevant variables $S$ using a generalized Bayesian posterior
with a sparsity inducing prior. We prove a sparsity oracle inequality which
shows that this procedure selects the optimal parameters $K$ and $S$. This
procedure is implemented using a Metropolis-Hastings algorithm, based on a
clustering-oriented greedy proposal, which makes the convergence to the
posterior very fast.
