  Dirichlet Process Mixture (DPM) models have been increasingly employed to
specify random partition models that take into account possible patterns within
the covariates. Furthermore, to deal with large numbers of covariates, methods
for selecting the most important covariates have been proposed. Commonly, the
covariates are chosen either for their importance in determining the clustering
of the observations or for their effect on the level of a response variable
(when a regression model is specified). Typically both strategies involve the
specification of latent indicators that regulate the inclusion of the
covariates in the model. Common examples involve the use of spike and slab
prior distributions. In this work we review the most relevant DPM models that
include covariate information in the induced partition of the observations and
we focus on available variable selection techniques for these models. We
highlight the main features of each model and demonstrate them in simulations
and in a real data application.
