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Latent class models rely on the conditional
independence assumption, i.e., it is assumed that the categorical
variables are independent given the cluster memberships.
Within the Bayesian framework, we propose a suitable specification of
priors for the latent class model to identify the clusters in
multivariate categorical data where the independence assumption is not
fulfilled.  Each cluster distribution is approximated by a latent
class model, leading overall to a mixture of latent class models.
The Bayesian approach allows to identify the clusters and fit their
cluster distributions using a one-step procedure. We provide suitable estimation and inference methods for the
mixture of latent class models and illustrate the performance of this
approach on artificial and real data.