  We consider the problem of inferring an unknown number of clusters in
replicated multinomial data. Under a model based clustering point of view, this
task can be treated by estimating finite mixtures of multinomial distributions
with or without covariates. Both Maximum Likelihood (ML) as well as Bayesian
estimation are taken into account. Under a Maximum Likelihood approach, we
provide an Expectation--Maximization (EM) algorithm which exploits a careful
initialization procedure combined with a ridge--stabilized implementation of
the Newton--Raphson method in the M--step. Under a Bayesian setup, a stochastic
gradient Markov chain Monte Carlo (MCMC) algorithm embedded within a prior
parallel tempering scheme is devised. The number of clusters is selected
according to the Integrated Completed Likelihood criterion in the ML approach
and estimating the number of non-empty components in overfitting mixture models
in the Bayesian case. Our method is illustrated in simulated data and applied
to two real datasets. An R package is available at
https://github.com/mqbssppe/multinomialLogitMix.
