  Recent work on overfitting Bayesian mixtures of distributions offers a
powerful framework for clustering multivariate data using a latent Gaussian
model which resembles the factor analysis model. The flexibility provided by
overfitting mixture models yields a simple and efficient way in order to
estimate the unknown number of clusters and model parameters by Markov chain
Monte Carlo (MCMC) sampling. The present study extends this approach by
considering a set of eight parameterizations, giving rise to parsimonious
representations of the covariance matrix per cluster. A Gibbs sampler combined
with a prior parallel tempering scheme is implemented in order to approximately
sample from the posterior distribution of the overfitting mixture. The
parameterization and number of factors is selected according to the Bayesian
Information Criterion. Identifiability issues related to label switching are
dealt by post-processing the simulated output with the Equivalence Classes
Representatives algorithm. The contributed method and software are demonstrated
and compared to similar models estimated using the Expectation-Maximization
algorithm on simulated and real datasets. The software is available online at
https://CRAN.R-project.org/package=fabMix.
