  There has been increased research interest in the subfield of sparse Bayesian
factor analysis with shrinkage priors, which achieve additional sparsity beyond
the natural parsimonity of factor models. In this spirit, we estimate the
number of common factors in the highly implemented sparse latent factor model
with spike-and-slab priors on the factor loadings matrix. Our framework leads
to a natural, efficient and simultaneous coupling of model estimation and
selection on one hand and model identification and rank estimation (number of
factors) on the other hand. More precisely, by embedding the unordered
generalized lower triangular loadings representation into overfitting sparse
factor modelling, we obtain posterior summaries regarding factor loadings,
common factors as well as the factor dimension via postprocessing draws from
our efficient and customized Markov chain Monte Carlo scheme.
