  Factor Analysis is a popular method for modeling dependence in multivariate
data. However, determining the number of factors and obtaining a sparse
orientation of the loadings are still major challenges. In this paper, we
propose a decision-theoretic approach that brings to light the relation between
a sparse representation of the loadings and factor dimension. This relation is
done through a summary from information contained in the multivariate
posterior. To construct such summary, we introduce a three-step approach. In
the first step, the model is fitted with a conservative factor dimension. In
the second step, a series of sparse point-estimates, with a decreasing number
of factors, is obtained by minimizing an expected predictive loss function. In
step three, the degradation in utility in relation to the sparse loadings and
factor dimensions is displayed in the posterior summary. The findings are
illustrated with applications in classical data from the Factor Analysis
literature. We used different prior choices and factor dimensions to
demonstrate the flexibility of the proposed method.
