Parsimonious Bayesian Factor Analysis when the Number of Factors is Unknown

We introduce a new and general set of identifiability conditions for factor models which handles the ordering problem associated with current common practice. In addition, the new class of parsimonious Bayesian factor analysis leads to a factor loading matrix representation which is an intuitive and easy to implement factor selection scheme. We argue that the structuring the factor loadings matrix is in concordance with recent trends in applied factor analysis. Our MCMC scheme for posterior inference makes several improvements over the existing alternatives while outlining various strategies for conditional posterior inference in a factor selection scenario. Four applications, two based on synthetic data and two based on well known real data, are introduced to illustrate the applicability and generality of the new class of parsimonious factor models, as well as to highlight features of the proposed sampling schemes.