Bayesian (non-)unique sparse factor modelling

Factor modelling extracts common information from a high-dimensional data set into few common components, where the latent factors usually explain a large share of data variation. Exploratory factor estimation induces sparsity into the loading matrix to associate units or series with those factors most strongly associated with them, eventually determining factor interpretation. We motivate geometrically under which circumstances it may be necessary to consider the existence of multiple sparse factor loading matrices with similar degrees of sparsity for a given data set. We propose two MCMC approaches for Bayesian inference and corresponding post-processing algorithms to uncover multiple sparse representations of the factor loadings matrix. We investigate both approaches in a simulation study. Applied to data on country-specific gross domestic product and U.S. price components series, we retrieve multiple sparse factor representations for each data set. Both approaches prove useful to discriminate between pervasive and weaker factors.