Bayesian factor analysis with uncertain functional constraints about factor loadings

Factor analysis with uncertain functional constraints about factor loading matrix is considered from a Bayesian viewpoint, in which the uncertain prior information is incorporated in the analysis. We propose a hierarchical screened scale mixture of normal factor (HSMF) model for flexible inference of the constrained factor loadings, factor scores, and specific variances as well as the covariance matrix of the factors. The proposed model makes provisions for robust factor analysis with uncertainty about the functional constraints. A number of inferential aspects of the proposed model are investigated in order to render the proposed analysis optimal. These include the closure properties of a class of rectangle-screened scale mixture of multivariate normal (RSMN) distributions which is useful for statistical inference of the HSMF model, eliciting the prior and posterior evolutions of the uncertainly constrained factor loadings, and providing the efficient Bayesian estimation procedure by using the MCMC methods. Empirical analysis for Bayesian factor models with synthetic data and real data applications is given to illustrate the usefulness of the proposed model.