Consistency of the Bayesian Information Criterion for Model Selection in Exploratory Factor Analysis

Abstract: We study model selection by the Bayesian information criterion (BIC) in fixed-dimensional exploratory factor analysis over a fixed finite family of compact covariance classes. Our main result shows that the BIC is strongly consistent for the pseudo-true factor order under misspecification, provided that all globally optimal models share a common pseudo-true covariance set, the population Gaussian criterion has a local quadratic margin away from that set, and the BIC complexity counts are order-separating at the pseudo-true order. The candidate models may have an unknown mean vector, exact-zero restrictions in the loading matrix, and either diagonal or spherical error covariance structures, and the selection target is the smallest candidate factor order that yields the best Gaussian approximation, in Kullback--Leibler divergence, to the data-generating covariance structure. The proof works directly in covariance space, so it does not require a regular loading parametrization and accommodates the familiar singularities caused by rotations and redundant factors. Under correct specification, the assumptions reduce to familiar properties of the true covariance matrix. More generally, the same argument applies to other information criteria whose penalties satisfy the same gap conditions, including several BIC-type modifications.