Bayesian Inference in Factor Analysis

Abstract

We propose a new method for analyzing factor analysis models using a Bayesian approach. Normal theory is used for the sampling distrib-ution, and we adopt a model with a full disturbance covariance matrix. Us-ing vague and natural conjugate priors for the parameters, we find that the marginal posterior distribution of the factor scores is approximately a matrix T-distribution, in large samples. This explicit result permits simple interval estimation and hypothesis testing of the factor scores. Explicit point and in-terval estimators of the factor score elements, in large samples, are obtained as means as means of the respective marginal posterior distributions. Factor loadings are estimated as joint modes (with factor scores), or alternatively as means or modes of the distribution of the factor loadings conditional upon the estimated factor scores. Disturbance variances and covariances are esti-mated conditional upon the estimated factor scores and factor loadings. This revision includes the correction of some typographical errors and some revised computations, plus an appendix that provides some intermediate results.