Default prior distributions and efficient posterior computation in Bayesian factor analysis

Abstract

Factor analytic models are widely used in social sciences. These models have also proven useful for sparse modeling of the covariance structure in multidimensional data. Normal prior distributions for factor loadings and inverse gamma prior distributions for residual variances are a popular choice because of their conditionally conjugate form. However, such prior distributions require elicitation of many hyperparameters and tend to result in poorly behaved Gibbs samplers. In addition, one must choose an informative specification, as high variance prior distributions face problems due to impropriety of the posterior distribution. This article proposes a default, heavy-tailed prior distribution specification, which is induced through parameter expansion while facilitating efficient posterior computation.We also develop an approach to allow uncertainty in the number of factors. The methods are illustrated through simulated examples and epidemiology and toxicology applications. Data sets and computer code used in this article are available online. © 2009 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America.