Abstract: Factor analysis has been one of the most powerful and flexible tools for assessment of multivariate dependence and codependence. Loosely speaking, it could be argued that the origin of its success rests in its very exploratory nature, where various kinds of data-relationships amongst the variables at study can be iteratively verified and/or refuted. Bayesian inference in factor analytic models has received renewed attention in recent years, partly due to computational advances but also partly to applied focuses generating factor structures as exemplified by recent work in financial time series modeling. The focus of our current work is on exploring questions of uncertainty about the number of latent factors in a multivariate factor model, combined with methodological and computational issues of model specification and model fitting. We explore reversible jump MCMC methods that build on sets of parallel Gibbs sampling-based analyses to generate suitable empirical proposal distributions and that address the challenging problem of finding efficient proposals in high-dimensional models. Alternative MCMC methods based on bridge sampling are discussed, and these fully Bayesian MCMC approaches are compared with a collection of popular model selection methods in empirical studies. Various additional computational issues are discussed, including situations where prior information is scarce, and the methods are explored in studies of some simulated data sets and an econometric time series example.
Key words and phrases: Bayes factors, Bayesian inference, bridge sampling, expected posterior prior, latent factor models, model selection criteria, model uncertainty, reversible jump MCMC.