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, and the methods explored in studies of some simulated data sets and an econometric time series example.

Keywords: Bayesian inference, Bridge sampling, Latent factor models, Model selection, Model uncertainty, Reversible jump MCMC.

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