| Summary: | We consider a factor analysis model that arises as some distribution form known up
to first and second moments. We propose a new Bayesian approach to determine if any latent factors exist and the number of factors. As opposed to current Bayesian
methodology for factor analysis, our approach only requires the specification of a
prior for the mean vector and the variance matrix for the manifest variables. We
compare the concentration of the prior and posterior about the various subsets of
parameter space specified by the hypothesized factor structures. We consider two priors here, one is conjugate type and the other is based on the correlation factorization of the covariance matrix. A computational problem associated with the use of the second prior is solved by the use of importance sampling for the posterior analysis.
If the data does not lead to a substantial increase in the concentration about
the relevant subset, of the posterior compared to the prior, then we have evidence
against the hypothesized factor structure. The hypothesis is assessed by computing
the observed relative surprise. This results in a considerable simplification of the
problem, especially with respect to the elicitation of the prior.
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