I have a question about the BSEM approach to CFA that was outlined in
Muthén, B. O., & Asparouhov, T. (2012). Bayesian structural equation modeling: A more flexible representation of substantive theory. Psychological Methods, 17(3), 313–335. doi:10.1037/a0026802
Having thought a bit about this approach, it seems to me that there are two uses to it; 1) to test the hypothesis that parameters (e.g. cross-loadings or residual covariances) are close to zero rather than exactly zero, and 2) to provide estimates for these parameters that can be used similarly to modification indices in ML estimation.
My question regards the first (1) issue: If one wants to conduct a stringent test for this hypothesis, wouldn't one want to specify the model so that no parameter estimate is allowed outside the boundary of the specified range that is thought to be trivial? For example, if one thinks residual correlations +- .20 are trivial, then one would want to estimate the model so that no parameter is estimated outside this range. Otherwise it seems hard to interpret model fit information?
If I understand your question correctly ... you are considering a prior distribution that doesn't have tails. This would be more readily accomplished for cross-loadings possibly with uniform priors, however, I don't really see the problem with interpretation even with normal prior that essentially has the range of -.2 to .2. If the posterior is out of that range the interpretation is clear - the parameter is non-trivial and the model does not fit. If the posterior is within that range you can interpret it as a trivial parameter that is approximately zero and is not a source of misfit. You can of course also use a uniform prior with a fixed range that does not allow the data to drive the parameter away and get a PPP value to determine the fit.
Advice on how to use the Inverse Wishart is given in the paper on our website:
Asparouhov, T., Muthén, B. & Morin, A. J. S. (2015). Bayesian structural equation modeling with cross-loadings and residual covariances: Comments on Stromeyer et al. Journal of Management, 41, 1561-1577.