Let's say we have a multimomial logit problem and we find the best beta coeffcients b* aggregating all units. The inverse of the Hessian at b* gives us a VCV matrix at this point which shows roughly how betas vary across units. The other way to get a VCV matrix is to use Hierarchical Bayes where you draw a VCV using inverse Wishart distribution using a Metropolis-Hastings algorithm. You get the average of your sampled VCVs and use that to see how betas change together across the units.

My questions are how off the first VCV is from the second? Are they talking about the same thing or they are innately different entities? Is there a case where these two VCVs become very similar/close to eachother?

Thank you very much