Well I guess my first approach would be to implement the two different approaches:
1- Approximate priors individually by univariate normals (uncorrelated)
2- Approximate priors collectively by multivariate normals (where correlation will be computed from previous optimizations trace)
and see if there are any differences. I guess this would already be obvious without fitting, if your parameters look correlated, probably second approach is closer to what you want.
These are of course subject to posteriors looking reasonably normal. I guess one can also check whether the collection of sampled parameters look like a multivariate normal right? Or if not try something else like student-t.
I am wondering, is there anything inherently wrong by taking your parameters from your trace and fitting couple multivariate distributions to it and taking the best one?