I bump this conversion, because there is a subtle issue with the model/data described previously. When I have a non-informative prior for the degrees of freedom, the posterior settles at around 9, but posterior prediction indicates that the off diagonal components vary too much.
This figure shows the histogram of the ratio between the minimum and maximum of the eigenvalues of the data (blue) and predictions (orange).
This figure is the trace of the data and the predictions respectively.
I guess as the d.o.f increases the magnitude of thediagonal increases too which decreases the likelihood, but for the off diagonal elements a higher d.o.f would be desirable. I wonder if there is a variant of the Wishart likelihood where the elements are normalized by the degrees of freedom?