Empirical Bayes is exactly what I was getting at. It's a pragmatic modelling choice, but it loses the theoretical guarantees about uncertainty quantification that pure Bayesianism gives us.
(Though if you have a reference for why empirical Bayes does give theoretical guarantees, I'll be happy to change my mind!)
(Though if you have a reference for why empirical Bayes does give theoretical guarantees, I'll be happy to change my mind!)