There is some frequentist procedure there, but it seems hard to not recognize the deep connection to Bayesian statistics and wonder if you should begin to question your baseline assumptions. Since the entire justification for using a shrinkage estimator has a whole lot more in common with the foundations of Bayesian statistics than it does with the foundations of frequentist stats.
Purist frequentists using a shrinkage estimator looks a lot like heliocentric Ptolemic astronomy.
Downvote me all you want. Bayesianism is misapplied much more frequently than frequentism. It just makes it way too easy to fudge p values. Sorry not sorry.
I do always laugh when I see a Bayesian object to p-values, then use a Bayesian procedure that is mathematically identical to treating p values as posterior probabilities.
Just saying the word "Bayesian" doesn't actually make it different
It’s mathematically identical but conceptually different. The things that go into the calculation are different, the numbers that get out of the calculation mean different things. Laughing is healthy though.
You can just shortcut all of that if you're a Bayesian and just plain say "p-values are posterior probabilities under a uniform (improper) prior" and save everyone a lot of time.
And if you're doing that, don't care complain that p-values can be misinterpreted, because you're basically just laundering the misinterpretation of p-values.
Sure, you are mathematically pure because you made an initial assumption that it can be so, rather than being confused, but the end result is the same.
I have no interest in laundering the misinterpretation of p-values. I don’t know whose time is saved, frequentists don’t care about the Bayesian interpretation anyway and Bayesians don’t need to restrict themselves to a particular prior. The fact that in some cases you can choose one particular prior to get a numerical value for one probability that is equal to the numerical value for the probability of a completely different thing calculated by someone else doesn’t help anyone much. Bayesians can get the same result on their own if they want.
