A couple of readers sent me notes about a recent post on (Normal Deviate)* that introduces the term “frequentist
“If we manipulate the data to get a posterior that mimics the frequentist answer, is this really a success for Bayesian inference? Is it really Bayesian inference at all? Similarly, if we choose a carefully constructed prior just to mimic a frequentist answer, is it really Bayesian inference? We call Bayesian inference which is carefully manipulated to force an answer with good frequentist behavior, frequentist pursuit. There is nothing wrong with it, but why bother?
If you want good frequentist properties just use the frequentist estimator.”(Robins and Wasserman)
I take it that the Bayesian response to the question (“why bother?”) is that the computations yield that magical posterior (never mind just how to interpret them).
Cox and Mayo 2010 say, about a particular example of “frequentist
“Reference priors yield inferences with some good frequentist properties, at least in one-dimensional problems – a feature usually called matching. … First, as is generally true in science, the fact that a theory can be made to match known successes does not redound as strongly to that theory as did the successes that emanated from first principles or basic foundations. This must be especially so where achieving the matches seems to impose swallowing violations of its initial basic theories or principles.
Even if there are some cases where good frequentist solutions are more neatly generated through Bayesian machinery, it would show only their technical value for goals that differ fundamentally from their own.” (301)
Imitation, some say, is the most sincere form of flattery. I don’t agree, but doubtless it is a good thing that we see a degree of self-imposed and/or subliminal frequentist constraints on much Bayesian work in practice. Some (many?) Bayesians suggest that this is merely a nice extra rather than necessary, forfeiting the (non-trivial) pursuit of frequentist (error statistical foundations) for Bayesian pursuits**.
*I had noticed this, but had no time to work through the thicket of the example he considers. I welcome a very simple upshot.
**At least some of them.
“I take it that the Bayesian response to the question (“why bother?”) is that the computations yield that magical posterior (never mind just how to interpret them).”
In this case, the actual Bayesian response was, “And what I [Chris Sims] did involved no ‘frequentist pursuit’ whatsoever. It is just straightforward infinite-dimensional Bayesian inference with the correct likellihood. The simple information-wasting Bayesian method I derived aimed at achieving simplicity by ignoring information, not at duplicating frequentist properties — it actually improves on Horwitz-Thompson without having started out with that objective.”
I can’t help you with a summary — I’m working on other things, so I haven’t gone through the arguments and counterarguments in detail.