To pick a frequentist algorithm is to pick a prior with a set of hypotheses, i.e. to make Bayes' Theorem computable and provide the unknowns on the r.h.s. above (as mentioned earlier you can in theory extract the prior and set of hypotheses from an algorithm by considering which outcome your algorithm would give when it saw a certain set of data, and then inverting Bayes' Theorem to find the unknowns.
Okay, this is the last thing I'll say here until/unless you engage with the Robins and Wasserman post that IlyaShpitser and I have been suggesting you look at. You can indeed pick a prior and hypotheses (and I guess a way to go from posterior to point estimation, e.g., MAP, posterior mean, etc.) so that your Bayesian procedure does the same thing as your non-Bayesian procedure for any realization of the data. The problem is that in the Robins-Ritov example, your prior may need to depend on the data to do this! Mechanically, this is no problem; philosophically, you're updating on the data twice and it's hard to argue that doing this is unproblematic. In other situations, you may need to do other unsavory things with your prior. If the non-Bayesian procedure that works well looks like a Bayesian procedure that makes insane assumptions, why should we look to Bayesian as a foundation for statistics?
(I may be willing to bite the bullet of poor frequentist performance in some cases for philosophical purity, but I damn well want to make sure I understand what I'm giving up. It is supremely dishonest to pretend there's no trade-off present in this situation. And a Bayes-first education doesn't even give you the concepts to see what you gain and what you lose by being a Bayesian.)
I have started to put together a sort of curriculum for learning the subjects that lend themselves to rationality. It includes things like experimental methodology and cognitive psychology (obviously), along with "support disciplines" like computer science and economics. I think (though maybe I'm wrong) that mathematics is one of the most important things to understand.
Eliezer said in the simple math of everything:
I want to have access to outlook-changing insights. So, what math do I need to know? What are the generally applicable mathematical principles that are most worth learning? The above quote seems to indicate at least calculus, and everyone is a fan of Bayesian statistics (which I know little about).
Secondarily, what are some of the most important of that "drop-dead basic fundamental embarrassingly simple mathematics" from different fields? What fields are mathematically based, other than physics and evolutionary biology, and economics?
What is the most important math for an educated person to be familiar with?
As someone who took an honors calculus class in high school, liked it, and did alright in the class, but who has probably forgotten most of it by now and needs to relearn it, how should I go about learning that math?