(Theoretical) Bayesian statistics is the study of probability flows under minimal assumptions - any quantity that behaves like we want a probability to behave can be described by Bayesian statistics.
But nobody, least of all Bayesian statistical practitioners, does this. They encounter data, get familiar with it, pick/invent a model, pick/invent a prior, run (possibly approximate) inference of the model against the data, verify if inference is doing something reasonable, and jump back to an earlier step and change something if it doesn't. After however long this takes (if they don't give up), they might make some decision based on the (possibly approximate) posterior distribution they end up with. This decision might involve taking some actions in the wider world and/or writing a paper.
This is essentially the same workflow a frequentist statistician would use, and it's only reasonable that a lot of the ideas that work in one of these settings would be useful, if not obvious or well-motivated, in the other.
I know that philosophical underpinnings and underlying frameworks matter but to quote from a recent review article by Reid and Cox (2014):
A healthy interplay between theory and application is crucial for statistics, as no doubt for other fields. This is particularly the case when by theory we mean foundations of statistical analysis, rather than the theoretical analysis of specific statistical methods. The very word foundations may, however, be a little misleading in that it suggests a solid base on which a large structure rests for its entire security. But foundations in the present context equally depend on and must be tested and revised in the light of experience and assessed by relevance to the very wide variety of contexts in which statistical considerations arise. It would be misleading to draw too close a parallel with the notion of a structure that would collapse if its foundations were destroyed.
But nobody, least of all Bayesian statistical practitioners, does this.
Well obviously. Same for physicists, nobody (other than some highly specialised teams working at particle accelerators) use the standard model to compute the predictions of their models. Or for computer science - most computer scientists don't write code at the binary level, or explicitly give commands to individual transistors. Or chemists - just how many of the reaction equations do you think are being checked by solving the quantum mechanics? But just because the underlying theory...
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?