I’d like to attempt a summary of the “divergence” from the original post (which I actually had nudged along) and then redirect the discussion with a quote that, in part, captures the kind of thing I was attempting to convey in starting this topic. However, I am not opposed to going back to the current discussion (I have enjoyed them greatly and have a long reading list thanks to the posters). I was just hoping to satisfy my initial question, even if the only “real” answer is along the lines of “go back and get a graduate degree in math and/or statistics”.
Summary
It seems that there is controversy in the prob/stats world regarding even a “simple” question such as “what is probability?”, which strikes me as a philosophical question at heart; see for example this forum topic. I don’t know whether the arguments between Frequentism, Bayesianism, and Likelihoodism all essentially stem from different “views” regarding how one should define “probability”, or whether there are purely methodological differences, but it seems to be academic to me (at times even dogmatic). There are even disagreements within “camps” (for example, see the topic @R_cubed linked to on whether P-values should be abolished or the “threshold” changed).
Coincidentally, there are several reviews of Deborah Mayo’s book (Statistical Inference as Severe Testing) on Gelman’s blog (also typed up in LaTeX and posted to arXiv if you prefer that format). In one of her responses she states that
The disagreements often grow out of hidden assumptions about the nature of scientific inference and the roles of probability in inference. Many of these are philosophical.
[I can’t comment on whether this is a true representation of the field, nor did I mention her book in support of her views as I only just learned of its existence.]
I think I agree with @f2harrell’s sentiment in his post on the topic on this forum: it is all interesting intellectually but not very useful in the real world. I am, ultimately, interested in understanding more and some excellent references have been shared already; I intend to read much of what was shared and appreciate everyone’s contributions. But beyond the intrinsic reward of accumulating knowledge and intellectual discussions, I don’t have much use for philosophy currently.
In any case, I still believe that a strong foundation in math (and to some extent theory) is required to properly appreciate such issues, even if it all boils down to differences in “philosophy”. I currently don’t have that foundation; think the Dunning-Kruger effect (I don’t even know all that I don’t know about the subject).
Back to Intuition
I recently came across an interesting historical comment by the late Sir David R. Cox recounting several “pioneers in modern statistical theory”. The quote that stood out to me was about John Tukey (Section 13 of the article):
Some 20 or more years later, he unexpectedly came to see me at home in London one Sunday. What was I working on? I told him; it was something that he was highly unlikely to have known about. After a few moments of thought he made ten suggestions. Six I had already considered, two would clearly not work, and the other two were strong ideas which had not occurred to me even after long thought on the topic. This small incident illustrates one of his many strengths: the ability to comment searchingly and swiftly on a very wide range of issues.
Although Sir David gives no indication as to what he was working on, the passage highlights what is, to me, the “ultimate goal”. That is, the ability to give a thoughtful (perhaps critical) appraisal of a problem based only on the basics and a bit of thought (and probably some questions about the problem/data/goal). Is this kind of ability even achievable for a “non-Tukey” (who, by all accounts I’ve seen, was regarded a brilliant man)? Or does it require years of experience, supported by a strong foundation in mathematics?
P.S. Gelman’s blog
The post on Gelman’s blog linked to above links to some papers that I haven’t had a chance to read yet but appear to be extremely interesting. There is an additional review by Christian Robert at another blog.
The first paper is a “monograph” by Robert Cousins. It looks quite interesting based on my brief reading of a few of the sections.
Another is a paper by Gelman and Christian Hennig (including extensive discussion after the article): Gelman & Hennig (2017). Beyond subjective and objective in statistics. J R Statist Soc A. I haven’t had a chance to go through it all yet.
He also links to another blog post of his discussing his article with Cosma Shalizi (Philosophy and the practice of Bayesian statistics) (along with responses).
P.P.S.
I also stumbled upon these lecture notes by Adam Caulton that look interesting. (I actually found when I was googling “savage ramsey finetti jeffreys jaynes” )
Apologies for how long this ended up!