Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or monograph that introduces probability and statistics at this level and still covers frequentist and bayesian views (philosophy?) in a balanced manner?

It appears to me (but please correct me if I am wrong -- as I have stated I haven't learned this yet) that introductions at this level usually fully adopt a frequentist view and don't really broach the subject. On the other hand, bayesian books appear to be pitched to a more experienced audience and/or are perhaps even more unbalanced, in the sense that they seem anti-frequentist as much as pro-bayesian.

To more fully describe my mathematical maturity, I am comfortable with the normal calculus sequence (although somewhat rusty), basic linear algebra, basic set theory, mathematical logic, computability theory, some abstract algebra, very little category theory. I am comfortable with the level of introductory analysis, but have not completed it, and I am not versed in measure theory (I expect I could handle measure theory, but knowledge of it shouldn't be assumed). I am often interested in foundational topics and a philosophical viewpoint, and for example I particularly enjoy reading Peter Smith (e.g. An Introduction to Gödel’s Theorems).


I would recommend the following 16-page article from 36 years ago, which is easily accessible to any upper-level undergraduate:

"Controversies in the Foundations of Statistics", Bradley Efron, The American Mathematical Monthly, vol. 85, 1978, pp. 231-246.

This won an MAA Writing Award.

In my inexpert opinion, it is quite unsettling. It does not offer a resolution.

Edit: The above-mentioned article impressed me when I was a student. But there has been much follow-up. For example:

"A Two-Hundred-and-Fifty-Year Argument", Bradley Efron, LASR 2011 — Next Generation Statistics in Biosciences (Proceedings, 30th Leeds Annual Statistical Research Workshop), 2011

"Bayes Theorem in the Twenty First Century", Bradley Efron, Science 340, June 7, 2013


Further edit: Oops, my ignorance is showing.

B. Efron was president of the ASA in 2004; see his presidential address "Bayesians, Frequentists, and Scientists" at https://www.asapresidentialpapers.info/documents/Efron_2004_BAK_01-05-10.pdf.

He recently won the Guy Medal in Gold, https://en.wikipedia.org/wiki/Guy_Medal.

See also http://statweb.stanford.edu/~ckirby/brad/EfronCV.pdf and https://en.wikipedia.org/wiki/Bradley_Efron and https://math.stackexchange.com/a/116444/18972.

  • 3
    $\begingroup$ Thank you. I have skimmed the articles and they look interesting. I'll read them more carefully when I have the chance. However, what I'm really looking for is a book-level introduction to probability and statistics that treats frequentist and bayesian viewpoints (and perhaps Fisherian? hadn't heard that one until I looked at the first article) in a balanced manner. Note that it doesn't need to be completely focused on this, just not completely ignore it and/or assume one of the other viewpoints. Perhaps I'm barking up the wrong tree here. $\endgroup$
    – joeA
    Jun 18 '13 at 13:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.