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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 viepoint, and for example I particularly enjoy reading Peter Smith (e.g. Introduction to Gödel’s Theorems).

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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

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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. –  joeA Jun 18 '13 at 13:33
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