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).