The Foundations of Statistics,   ISBN:9780486623498

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One of the earliest books/texts to present the Bayesian paradigm. You need at least three semesters of calculus to do the math - Savage was one of the statisticians who worked on the Manhattan Project and this text is very rigorous.

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John Maynard Keynes's A Treatise on Probability(1921;TP)was given a five star rating by this reviewer because he dealt clearly and succinctly with what Ellsberg later called "ambiguity".Keynes did not use the terminology of Ellsberg.In the TP Keynes analyzed the problem of ambiguity using the term "weight of the evidence".In the General Theory(GT)Keynes used the term"uncertainty".Keynes defined an index to measure the weight of the evidence,denoted by the variable w.By definition,0<=w<=1(See p.315,TP,chapter 26).Keynes then incorporated w into a decision rule which he called a conventional coefficient of weight and risk c.Given outcome A,the decisionmaker's goal is to maximize cA,as opposed to pA or pU(A),where p equals a probability(p+q=1)and U(A)is a utility function.c=p(1/(1+q)[2w/1+w].Savage did discuss the problem of ambiguity or uncertainty or lack of evidential weight using the term vagueness.Unfortunately,after discussing the problem of vagueness,Savage decided not to deal with this problem technically.Savage's technical analysis of decision making under risk is certainly very interesting.However,his failure to deal with the problem that Keynes had already solved in his TP means that I can only give Savage four stars.

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Savage rigorously reworks the approach of Frank Plumpton Ramsey to give us a personalist (subjectivist) probability construct, and uses this to found a theory of statistics. Even many of us who disagree strongly with Savage are greatly impressed by this remarkable work.