An excellent book. Three things I like: (1) it is correct (so many others are not), (2) it can be read by someone who does not have a PhD in math, (3) they don't pull punches. Appendix B explains directly why all alternative theories are nonsense.

Bernardo and Smith(BS)have written a book that assumes that Frank Ramsey, Bruno De Finetti,and Leonard Savage solved all of the major problems concerning the foundations of probability and decision theory in the period between 1931,the year Ramsey's major essay on probability was published,and 1954,the year that Savage published his book.All that remains is a mopping up effort at minor,residual anomalies.The basic point made by BS is that all probabilities are precise,single number, point estimates or that they can be treated "as if" they were.Unfortunately,this is not the case.The subjectivist approach is applicable only in those situations where the purely deductive,mathematical laws of probability(the addition and multiplication rules for conjunction and disjunction)apply.This requires that a)there exists a complete sample space of all possible outcomes representing the choice problem before any probability is calculated;b)a complete preference ordering of all possible outcomes exists for the problem or c)a single,unique probability distribution is defined for the problem.Under these conditions,the probability calculus serves as a consistency and coherence check for the rational decision maker who is willing to bet on one side or another of all propositions.The subjectivist approach is a special theory with limited applicability.It is this failure to recognize that the subjective approach is a limiting case, that conflates the concepts of probability,logical probability,inductive probability,and degree of belief with mathematical probability, that is the source of much of the criticism of the subjectivist approach.There are many assertions made throughout the book that are highly dubious and/or unsupported.The rest of the review will be devoted to correcting these assertions.First,it is not the case that the Allais paradox choices are mistaken.It is strange to see it argued that such choices are similar to"...individuals(who)can often be shown to perform badly at deduction or long division"(BS,P.97).The real problem is that many/some decision makers have nonlinear probability preferences,as opposed to the linear probability preferences axiomatised by the subjectivists.The BS claim is similar to the claim made by many proponents of Euclidean geometry in the 18th and 19th centuries that non Euclidean geometries were erroneous and/or could not exist.Second,it is not the case that the Raiffa(1961) and Roberts(1963)replies to Ellsberg provide"...clear and convincing rejoinders to the Ellsberg criticisms"(BS,P.98).Both Raiffa and Roberts,like Savage in his belated reply to Allais,simply restructured and changed the problem on which they commented.Third,the claim that the Ellsberg problems and/or examples(the two color and three color urn ball problems)are"...optical or magical illusions..." makes no sense.Fourth,the claim that "The logical(emphasis added)view is entirely lacking in operational content." (BS,p.100),has no support at all.It is impossible to even talk about scientific theories unless an underlying logical conceptualization of probability is already in place beforehand.Fifth,the claim that John Maynard Keynes changed his view in 1931 and accepted the primacy of the subjectivist interpretation of F.Ramsey is erroneous.Keynes accepted Ramsey's dutch book argument claim only if the deductive,purely mathematical laws of probability("...the calculus of probability...") were completely operational.Keynes completely rejected Ramsey's assertions that habits and memory alone were the only foundations for induction and analogy.Sixth,BS are completely and totally ignorant about Keynes's establishment of the interval estimate approach to probability in this century.It is a widespread misbelief on the part of many economists,philosophers,psychologists,etc.,that only partial, ordinal rankings,that could be made only part of the time,represents the main outcome of Keynes's 16 years of study of probability.Nothing could be further from the truth.In fact,this misbelief is due to the acceptance by most scholars of the conclusions arrived at in the horrible mess made of Keynes's book by Ramsey in both his 1922 and 1926 reviews,respectively.Ramsey's unsupported claims about Keynes's strange nonnumerical probabilities and mysterious logical relations are just that,unsupported.Most Keynesian probabilities have an upper and a lower bound or limit. It is in chapters 15 and 17 of Keynes's 1921 A Treatise on Probability(TP) that BS can find Keynes's "approximation" approach worked out in great detail.A number of problems are worked out by Keynes on pp.161-163 and pp.186-194 of the TP.All of these problems can now be solved using easier integer-mixed integer linear programming techniques.Keynes's approach is fully operational.Seventh,the claim that Keynes's logical approach provides "...no operational guidance as to how to choose..."(BS,p.99)makes it crystal clear to this reviewer that BS have never read Keynes's TP.It is a great tragedy that books can be written on probability by authors that are grossly ignorant of basic literature.

[1] It is an excellent book on the classical Bayesian theory. The first author is a famous mathematician, who held several international conferences on Bayesian statistics.
[2] Similar to Berger's book, it is also built on Statistical Decision Theory. In my opinion, Berger's is a little better.
[3] The part of Bayesian foundation is heavy, maybe a topos today. But in the bookshelf, we indeed need such work.
[4] Think about the thickness of the bibliography --- the reference is awesome!
[5] The history of Bayesian statistics is well overviewed.
[6] To learn more about the Bayesian computation, you need some complement books, such as Liu's, Tanner's, Gelman's, etc.