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The use of Bayesian probabilities as the basis of Bayesian inference has been supported by several arguments, such as Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem.

A decision-theoretic justification of the use of Bayesian inference (and hence of Bayesian probabilities) was given by Abraham Wald, who proved that every admissible statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures.[26] Conversely, every Bayesian procedure is admissible.[27]

Wikipedia: Bayesian Probability - Justification of Bayesian Probabilities

What's the best book on the decision-theoretic justification of bayesian probability? I found a few books on the topic:

  • The Bayesian choice - Christian P Robert
  • Bayesian Theory - Adrian Smith and José-Miguel Bernardo
  • Statistical Decision Theory and Bayesian Analysis - Jim Berger

Which one of these books is the best and most relevant to the topic? Are there any other books that are better than these?

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    $\begingroup$ Check out Principles of Uncertainty by Joesph Kadane. $\endgroup$
    – aduh
    Nov 12, 2020 at 8:59
  • $\begingroup$ Thank you for your reply. I think that book is about subjective-bayesian/deFinetti probability theory. Are there any books that justify probability theory purely using decision theory? $\endgroup$
    – user487586
    Nov 12, 2020 at 9:50
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    $\begingroup$ What do you think "deFinetti probability theory" is? $\endgroup$
    – aduh
    Nov 12, 2020 at 10:18
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    $\begingroup$ I was being a bit glib. My point was that de Finetti's approach is decision-theoretic. But never mind. If you're interested in Wald's work in particular, then why not just read his books and papers first? $\endgroup$
    – aduh
    Nov 12, 2020 at 19:42
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    $\begingroup$ Your question just isn't very clearly stated, and your comments aren't clarifying things. What do you mean by "decision-theoretic foundations"? What does it mean to "prove probability theory using decision theory"? $\endgroup$
    – aduh
    Nov 12, 2020 at 19:44

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I know the third you mentioned and it is a good textbook, even if my favourite is Box Tiao

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  • $\begingroup$ Thank you. Does Berger justify/derive bayesian probability theory using decision theory or does he focus on the applications of bayesian methods in making decisions? $\endgroup$
    – user487586
    Nov 12, 2020 at 10:00
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    $\begingroup$ @HaziqMuhammad : it's a very complete book including Bayesian Analysis (Chapers 1-4), Bayesian methods in making decision under uncertainty (i.e. minimax, and Bayesian sequential analysis) in Chapters 5-8 $\endgroup$
    – tommik
    Nov 12, 2020 at 10:23
  • $\begingroup$ I'm sorry, I'm still a little confused :). I'm looking for a book that proves probability theory using decision theory. Is that what you meant by Bayesian Analysis? $\endgroup$
    – user487586
    Nov 12, 2020 at 11:23
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    $\begingroup$ @HaziqMuhammad : it's included over there... but I do not know a book with specific focus on that, sorry $\endgroup$
    – tommik
    Nov 12, 2020 at 11:26
  • $\begingroup$ No worries. Thanks a lot for your help! Much appreciated $\endgroup$
    – user487586
    Nov 12, 2020 at 11:39

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