Editing Bayesian probability (section)

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

===Axiomatic approach===
[[Richard Threlkeld Cox|Richard T. Cox]] showed that Bayesian updating follows from several axioms, including two [[functional equations]] and a hypothesis of differentiability.<ref name = "vkdmsn" /><ref>{{cite book |first1=C. Ray |last1=Smith |first2=Gary |last2=Erickson |chapter=From Rationality and Consistency to Bayesian Probability |pages=29–44 |title=Maximum Entropy and Bayesian Methods |editor-first=John |editor-last=Skilling |location=Dordrecht |publisher=Kluwer |year=1989 |isbn=0-7923-0224-9 |doi=10.1007/978-94-015-7860-8_2 }}</ref> The assumption of differentiability or even continuity is controversial; Halpern found a counterexample based on his observation that the Boolean algebra of statements may be finite.<ref>{{cite journal |author=Halpern, J. |title=A counterexample to theorems of Cox and Fine |journal=Journal of Artificial Intelligence Research |volume=10 |pages=67–85|url=http://www.cs.cornell.edu/info/people/halpern/papers/cox.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.cs.cornell.edu/info/people/halpern/papers/cox.pdf |archive-date=2022-10-09 |url-status=live|doi=10.1613/jair.536 |year=1999 |s2cid=1538503 |doi-access=free }}</ref> Other axiomatizations have been suggested by various authors with the purpose of making the theory more rigorous.<ref name="rbp">{{cite journal |author1=Dupré, Maurice J. |author2=Tipler, Frank J. |url=http://projecteuclid.org/download/pdf_1/euclid.ba/1340369856 |title=New axioms for rigorous Bayesian probability |journal=Bayesian Analysis |volume=4 |year=2009 |issue=3 |pages=599–606|doi=10.1214/09-BA422 |citeseerx=10.1.1.612.3036 }}</ref>

===Dutch book approach===
{{main|Dutch book}}
[[Bruno de Finetti]] proposed the Dutch book argument based on betting. A clever [[bookmaker]] makes a [[Dutch book]] by setting the [[odds]] and bets to ensure that the bookmaker profits—at the expense of the gamblers—regardless of the outcome of the event (a horse race, for example) on which the gamblers bet. It is associated with [[probability|probabilities]] implied by the odds not being [[Coherence (philosophical gambling strategy)|coherent]].

However, [[Ian Hacking]] noted that traditional Dutch book arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books. For example, [[Ian Hacking|Hacking]] writes<ref>Hacking (1967), Section 3, page 316</ref><ref>Hacking (1988, page 124)</ref> "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour."

In fact, there are non-Bayesian updating rules that also avoid Dutch books (as discussed in the literature on "[[probability kinematics]]"<ref>{{cite journal |last=Skyrms |first=Brian |date=1987-01-01 |title=Dynamic Coherence and Probability Kinematics |journal=Philosophy of Science |volume=54 |issue=1 |pages=1–20 |doi=10.1086/289350 |jstor=187470 |citeseerx=10.1.1.395.5723 |s2cid=120881078 |df=dmy-all}}</ref> following the publication of [[Richard Jeffrey|Richard C. Jeffrey]]'s rule, which is itself regarded as Bayesian<ref>{{cite encyclopedia |url=http://plato.stanford.edu/entries/bayes-theorem/ |title=Bayes' Theorem |publisher =stanford.edu |df=dmy-all|last = Joyce|first = James|encyclopedia= The Stanford Encyclopedia of Philosophy |date =30 September 2003 }}</ref>). The additional hypotheses sufficient to (uniquely) specify Bayesian updating are substantial<ref>{{Cite book |title=Probability in Physics |url=https://archive.org/details/probabilityphysi00benm |url-access=limited |last1=Fuchs |first1=Christopher A. |last2=Schack |first2=Rüdiger |chapter=Bayesian Conditioning, the Reflection Principle, and Quantum Decoherence |date=2012-01-01 |publisher=Springer Berlin Heidelberg |isbn=9783642213281 |editor-last1=Ben-Menahem |editor-first1=Yemima |series=The Frontiers Collection |pages=[https://archive.org/details/probabilityphysi00benm/page/n245 233]–247 |language=en |arxiv=1103.5950 |doi=10.1007/978-3-642-21329-8_15 |s2cid=119215115 |editor-last2=Hemmo |editor-first2=Meir |df=dmy-all}}</ref> and not universally seen as satisfactory.<ref>{{cite book |author-link=Bas van Fraassen |last=van Frassen |first=Bas |year=1989 |title=Laws and Symmetry |publisher=Oxford University Press |isbn=0-19-824860-1}}</ref>

===Decision theory approach===
A [[statistical decision theory|decision-theoretic]] justification of the use of Bayesian inference (and hence of Bayesian probabilities) was given by [[Abraham Wald]], who proved that every [[admissible decision rule|admissible]] statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures.<ref>{{cite book |author=Wald, Abraham |title=Statistical Decision Functions |publisher=Wiley |year=1950}}</ref> Conversely, every Bayesian procedure is [[admissible decision rule|admissible]].<ref>{{cite book |author1=Bernardo, José M. |author2=Smith, Adrian F.M. |title=Bayesian Theory |publisher=John Wiley |year=1994 |isbn=0-471-92416-4}}</ref>