Editing Bayesian probability (section)

==History==
{{Main|History of statistics#Bayesian statistics}}

The term ''Bayesian'' derives from [[Thomas Bayes]] (1702–1761), who proved a special case of what is now called [[Bayes' theorem]] in a paper titled "[[An Essay Towards Solving a Problem in the Doctrine of Chances]]".<ref>{{cite book |author=McGrayne, Sharon Bertsch |year=2011 |title=The Theory that Would not Die |url=https://archive.org/details/theorythatwouldn0000mcgr |url-access=registration |at={{Google books|_Kx5xVGuLRIC|&nbsp;|page=[https://archive.org/details/theorythatwouldn0000mcgr/page/10 10]}} }}</ref> In that special case, the prior and posterior distributions were [[beta distribution]]s and the data came from [[Bernoulli trial]]s. It was [[Pierre-Simon Laplace]] (1749–1827) who introduced a general version of the theorem and used it to approach problems in [[celestial mechanics]], medical statistics, [[Reliability (statistics)|reliability]], and [[jurisprudence]].<ref>{{cite book |author=Stigler, Stephen M. |year=1986 |title=The History of Statistics |chapter-url=https://archive.org/details/historyofstatist00stig |chapter-url-access=registration |publisher=Harvard University Press |chapter=Chapter&nbsp;3|isbn=9780674403406 }}</ref> Early Bayesian inference, which used uniform priors following Laplace's [[principle of insufficient reason]], was called "[[inverse probability]]" (because it [[Inductive reasoning|infer]]s backwards from observations to parameters, or from effects to causes).<ref name=Fienberg2006>{{cite journal |author=Fienberg, Stephen. E. |year=2006 |url=http://ba.stat.cmu.edu/journal/2006/vol01/issue01/fienberg.pdf |title=When did Bayesian Inference become "Bayesian"? |archive-url=https://web.archive.org/web/20140910070556/http://ba.stat.cmu.edu/journal/2006/vol01/issue01/fienberg.pdf |archive-date=10 September 2014 |journal=Bayesian Analysis |volume=1 |issue=1 |pages=5, 1–40|doi=10.1214/06-BA101 |doi-access=free }}</ref> After the 1920s, "inverse probability" was largely supplanted by a collection of methods that came to be called [[frequentist statistics]].<ref name=Fienberg2006/>

In the 20th century, the ideas of Laplace developed in two directions, giving rise to ''objective'' and ''subjective'' currents in Bayesian practice.
[[Harold Jeffreys]]' ''Theory of Probability'' (first published in 1939) played an important role in the revival of the Bayesian view of probability, followed by works by [[Abraham Wald]] (1950) and [[Leonard J. Savage]] (1954). The adjective ''Bayesian'' itself dates to the 1950s; the derived ''Bayesianism'', ''neo-Bayesianism'' is of 1960s coinage.<ref>{{cite journal |quote=The works of [[Abraham Wald|Wald]], ''Statistical Decision Functions'' (1950) and [[Leonard J. Savage|Savage]], ''The Foundation of Statistics'' (1954) are commonly regarded starting points for current Bayesian approaches |title=Recent developments of the so-called Bayesian approach to statistics |first=Marshall Dees |last=Harris |journal=Legal-Economic Research |publisher=University of Iowa |department=Agricultural Law Center |year=1959 |pages=125 (fn. #52), 126}}</ref><ref>{{cite book |quote=This revolution, which may or may not succeed, is neo-Bayesianism. Jeffreys tried to introduce this approach, but did not succeed at the time in giving it general appeal. |title=Annals of the Computation Laboratory of Harvard University |volume=31 |year=1962 |page=180}}</ref><ref>{{cite conference |quote=It is curious that even in its activities unrelated to ethics, humanity searches for a religion. At the present time, the religion being 'pushed' the hardest is Bayesianism. |first=Oscar |last=Kempthorne |title=The Classical Problem of Inference—Goodness of Fit |conference=Fifth Berkeley Symposium on Mathematical Statistics and Probability |year=1967 |url=https://books.google.com/books?id=IC4Ku_7dBFUC&pg=PA235 |page=235}}</ref> In the objectivist stream, the statistical analysis depends on only the model assumed and the data analysed.<ref name="Bernardo">{{cite book |last1=Bernardo |first1=J.M. |title=Bayesian Thinking - Modeling and Computation |publisher=Handbook of Statistics |year=2005 |isbn=9780444515391 |volume=25 |pages=17–90 |chapter=Reference analysis |series=Handbook of Statistics |doi=10.1016/S0169-7161(05)25002-2 |author-link=José-Miguel Bernardo}}</ref> No subjective decisions need to be involved. In contrast, "subjectivist" statisticians deny the possibility of fully objective analysis for the general case.

In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of [[Markov chain Monte Carlo]] methods and the consequent removal of many of the computational problems, and to an increasing interest in nonstandard, complex applications.<ref>{{cite journal |author=Wolpert, R.L. |year=2004 |title=A conversation with James O. Berger |journal=Statistical Science |volume=9 |pages=205–218|doi=10.1214/088342304000000053 |doi-access=free }}</ref> While frequentist statistics remains strong (as demonstrated by the fact that much of undergraduate teaching is based on it <ref>{{cite conference |author-link=José-Miguel Bernardo |author=Bernardo, José M. |year=2006 |url=http://www.ime.usp.br/~abe/ICOTS7/Proceedings/PDFs/InvitedPapers/3I2_BERN.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.ime.usp.br/~abe/ICOTS7/Proceedings/PDFs/InvitedPapers/3I2_BERN.pdf |archive-date=2022-10-09 |url-status=live |title=A Bayesian mathematical statistics primer |conference=ICOTS-7 |location=Bern}}</ref>), Bayesian methods are widely accepted and used, e.g., in the field of [[machine learning]].<ref name="ReferenceA">{{cite book |author=Bishop, C.M. |title=Pattern Recognition and Machine Learning |publisher=Springer |year=2007}}</ref>