Editing Bayesian probability (section)

{{broader|Bayesian statistics}}
{{Bayesian statistics}}{{Short description|Interpretation of probability}}'''Bayesian probability''' ({{IPAc-en|ˈ|b|eɪ|z|i|ə|n}} {{respell|BAY|zee|ən}} or {{IPAc-en|ˈ|b|eɪ|ʒ|ən}} {{respell|BAY|zhən}}){{refn|{{MerriamWebsterDictionary|=2023-08-12|Bayesian}}}} is an [[Probability interpretations|interpretation of the concept of probability]], in which, instead of [[frequentist probability|frequency]] or [[propensity probability|propensity]] of some phenomenon, probability is interpreted as reasonable expectation<ref>{{Cite journal |last=Cox |first=R.T. |author-link=Richard Threlkeld Cox |doi=10.1119/1.1990764 |title=Probability, Frequency, and Reasonable Expectation |journal=American Journal of Physics |volume=14 |issue=1 |pages=1–10 |year=1946 |bibcode=1946AmJPh..14....1C }}</ref> representing a state of knowledge<ref name="ghxaib">{{cite book |author=Jaynes, E.T. |year=1986 |contribution=Bayesian Methods: General Background |title=Maximum-Entropy and Bayesian Methods in Applied Statistics |editor=Justice, J. H. |location=Cambridge |publisher=Cambridge University Press|bibcode=1986mebm.conf.....J |citeseerx=10.1.1.41.1055 }}</ref> or as quantification of a personal belief.<ref name="Finetti, B. 1974">{{cite book |last1=de Finetti |first1=Bruno |title=Theory of Probability: A critical introductory treatment |year=2017 |publisher=John Wiley & Sons Ltd. |location=Chichester|isbn=9781119286370}}</ref>

The Bayesian interpretation of probability can be seen as an extension of [[propositional logic]] that enables reasoning with [[Hypothesis|hypotheses]];<ref name="Hailperin, T. 1996">{{cite book |last1=Hailperin |first1=Theodore |title=Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications |year=1996 |publisher=Associated University Presses|location=London|isbn=0934223459}}</ref><ref>{{cite book |first=Colin |last=Howson |chapter=The Logic of Bayesian Probability |pages=137–159 |editor-first=D. |editor-last=Corfield |editor2-first=J. |editor2-last=Williamson |title=Foundations of Bayesianism |location=Dordrecht |publisher=Kluwer |year=2001 |isbn=1-4020-0223-8 }}</ref> that is, with propositions whose [[truth value|truth or falsity]] is unknown. In the Bayesian view, a probability is assigned to a hypothesis, whereas under [[frequentist inference]], a hypothesis is typically tested without being assigned a probability.

Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies a [[prior probability]]. This, in turn, is then updated to a [[posterior probability]] in the light of new, relevant [[data]] (evidence).<ref name="paulos">{{cite web |author-link=John Allen Paulos |last=Paulos |first=John Allen |url=https://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html |archive-url=https://ghostarchive.org/archive/20220101/https://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html |archive-date=2022-01-01 |url-access=limited |title=The Mathematics of Changing Your Mind [by Sharon Bertsch McGrayne] |department=Book Review |newspaper=New York Times |date=5 August 2011 |access-date=2011-08-06}}{{cbignore}}</ref> The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation.

The term ''Bayesian'' derives from the 18th-century mathematician and theologian [[Thomas Bayes]], who provided the first mathematical treatment of a non-trivial problem of statistical [[data analysis]] using what is now known as [[Bayesian inference]].<ref name="HOS">{{cite book |last1=Stigler |first1=Stephen M. |title=The history of statistics |date=March 1990 |publisher=Harvard University Press |isbn=9780674403413}}</ref>{{rp|131}} Mathematician [[Pierre-Simon Laplace]] pioneered and popularized what is now called Bayesian probability.<ref name="HOS" />{{rp|97–98}}