Editing Bayesian statistics (section)

{{Short description|Theory and paradigm of statistics}}
{{Bayesian statistics}}

'''Bayesian statistics''' ({{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 a theory in the field of [[statistics]] based on the [[Bayesian probability|Bayesian interpretation of probability]], where [[probability]] expresses a ''degree of belief'' in an [[Event (probability theory)|event]]. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other [[Probability interpretations|interpretations of probability]], such as the [[Frequentist probability|frequentist]] interpretation, which views probability as the [[Limit of a sequence|limit]] of the relative frequency of an event after many trials.<ref name="bda">{{Cite book|last1=Gelman|first1=Andrew|title=Bayesian Data Analysis |edition=Third|last2=Carlin|first2=John B.|last3=Stern|first3=Hal S.|last4=Dunson|first4=David B.|last5=Vehtari|first5=Aki|last6=Rubin|first6=Donald B.|publisher=Chapman and Hall/CRC|year=2013|isbn=978-1-4398-4095-5|author-link1=Andrew Gelman|author-link2=John Carlin (professor)|author-link6=Donald Rubin}}</ref> More concretely, analysis in Bayesian methods codifies prior knowledge in the form of a [[prior distribution]].

Bayesian statistical methods use [[Bayes' theorem]] to compute and update probabilities after obtaining new data. Bayes' theorem describes the [[conditional probability]] of an event based on data as well as prior information or beliefs about the event or conditions related to the event.<ref name="rethinking">{{Cite book| title = Statistical Rethinking : A Bayesian Course with Examples in R and Stan |edition=2nd | publisher = Chapman and Hall/CRC | year = 2020 | isbn = 978-0-367-13991-9 |last1=McElreath|first1=Richard|author-link1=Richard McElreath}}</ref><ref>{{cite book |first=John |last=Kruschke |author-link=John K. Kruschke |title=Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan |location= |publisher=Academic Press |edition=2nd |year=2014 |isbn=978-0-12-405888-0 }}</ref> For example, in [[Bayesian inference]], Bayes' theorem can be used to estimate the parameters of a [[probability distribution]] or [[statistical model]]. Since Bayesian statistics treats probability as a degree of belief, Bayes' theorem can directly assign a probability distribution that quantifies the belief to the parameter or set of parameters.<ref name="bda" /><ref name="rethinking" />

Bayesian statistics is named after [[Thomas Bayes]], who formulated a specific case of Bayes' theorem in [[An Essay Towards Solving a Problem in the Doctrine of Chances|a paper]] published in 1763. In several papers spanning from the late 18th to the early 19th centuries, [[Pierre-Simon Laplace]] developed the Bayesian interpretation of probability.<ref>{{Cite book| title = The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy |edition=First | publisher = Chapman and Hall/CRC | year = 2012 | isbn = 978-0-3001-8822-6|last1=McGrayne|first1=Sharon|author-link1=Richard McElreath}}</ref> Laplace used methods now considered Bayesian to solve a number of statistical problems. While many Bayesian methods were developed by later authors, the term "Bayesian" was not commonly used to describe these methods until the 1950s. Throughout much of the 20th century, Bayesian methods were viewed unfavorably by many statisticians due to philosophical and practical considerations. Many of these methods required much computation, and most widely used approaches during that time were based on the frequentist interpretation. However, with the advent of powerful computers and new [[algorithm]]s like [[Markov chain Monte Carlo]], Bayesian methods have gained increasing prominence in statistics in the 21st century.<ref name="bda" /><ref>{{cite journal |last1=Fienberg |first1=Stephen E. |title=When Did Bayesian Inference Become "Bayesian"? |date=2006|journal=Bayesian Analysis|volume=1|issue=1|pages=1–40 |doi=10.1214/06-BA101 |doi-access=free }}</ref>