Bayesian statistics information

Bayesian statistics (/ˈbeɪziən/ BAY-zee-ən or /ˈbeɪʒən/ BAY-zhən)^[1] is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a degree of belief in an 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 interpretations of probability, such as the frequentist interpretation, which views probability as the limit of the relative frequency of an event after many trials.^[2] 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.^[3]^[4] 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.^[2]^[3]

Bayesian statistics is named after Thomas Bayes, who formulated a specific case of Bayes' theorem in 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.^[5] Laplace used methods that would now be considered Bayesian to solve a number of statistical problems. Many Bayesian methods were developed by later authors, but the term was not commonly used to describe such methods until the 1950s. During much of the 20th century, Bayesian methods were viewed unfavorably by many statisticians due to philosophical and practical considerations. Many Bayesian methods required much computation to complete, and most methods that were widely used during the century were based on the frequentist interpretation. However, with the advent of powerful computers and new algorithms like Markov chain Monte Carlo, Bayesian methods have seen increasing use within statistics in the 21st century.^[2]^[6]

^ "Bayesian". Merriam-Webster.com Dictionary.
^ ^a ^b ^c Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B. (2013). Bayesian Data Analysis (Third ed.). Chapman and Hall/CRC. ISBN 978-1-4398-4095-5.
^ ^a ^b McElreath, Richard (2020). Statistical Rethinking : A Bayesian Course with Examples in R and Stan (2nd ed.). Chapman and Hall/CRC. ISBN 978-0-367-13991-9.
^ Kruschke, John (2014). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan (2nd ed.). Academic Press. ISBN 978-0-12-405888-0.
^ McGrayne, Sharon (2012). 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 (First ed.). Chapman and Hall/CRC. ISBN 978-0-3001-8822-6.
^ Fienberg, Stephen E. (2006). "When Did Bayesian Inference Become "Bayesian"?". Bayesian Analysis. 1 (1): 1–40. doi:10.1214/06-BA101.

[1] "Bayesian". Merriam-Webster.com Dictionary.

[bda-2] Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B. (2013). Bayesian Data Analysis (Third ed.). Chapman and Hall/CRC. ISBN 978-1-4398-4095-5.

[rethinking-3] McElreath, Richard (2020). Statistical Rethinking : A Bayesian Course with Examples in R and Stan (2nd ed.). Chapman and Hall/CRC. ISBN 978-0-367-13991-9.

[4] Kruschke, John (2014). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan (2nd ed.). Academic Press. ISBN 978-0-12-405888-0.

[5] McGrayne, Sharon (2012). 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 (First ed.). Chapman and Hall/CRC. ISBN 978-0-3001-8822-6.

[6] Fienberg, Stephen E. (2006). "When Did Bayesian Inference Become "Bayesian"?". Bayesian Analysis. 1 (1): 1–40. doi:10.1214/06-BA101.

Bayesian statistics information

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