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In [[probability theory]] and [[statistics]], '''Bayes' theorem''' (alternatively '''Bayes' law''' or '''Bayes' rule''') describes the [[probability]] of an [[Event (probability theory)|event]], based on conditions that might be related to the event.

== Description ==

For example, suppose one is interested in whether a person has cancer, and knows the person's age. If cancer is related to age, then, using Bayes' theorem, information about the person's age can be used to more accurately assess the probability that they have cancer.

When applied, the probabilities involved in Bayes' theorem may have different probability interpretations. In one of these interpretations, the theorem is used directly as part of a particular approach to statistical inference. With the Bayesian probability interpretation the theorem expresses how a subjective degree of belief should rationally change to account for evidence: this is Bayesian inference, which is fundamental to Bayesian statistics.

However, Bayes' theorem has applications in a wide range of calculations involving probabilities, not just in Bayesian inference.

Bayes' theorem is named after Rev. [[Thomas Bayes]] (1701–1761), who first provided an equation that allows new evidence to update beliefs. It was further developed by [[Pierre-Simon Laplace]], who first published the modern formulation in his 1812 ''Théorie analytique des probabilités''. Sir [[Harold Jeffreys]] put Bayes' algorithm and Laplace's formulation on an axiomatic basis. Jeffreys wrote that Bayes' theorem "is to the theory of [[probability]] what the [[Pythagorean theorem]] is to geometry".

== See also ==

* [[Bayesian inference]]
* [[Bayesian probability]]
* [[Event (probability theory)]]
* [[Inductive probability]]
* [[Probability]]
* [[Statistics]]

== External links ==

* [https://en.wikipedia.org/wiki/Bayes%27_theorem Bayes' theorem] @ Wikipedia.org 

[[Category:Mathematics]]
[[Category:Probability]]