# Difference between revisions of "Bayes' theorem"

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− | A law of probability that describes the proper way to incorporate new evidence into [[priors|prior probabilities]] to form an [[belief update|updated]] probability estimate. [[Bayesian rationality]] takes its name from this theorem, as it is regarded as the foundation of consistent rational reasoning under uncertainty. | + | A law of probability that describes the proper way to incorporate new [[evidence]] into [[priors|prior probabilities]] to form an [[belief update|updated]] probability estimate. [[Bayesian rationality]] takes its name from this theorem, as it is regarded as the foundation of consistent rational reasoning under uncertainty. |

The theorem commonly takes the form: | The theorem commonly takes the form: | ||

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where A is the proposition of interest, B is the observed evidence, P(A) and P(B) are prior probabilities, and P(A|B) is the posterior probability of A. | where A is the proposition of interest, B is the observed evidence, P(A) and P(B) are prior probabilities, and P(A|B) is the posterior probability of A. | ||

− | With the posterior odds, the prior odds and the likelihood ratio written explicitly, the theorem reads: | + | With the posterior odds, the prior odds and the [[likelihood ratio]] written explicitly, the theorem reads: |

:<math>\frac{P(A|B)}{P(\neg A|B)} = \frac{P(A)}{P(\neg A)} \cdot \frac{P(B | A)}{P(B|\neg A)}</math> | :<math>\frac{P(A|B)}{P(\neg A|B)} = \frac{P(A)}{P(\neg A)} \cdot \frac{P(B | A)}{P(B|\neg A)}</math> | ||

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==See also== | ==See also== | ||

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*[[Bayesian probability]] | *[[Bayesian probability]] | ||

==External references== | ==External references== | ||

+ | |||

*[http://yudkowsky.net/rational/bayes An Intuitive Explanation of Bayes' Theorem] by [[Eliezer Yudkowsky]] | *[http://yudkowsky.net/rational/bayes An Intuitive Explanation of Bayes' Theorem] by [[Eliezer Yudkowsky]] | ||

*[http://causalityrelay.wordpress.com/2008/06/23/odds-and-intuitive-bayes/ Odds, evidence, and an intuitive form of Bayes’ theorem] by [[Vladimir Nesov]] | *[http://causalityrelay.wordpress.com/2008/06/23/odds-and-intuitive-bayes/ Odds, evidence, and an intuitive form of Bayes’ theorem] by [[Vladimir Nesov]] | ||

[[Category:Theorems]] | [[Category:Theorems]] |

## Revision as of 10:33, 8 July 2009

A law of probability that describes the proper way to incorporate new evidence into prior probabilities to form an updated probability estimate. Bayesian rationality takes its name from this theorem, as it is regarded as the foundation of consistent rational reasoning under uncertainty.

The theorem commonly takes the form:

where A is the proposition of interest, B is the observed evidence, P(A) and P(B) are prior probabilities, and P(A|B) is the posterior probability of A.

With the posterior odds, the prior odds and the likelihood ratio written explicitly, the theorem reads: