# Difference between revisions of "Bayes' theorem"

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− | {{ | + | {{arbitallink|https://arbital.com/p/bayes_rule_proof/|Proof of Bayes' Rule}} |

+ | {{wikilink}} | ||

+ | A law of probability that describes the proper way to incorporate new [[evidence]] into [[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.k.a. "Bayes's Theorem" or "Bayes's Rule". | ||

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

− | :<math>P(A|B) = \frac{P(B | A)\, P(A)}{P(B)} | + | :<math>P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}</math> |

+ | 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: | ||

+ | :<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> | ||

− | = | + | or in words, <math>Posterior ~ odds = Prior ~ odds \times Likelihood ~ ratio</math>. |

− | < | ||

− | == | + | ==Visualization== |

− | |||

− | |||

− | + | [[Image:Bayes.png]] | |

− | |||

− | ==== | + | ==Blog posts== |

− | |||

− | |||

− | + | *[http://lesswrong.com/lw/2b0/bayes_theorem_illustrated_my_way/ Bayes' Theorem Illustrated (My Way)] by [[User:Komponisto|komponisto]]. | |

− | + | *[http://blog.oscarbonilla.com/2009/05/visualizing-bayes-theorem/ Visualizing Bayes' theorem] by Oscar Bonilla | |

− | + | *[http://oracleaide.wordpress.com/2012/12/26/a-venn-pie/ Using Venn pies to illustrate Bayes' theorem] by [[User:oracleaide|oracleaide]] | |

− | ==== | + | ==External links== |

− | + | ||

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

+ | *[https://arbital.com/p/bayes_rule_guide/ Arbital Guide to Bayes' Rule] | ||

+ | *[http://kruel.co/2010/02/27/a-guide-to-bayes-theorem-a-few-links/ A Guide to Bayes’ Theorem – A few links] by Alexander Kruel | ||

+ | |||

+ | ==See also== | ||

+ | |||

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

+ | *[[Priors]] | ||

+ | *[[Prior probability]] | ||

+ | *[[Likelihood ratio]] | ||

+ | *[[Posterior probability]] | ||

+ | *[[Belief update]] | ||

+ | |||

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

+ | [[Category:Bayesian]] |

## Latest revision as of 19:48, 7 February 2020

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. A.k.a. "Bayes's Theorem" or "Bayes's Rule".

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:

or in words, .

## Visualization

## Blog posts

- Bayes' Theorem Illustrated (My Way) by komponisto.
- Visualizing Bayes' theorem by Oscar Bonilla
- Using Venn pies to illustrate Bayes' theorem by oracleaide

## External links

- An Intuitive Explanation of Bayes' Theorem
- Arbital Guide to Bayes' Rule
- A Guide to Bayes’ Theorem – A few links by Alexander Kruel