Difference between revisions of "Bayes' theorem"

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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".
A law of probability that describes the proper way to incorporate new evidence into [[priors|prior probabilities]] to form an [[update]]d 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.
  
==References==
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With the posterior odds, the prior odds and the [[likelihood ratio]] written explicitly, the theorem reads:
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:<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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or in words, <math>Posterior ~ odds = Prior ~ odds \times Likelihood ~ ratio</math>.
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==Visualization==
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[[Image:Bayes.png]]
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==Blog posts==
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*[http://lesswrong.com/lw/2b0/bayes_theorem_illustrated_my_way/ Bayes' Theorem Illustrated (My Way)] by [[User:Komponisto|komponisto]].
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*[http://blog.oscarbonilla.com/2009/05/visualizing-bayes-theorem/ Visualizing Bayes' theorem] by Oscar Bonilla
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*[http://oracleaide.wordpress.com/2012/12/26/a-venn-pie/ Using Venn pies to illustrate Bayes' theorem] by [[User:oracleaide|oracleaide]]
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==External links==
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*[http://yudkowsky.net/rational/bayes An Intuitive Explanation of Bayes' Theorem]
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*[https://arbital.com/p/bayes_rule_guide/ Arbital Guide to Bayes' Rule]
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*[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
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==See also==
  
=====Other Resources=====
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*[[Bayesian probability]]
* [http://yudkowsky.net/rational/bayes An Intuitive Explanation of Bayes' Theorem] by [[Eliezer Yudkowsky]]
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*[[Priors]]
* [http://causalityrelay.wordpress.com/2008/06/23/odds-and-intuitive-bayes/ Odds, evidence, and an intuitive form of Bayes’ theorem] by [[Vladimir Nesov]]
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*[[Prior probability]]
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*[[Likelihood ratio]]
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*[[Posterior probability]]
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*[[Belief update]]
  
 
[[Category:Theorems]]
 
[[Category:Theorems]]
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[[Category:Bayesian]]

Latest revision as of 18:48, 7 February 2020

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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

Bayes.png

Blog posts

External links

See also