Bayes' theorem

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:

$P(A|B)={\frac {P(B|A)\,P(A)}{P(B)}}$

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:

${\frac {P(A|B)}{P(\neg A|B)}}={\frac {P(A)}{P(\neg A)}}\cdot {\frac {P(B|A)}{P(B|\neg A)}}$

or in words, $Posterior~odds=Prior~odds\times Likelihood~ratio$.

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

See also

• Bayesian probability
• Priors
• Prior probability
• Likelihood ratio
• Posterior probability
• Belief update