Bayes' Theorem (also known as Bayes' Rule or Bayes' Law) is a law of probability that describes the proper way to incorporate new evidence into prior probabilities to form an updated probability estimate. It is commonly regarded as the foundation of consistent rational reasoning under uncertainty. Bayes' Theorem is named after Reverend Thomas Bayes, who proved the theorem in 1763.

Bayes' 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:

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

Visualization of Bayes' Rule

Further reading

Alexander Kruel (2010) A guide to Bayes’ theorem – A few links, Alexander Kruel’s Blog, February 27.

Arbital (2021) Bayes’ rule: Guide, Arbital.

Bonilla, Oscar (2009) Visualizing Bayes theorem, Oscar Bonilla’s Blog, May 1.

Joyce, James (2003) Bayes’ theorem, The Stanford Encyclopedia of Philosophy, June 28 (updated 12 August 2021).

Oracle Aide (2012) A Venn pie (using Venn pies to illustrate Bayes’ theorem), Oracle Aide, December 26.

Wikipedia (2002) Bayes’ theorem, Wikipedia, April 18 (updated 3 August 2021‎).

Wikipedia (2004) Base rate fallacy, Wikipedia, June 17 (updated 17 June 2021‎).

Yudkowsky, Eliezer S. (2003) An intuitive explanation of Bayes’ theorem, Eliezer S. Yudkowsky’s Website, (updated 4 June 2006).

Related entries

Bayesian epistemology | credence | epistemology