Bayes' Theorem (also known as 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:

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)}

Visualisation of Bayes' Rule