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. Bayesianism is the broader philosophy inspired by the theorem. The core claim behind all varieties of Bayesianism is that probabilities are subjective degrees of belief -- often operationalized as willingness to bet.

This stands in contrast to other interpretations of probability, which attempt greater objectivity. The frequentist interpretation of probability has a focus on repeatable experiments; probabilities are the limiting frequency of an event if you performed the experiment an infinite number of times.

Another contender is the propensity interpretation, which grounds probability in the propensity for things to happen. A perfectly balanced 6-sided die would have a 1/6 propensity to land on each side. A propensity theorist sees this as a basic fact about dice not derived from infinite sequences of experiments or subjective viewpoints.

Note how both of these alternative interpretations ground the meaning of probability in an external objective fact which cannot be directly accessed.

As a consequence of the subjective interpretation of probability theory, Bayesians are more inclined to apply Bayes' Theorem in practical statistical inference. The primary example of this is statistical hypothesis testing. Frequentists take the application of Bayes' Theorem to be inappropriate, because "the probability of a hypothesis" is meaningless: a hypothesis is either true or false; you cannot define a repeated experiment in which it is sometimes true and sometimes false, so you cannot assign it an intermediate probability.

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

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