Bayes' Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics, and inductive logic. Subjectivists, who maintain that rational belief is governed by the laws of probability, lean heavily on conditional probabilities in their theories of evidence and their models of empirical learning. Bayes' Theorem is central to these enterprises both because it simplifies the calculation of conditional probabilities and because it clarifies significant features of subjectivist position. Indeed, the Theorem's central insight — that a hypothesis is confirmed by any body of data that its truth renders probable — is the cornerstone of all subjectivist methodology.
Bayes' theorem relates the conditional and marginal probabilities of events A and B, where B has a non-vanishing probability:
Each term in Bayes' theorem has a conventional name:
P(A) is the prior probability or marginal probability of A. It is "prior" in the sense that it does not take into account any information about B.
P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.
P(B|A) is the conditional probability of B given A.
P(B) is the prior or marginal probability of B, and acts as a normalizing constant.
Intuitively, Bayes' theorem in this form describes the way in which one's beliefs about observing 'A' are updated by having observed 'B'
