Theorem in probability theory. Thomas Bayes (1702–61) was an English clergyman, whose An Essay towards Solving a Problem in the Doctrine of Chances occurs in two memoirs presented by Price (Bayes having died), in Philosophical Transactions of 1763 and 1764. Bayes gave a result for the probability that the chance of an event on a single trial is within a certain interval, given the number of times the event has occurred and the number it has failed. But the form in which his theorem is remembered is as an expression for the posterior probability of a hypothesis (its probability after evidence is obtained). This is a product of

1. (i) its probability before the evidence, or prior probability, and

2. (ii) the probability of the evidence being as it is, given the hypothesis, divided by

3. (iii) the prior probability of the evidence (often expressed as the probability of the evidence considered in the light of all the different possible hypotheses).