Bayes Theorem

Bayes' Theorem is a mathematical identity relating conditional probabilities: the probability of hypothesis H given evidence E equals the probability of E given H, multiplied by the prior probability of H, divided by the marginal probability of E. In formal notation: P(H|E) = P(E|H)·P(H) / P(E). The theorem is a tautology in the axiomatic theory of probability — it follows directly from the definition of conditional probability and is not empirically contestable. What is contested, and what generates the deep dispute between Bayesian statistics and frequentist statistics, is whether the theorem licenses the use of probability to represent degrees of belief in hypotheses. The identity is uncontroversial; its interpretation as a rational updating rule for scientific inference is the central epistemological question it raises.