bayes theorem

Bayes theorem is a human-made mathematical theorem in probability theory that describes how to update the probability of a hypothesis in light of new evidence. Formally, for events A and B with P(B) > 0, the theorem states: P(A|B) = P(B|A) · P(A) / P(B), where P(A|B) is the posterior probability of A given B, P(B|A) is the likelihood, P(A) is the prior probability of A, and P(B) is the marginal likelihood (evidence). The theorem persists through rigorous mathematical formalism in probability theory, statistics, and decision theory. It was first formulated by Thomas Bayes (1763, posthumously) and rigorously established by Pierre-Simon Laplace (1774-1812), who independently derived and applied it to diverse problems in astronomy, medicine, and jurisprudence. The theorem provides a principled computational framework for inductive inference — the process of generalizing from observed data to unobserved hypotheses — and underlies Bayesian statistics, probabilistic machine learning (naive Bayes classifiers, Bayesian networks), and decision theory under uncertainty. The mathematical framework extends naturally to continuous variables via Bayes' rule with probability density functions and to multiple hypotheses via the law of total probability. [formal: bayes | substrate: mind | horizon: centuries | explicit: yes | epoch: 0.12]