Bayes' Theorem

The theorem stating how to update prior beliefs in light of new evidence — providing the mathematical foundation for rational belief revision.

P(H|E) = P(E|H) × P(H) / P(E): the posterior probability of hypothesis H given evidence E equals the prior probability of H multiplied by the likelihood of E given H, divided by the total probability of E. Bayesian inference: a complete framework for reasoning under uncertainty — start with a prior, update on evidence, derive posterior. Bayesian epistemology: rational belief is probabilistic, and rational learning is conditionalization on evidence. The statistical foundation for machine learning, signal processing, medical diagnosis, and decision theory.