Bayes' Theorem is a fundamental mathematical rule in probability theory and statistics that updates the probability of a hypothesis as more evidence or data becomes available by combining prior beliefs with new observations via conditional probabilities. It formally defines posterior probability through an equation relating it to likelihood, marginal probability (evidence), and prior probability, establishing a rigorous framework for Bayesian inference within deductive reasoning contexts. This principle serves as the cornerstone for probabilistic modeling in decision-making processes where uncertainty must be quantified and revised iteratively based on new information inputs.
Visual derivation of Bayes' theorem: priors, likelihoods, and posteriors as areas in a probability square, updating beliefs on evidence, and why conditional probabilities invert the way they do.