In 1763, two years after his death, an English Presbyterian minister named Thomas Bayes had a paper on probability published by a friend who thought it deserved attention. The mathematics was modest. The conceptual move was not. Bayes argued that belief itself was a quantity — specifically, a ratio of evidence to prior expectation — and that one could update beliefs with formal precision as new data arrived. The world's first mathematical theory of how to change one's mind was published anonymously, ignored for fifty years, and then quietly absorbed into the foundations of every science that learns from data.
Bayes' theorem says: the posterior odds (your belief after the evidence) equal the prior odds (what you thought before) times the likelihood ratio (how much more likely the evidence is under one hypothesis than the other). The mechanics are an afternoon's algebra. The discipline is a lifetime's: most reasoning errors are prior errors — beliefs that haven't been updated when they should have, or have been updated when they shouldn't. The theorem also explains the base-rate fallacy: a 99%-accurate cancer test is not 99% likely to mean you have cancer, because the prior probability of cancer is low. The statistical wars of the twentieth century — frequentist vs. Bayesian — were largely a fight over whether prior beliefs could legitimately enter scientific reasoning. The Bayesians won the practical fight, partly because computers can finally do the integrals, and partly because the world is full of problems where you cannot collect more data and must reason from what you have.
Modern AI is thoroughly Bayesian in spirit if not always in formalism: spam filters, medical diagnostics, recommender systems, autonomous vehicle perception, large-language-model fine-tuning all run on probabilistic updating. The replication crisis in social science was, in part, a Bayesian critique of frequentist null-hypothesis testing finally reaching critical mass. Whether ordinary public reasoning ever becomes more Bayesian — whether people learn to update on evidence in the way the theorem prescribes — is a separate and unsettled question.