{"id":"bayes-theorem","domain":"probability-statistics","type":"Theorem","title":"Bayes' Theorem","slug":"bayes-theorem","url":"/mathematics/probability-statistics/bayes-theorem/","published":"1763 (posthumous), Thomas Bayes","formula":"P(A|B) = P(B|A)P(A)/P(B)","summary":"Rule for updating beliefs given new evidence. Basis of medical diagnosis interpretation, spam filters, Bayesian statistics, Bayesian networks in AI.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society, 53, 370-418."},{"tier":1,"citation":"Gelman, A., Carlin, J.B., Stern, H.S. et al. (2013). Bayesian Data Analysis. 3rd ed. CRC Press."},{"tier":2,"citation":"McGrayne, S.B. (2011). The Theory That Would Not Die. Yale University Press."},{"tier":3,"citation":"Silver, N. (2012). The Signal and the Noise. Penguin Press."}],"relationships":[{"type":"built_from","target":"probability","label":"Probability"},{"type":"related_to","target":"random-variable","label":"Random Variable"},{"type":"related_to","target":"philosophy-of-mathematics","label":"Philosophy of Mathematics — Bayesian vs frequentist"},{"type":"uses","target":"graph-theory","label":"Graph Theory — Bayesian networks"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.8"}