Bayes' theorem
Quantitative Finance glossary · Probability, Statistics & Linear Algebra
Definition
P(A,|,B) = (P(B,|,A),P(A))/(P(B)), or in continuous form p(θ,|,x) propto p(x,|,θ),p(θ) — posterior is proportional to likelihood times prior. The basis of Bayesian inference (Black-Litterman, Kalman filter, particle filter, MCMC). The denominator P(B) = int p(x,|,θ)p(θ),dθ is the marginal likelihood used in model selection; its intractability motivates variational and Monte Carlo posterior approximations.
Bayes' theorem in plain English
Bayes' theorem is the math of updating a belief when new evidence shows up. You start with a hunch (the "prior"), you see some data, and the formula tells you the smart new belief (the "posterior") by weighing how well that data fits each possibility. In plain terms: new belief propto how well the data matches × old belief. Everyday version: you think a friend is probably running late (prior); then they text "almost there" (evidence); you sensibly update toward "arriving soon." It's the engine behind tons of finance and AI tools that keep refining their estimates as fresh numbers arrive.
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