Bayes' Theorem

Probability · Axiom Academy

Updating beliefs with new evidence: from prior to posterior Often we know P(B|A) but need P(A|B). For example, we might know the probability of symptoms given a disease, but want the probability of disease given symptoms. Bayes' Theorem makes this reversal possible. We start from the multiplication rule, which gives us two expressions for P(A∩B). Setting these equal and rearranging yields Bayes' Theorem. Using the Law of Total Probability in the denominator, we can express Bayes' Theorem entirely in terms of conditional probabilities and a partition of the sample space. 4. Prior, Likelihood, and Posterior Bayes' Theorem has a natural interpretation: P(A) is the prior probability (before observing B), P(B|A) is the likelihood (how probable B is if A is true), and P(A|B) is the posterior probability (after observing B).

This is the written version of the interactive lesson above. See the full Probability course.