principles.fyi · the brain · concept

Bayes' rule

How beliefs update: start from the prior, add what the evidence is worth.

posterior odds = prior odds × likelihood ratio

Bayes' rule connects the probability you had before the evidence (the prior) to the one you should hold after (the posterior). Its textbook fraction form, P(H|E) = P(E|H)·P(H)/P(E), hides the intuition; in odds form it collapses to posterior odds = prior odds × likelihood ratio, and on the log-odds ruler multiplication becomes addition: posterior = prior + evidence. That last face makes updating feel like arithmetic — strong evidence is a big stride, a rare hypothesis is a start deep below zero, and stacking independent evidence is just more addition. It is the normative core of reasoning under uncertainty, and the reason base-rate 'paradoxes' dissolve once the starting point is honest.