Bayes' Theorem: Prior, Likelihood and Posterior
~13 min read
Bayes' theorem is a formula for updating a belief once you see new evidence — turning 'how likely is the evidence given my belief' into 'how likely is my belief given the evidence.'
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Key points
- •Bayes' theorem updates a belief (prior) into a new belief (posterior) once evidence (likelihood) arrives: P(A|evidence) = P(evidence|A) x P(A) / P(evidence)
- •Prior = what you believed before evidence; likelihood = how probable the evidence is if the belief were true; posterior = your updated belief
- •It flips the question you're often GIVEN (evidence given belief) into the question you actually WANT (belief given evidence) — these differ, often hugely
- •Even a highly accurate test can yield a low posterior probability when the underlying event is rare — false positives from the large 'unlikely' group can dominate
- •LLMs implicitly do this: a 'prior' over the next token gets updated into a 'posterior' by the evidence of the prompt/context so far