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Concept

Bayes Theorem

Bayes theorem turns likelihoods and base rates into posterior probabilities.

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Bayes theorem

Formula

Bayes theorem reverses conditioning. The denominator is usually found with the law of total probability.

Bayes theorem
P(A∣B)=P(B∣A)P(A)P(B)P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}
Law of total probability denominator
P(B)=∑iP(B∣Ai)P(Ai)P(B)=\sum_i P(B\mid A_i)P(A_i)

Worked Example

If 2% of risks are high-risk, a flag catches 80% of high-risk policies, and falsely flags 10% of others, then P(high-risk | flagged)=0.8(0.02)/(0.8(0.02)+0.1(0.98))=0.140.

Common Mistake

Do not ignore the base rate. A strong signal can still produce a modest posterior probability when the original event is rare.

References And Official Sources