AHL 4.13 (HL)—Bayes' theorem

Bayes updates a probability after new evidence.

Bayes’ theorem reverses a conditional relationship by weighting the evidence with the prior probabilities of its possible causes.

Example

If a rare condition affects 1% and a test detects it 90% of the time but has a 5% false-positive rate, a positive result is not a 90% diagnosis.

Build the joint probabilities first, then divide the relevant cause-and-evidence cell by all positive evidence.

Sensitivity is P(positive|condition), not P(condition|positive).

For mutually exclusive and exhaustive causes $C_1,\ldots,C_n$ with $n\le3$, $P(C_i|E)=P(E|C_i)P(C_i)/\sum_j P(E|C_j)P(C_j)$. In the test example, $P(positive)=0.90(0.01)+0.05(0.99)=0.0585$, so $P(condition|positive)=0.009/0.0585\approx0.154$. Thus about 15.4%, not 90%, of positive tests correspond to the condition. Independence would instead mean the evidence leaves the prior unchanged.