Last lesson: posterior odds = prior odds × likelihood ratio (LR), where the LR of a piece of evidence E for hypothesis H is P(E | H) / P(E | ¬H) — how many times more consistent the evidence is with H than with not-H.
A patient gets two different tests for a rare disease, run by two independent labs using unrelated methods (one’s a blood antibody assay, the other’s a genetic marker panel):
- Base rate: the disease affects 1 in 1,000 people, so prior odds = 1/999 ≈ 0.001.
- Test A comes back positive. Its likelihood ratio for a positive result is LR_A = 20.
- Test B — run independently, unaware of Test A’s result — also comes back positive. Its likelihood ratio is LR_B = 15.
Because the two tests use unrelated biology, a crucial assumption holds: conditional independence given disease status. That means: among people who have the disease, whether Test A comes back positive tells you nothing extra about whether Test B will (its errors don’t share a cause); the same holds among people who don’t have the disease. Each test’s LR is exactly as strong in combination as it is alone.
Under that assumption, chaining evidence is just chaining multiplications:
posterior odds = prior odds × LR_A × LR_B
Compute the posterior odds of disease after both positives. (You can convert to a probability with P = odds/(1+odds) to build intuition, but the answer wanted is the odds.)