Questions / Quant trading, Information & Bayes
Dependence bounds for two positives
H has prior probability 0.2. Each of two tests is positive with probability 0.8 under H and 0.2 under not-H. Their conditional dependence is unspecified. Given two positives, what is the smallest possible posterior P(H)? Enter a percentage.
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Answer: 42.86% (3/7)
- Under H the joint-positive probability is at least 0.8 + 0.8 − 1 = 0.6. Under not-H it is at most 0.2, so the smallest posterior is 0.2 × 0.6/(0.2 × 0.6 + 0.8 × 0.2) = 3/7.
Interview shortcut. Worst case: the smallest joint under H (0.6) and the largest under not-H (0.2).
In the app this has a follow-up where one thing about the setup changes. Try the follow-up.