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gjm comments on Hearsay, Double Hearsay, and Bayesian Updates - Less Wrong
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If a random 80% of suspects are guilty, the appropriate naive predictor is one that always votes "guilty", not one that tries to match probabilities by choosing a random 80% of suspects to call guilty. Then you get an accurate result 80% of the time, which is a lot better than 68%. That seems to me a more appropriate benchmark.
(Alternatively, you might consider a predictor that matches its probabilities not to the proportion of defendants who are guilty but to the proportion who are convicted. There might be something to be said for that.)
I think the intended question is whether the legal system adds anything beyond a pure chance element. Somehow we'd need a gold standard of actually guilty and innocent suspects, then we'd need to measure whether p(guilty|convicted) > 80%. You could also ask if p(innocent|acquitted) > 20%, but that's the same question.
Thank you! Intended or not, it's a fantastic question, and I don't know where to look up the answer. I'm not even sure that anyone has seriously tried to answer that question. If they haven't, then I want to. I'll look into it.
I don't see how those are "the same question". If out of 8 accused 4 are guilty and two of them are convicted, the rest acquitted. Than p(guilty|convicted) = 1 and p(innocent|acquitted) = 2/3.
The assumption was that 80% of defendants are guilty, which is more than 4 of 8. Under this assumption, asking whether p(guilty|convicted) > 80% is just asking whether conviction positively correlates with guilt. Asking if p(innocent|acquitted) > 20% is just asking if acquittal positively correlates with innocence. These are really the same question, because P correlates with Q iff ¬P correlates with ¬Q.
Perfect. Thanks.