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ESRogs comments on How should Eliezer and Nick's extra $20 be split - Less Wrong
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For 4 could you explain where that formula comes from? I thought I understood how to apply Bayes' theorem, but I'm getting stuck on what P(B) or P(B|A) should be (where A is: $20 is Eliezer's, and B is: Nick says 85%).
Sure! You start with a likelihood ratio of 1:1 of the 20 belonging to EY or NB. EY says that it is his with probability 20%. This is evidence with a ratio of 20:80 or 1:4. After you update to account for EYs claim, you assign a 20% chance that it belongs to EY. Then, you have NBs claim, which is evidence in the ratio of 85:15 or 17:3. You multiply the 1 by 17 and the 4 by 3 to get 17:12. This means that EY should get 17/29 of the money, which is $11.72.
This is using the notation of likelihood ratios, which is easier to work with. Trying to attack it with Bayes Theorem directly is a more confusing. The reason is because we are trusting EY and NBs claims as evidence, without actually specifying some event that caused that evidence. A good way to think about it is that we start by taking EY into account and thinking that the probabiltiy of A is 20%. Then we say that NB has a sensor that tells him whether or not the 20 is his and gets it right 85% of the time. We let B be the event that NBs sensor tells him that the 20 belongs to EY. Then, P(B|A)=85%, and B(B)=85% of 20% plus 15% of 80%, which is 29%. Therefore, we get P(A|B)=P(A)P(B|A)/P(B)=20%85%/(29%)=17/29.
Does that make sense?
Yes, that does make sense. The problem was that I was thinking of B as the event that Nick says a certain percentage, rather than the event that he says the bill is Eliezer's with the percentage being the probability that he's right. Thanks!