Tyrrell, right, thanks. :) Your formalization makes clear that P1/P2 = p(M(B1) predicts B2 | M flawed) / p(M(B1) predicts B2), which is a stronger result than I thought. Argh, I wish I were able to see this sort of thing immediately.
One small nitpick: It could be more explicit that in Assumption 2, B1 and B2 range over actual observation, whereas in Assumption 1, B ranges over all possible observations. :)
Anna, right, I think we need some sort of “other things being equal” proviso to Tyrrell’s solution. If experiments 11..20 were chosen by scientist 1, experiment 21 is chosen by scientist 2, and experiments 1..10 were chosen by a third party, and scientist 2 knows scientist 1′s theory, for example, we could speculate that scientist 2 has found a strange edge case in 1′s formalization that 1 did not expect. I think I was implicitly taking the question to refer to a case where all 21 experiments are of the same sort and chosen independently—say, lowest temperatures at the magnetic north pole in consecutive years, that sort of thing.
Tyrrell, right, thanks. :) Your formalization makes clear that P1/P2 = p(M(B1) predicts B2 | M flawed) / p(M(B1) predicts B2), which is a stronger result than I thought. Argh, I wish I were able to see this sort of thing immediately.
One small nitpick: It could be more explicit that in Assumption 2, B1 and B2 range over actual observation, whereas in Assumption 1, B ranges over all possible observations. :)
Anna, right, I think we need some sort of “other things being equal” proviso to Tyrrell’s solution. If experiments 11..20 were chosen by scientist 1, experiment 21 is chosen by scientist 2, and experiments 1..10 were chosen by a third party, and scientist 2 knows scientist 1′s theory, for example, we could speculate that scientist 2 has found a strange edge case in 1′s formalization that 1 did not expect. I think I was implicitly taking the question to refer to a case where all 21 experiments are of the same sort and chosen independently—say, lowest temperatures at the magnetic north pole in consecutive years, that sort of thing.