Intuitively it makes sense to me that if someone thinks they got abducted by aliens, it’s more likely they’re hallucinating than that they actually got abducted by aliens. It’s true that aliens actually abducting people wouldn’t mean people stop having hallucinations. But adding P(B|¬A) - the rate of false positives—to P(B|A) - the rate of true positives—seems like some kind of weird double counting. What am I misunderstanding here?
Well, to start with, if you don’t include the false positive rate in P(B|A), and work through the numbers, then, as I said, you’ll find that having the abduction experience will drastically lower your probability estimate of aliens. You would have:
So that’s clearly very wrong—even if it doesn’t tell you what the right answer should be.
But, intuitively… well, I explained it already: if aliens exist and abduct people, some people will still take drugs or go crazy or whatever, and hallucinate being abducted. Bob could be one of those people. It’s not double-counting because the A is “aliens exist and abduct people”, not “Bob was abducted by aliens”. (Otherwise P(A) could not possibly have started as high as 0.01—that would’ve been wrong by many orders of magnitude as a prior!) (This is essentially @clone of saturn’s explanation, so see his sibling comment for more on this point.)
Well, to start with, if you don’t include the false positive rate in P(B|A), and work through the numbers, then, as I said, you’ll find that having the abduction experience will drastically lower your probability estimate of aliens. You would have:
P(A|B) = (0.000005 · 0.01) / ((0.000005 · 0.01) + (0.001 · 0.99)) = 0.00000005 / (0.00000005 + 0.00099) = 0.00000005 / 0.00099005 = ~0.0000505025 = 0.00505025%.
So that’s clearly very wrong—even if it doesn’t tell you what the right answer should be.
But, intuitively… well, I explained it already: if aliens exist and abduct people, some people will still take drugs or go crazy or whatever, and hallucinate being abducted. Bob could be one of those people. It’s not double-counting because the A is “aliens exist and abduct people”, not “Bob was abducted by aliens”. (Otherwise P(A) could not possibly have started as high as 0.01—that would’ve been wrong by many orders of magnitude as a prior!) (This is essentially @clone of saturn’s explanation, so see his sibling comment for more on this point.)