For each 80-digit binary number X, let N(X) be the assertion “Unknowns picked an 80-digit number at random, and it was X.” In my ledger of probabilities, I dutifully fill in, for each of these statements X, 2^{-80} in the P column. Now for a particular 80-digit number Y, I am told that “Unknowns claims he picked an 80-digit number at random, and it was Y”—call that statement U(Y) -- and am asked for P(N(Y)|U(Y)).
My answer: pretty high by Bayes formula. P(U|N(Y)) is pretty high because Unknowns is trustworthy, and my ledger has P(U(Y)) = number on the same order as two-to-the-minus-eighty. (Caveat: P(U(Y)) is a lot higher for highly structured things like the sequence of all 1′s. But for the vast majority of Y I have P(U(Y)) = 2^-80 times something between (say) 10^-1 and 10^-6). So P(N(Y)|U(Y)) = P(U(Y)|N(Y)) x [P(N(Y))/P(U(Y))] is a big probability times a medium-sized probability
What’s your answer?
Reincarnation is explained to me, and I am asked for my opinion of how likely it is. I respond with P(R), a good faith estimate based on my experience and judgement. I am then told that hundreds of millions of people believe in reincarnation—call that statement B, and assume that I was ignorant of it before—and am asked for P(R|B). Your claim is that no matter how small P(R) is, P(R|B) should be larger than some threshold t. Correct?
Some manipulation with Bayes formula shows that your claim (what I understand to be your claim) is equivalent to this inequality:
P(B) < P(R) / t
That is, I am “overconfident” if I think that the probability of someone believing in reincarnation is larger than some fixed multiple of the probability that reincarnation is actually true. Moreover, though I assume (sic) you think t is sensitive to the quantity “hundreds of millions”—e.g. that it would be smaller if it were just “hundreds”—you do not think that t is sensitive to the statement R. R could be replaced by another religious claim, or by the claim that I just flipped a coin 80 times and the sequence of heads and tails was [whatever].
My position: I think it’s perfectly reasonable to assume that P(B) is quite a lot larger than P(R). What’s your position?
Your analysis is basically correct, i.e. I think it is overconfident to say that the probability P(B) is greater than P(R) by more than a certain factor, in particular because if you make it much greater, there is basically no way for you to be well calibrated in your opinions—because you are just as human as the people who believe those things. More on that later.
For now, I would like to see your response to question on my comment to komponisto (i.e. how many 1′s do you wait for.)
Let’s consider two situations:
For each 80-digit binary number X, let N(X) be the assertion “Unknowns picked an 80-digit number at random, and it was X.” In my ledger of probabilities, I dutifully fill in, for each of these statements X, 2^{-80} in the P column. Now for a particular 80-digit number Y, I am told that “Unknowns claims he picked an 80-digit number at random, and it was Y”—call that statement U(Y) -- and am asked for P(N(Y)|U(Y)).
My answer: pretty high by Bayes formula. P(U|N(Y)) is pretty high because Unknowns is trustworthy, and my ledger has P(U(Y)) = number on the same order as two-to-the-minus-eighty. (Caveat: P(U(Y)) is a lot higher for highly structured things like the sequence of all 1′s. But for the vast majority of Y I have P(U(Y)) = 2^-80 times something between (say) 10^-1 and 10^-6). So P(N(Y)|U(Y)) = P(U(Y)|N(Y)) x [P(N(Y))/P(U(Y))] is a big probability times a medium-sized probability
What’s your answer?
Reincarnation is explained to me, and I am asked for my opinion of how likely it is. I respond with P(R), a good faith estimate based on my experience and judgement. I am then told that hundreds of millions of people believe in reincarnation—call that statement B, and assume that I was ignorant of it before—and am asked for P(R|B). Your claim is that no matter how small P(R) is, P(R|B) should be larger than some threshold t. Correct?
Some manipulation with Bayes formula shows that your claim (what I understand to be your claim) is equivalent to this inequality:
P(B) < P(R) / t
That is, I am “overconfident” if I think that the probability of someone believing in reincarnation is larger than some fixed multiple of the probability that reincarnation is actually true. Moreover, though I assume (sic) you think t is sensitive to the quantity “hundreds of millions”—e.g. that it would be smaller if it were just “hundreds”—you do not think that t is sensitive to the statement R. R could be replaced by another religious claim, or by the claim that I just flipped a coin 80 times and the sequence of heads and tails was [whatever].
My position: I think it’s perfectly reasonable to assume that P(B) is quite a lot larger than P(R). What’s your position?
Your analysis is basically correct, i.e. I think it is overconfident to say that the probability P(B) is greater than P(R) by more than a certain factor, in particular because if you make it much greater, there is basically no way for you to be well calibrated in your opinions—because you are just as human as the people who believe those things. More on that later.
For now, I would like to see your response to question on my comment to komponisto (i.e. how many 1′s do you wait for.)