I did misunderstand you, and it might change things; I will have to think. But now your positions seem less coherent to me, and I no longer have a model of how you came to believe them. Tell me more:
Let CM(n) be the assertion “one day I’ll play guitar on the moon, and then flip an n-sided coin and it will come up heads.” The point being that P(CM(n)) is proportional to 1/n. Consider the following ratios:
R1(n) = P(CM(n)|CM(n) is raised as a possibility)/P(CM(n))
R2(n) = P(CM(n)|CM(n) is raised as a possibility by a significant number of people)/P(CM(n))
R3(n) = P(CM(n)|CM(n) is believed by one person)/P(CM(n))
R4(n) = P(CM(n)|CM(n) is believed by a significant number of people)/P(CM(n))
How do you think these ratios change as n grows? Before I had assumed you thought that ratios 1. and 4. grew to infinity as n did. I still understand you to be saying that for 4. Are you now denying it for 1., or just saying that 1. grows more slowly? I can’t guess what you believe about 2. and 3.
First we need to decide on the meaning of “flip an n-sided coin and it will come up heads”. You might mean this as:
1) a real world claim; or
2) a fixed probability device
To illustrate: if I assert, “I happen to know that I will win the lottery tomorrow,” this greatly increases the chance that it will happen, among other reasons, because of the possibility that I am saying this because I happen to have cheated and fixed things so that I will win. This would be an example of a real world claim.
On the other hand, if it is given that I will play the lottery, and given that the chance of winning is one in a trillion, as a fixed fact, then if I say, “I will win,” the probability is precisely one in a trillion, by definition. This is a fixed probability device.
In the real world there are no fixed probability devices, but there are situations where things are close enough to that situation that I can mathematically calculate a probability, even one which will break the bound of one in a billion, and even when people believe it. This is why I qualified my original claim with “Unless you have actually calculated the probability...” So in order to discuss my claim at all, we need to exclude the fixed probability device and only consider real world claims. In this case, the probability of P(CM(n)) is not exactly proportional to 1/n. However, it is true that this probability goes to zero as n goes to infinity.
In fact, all of these probabilities go to zero as n goes to infinity:
Given this fact (that all the probabilities go to zero), I am unsure about the behavior of your cases 1 & 2. I’ll leave 3 for another time, and say that case 4, again remembering that we take it as a real world claim, does go to infinity, since the numerator remains at no less than 1 in a billion, while the denominator goes to zero.
One more note about my original claim: if you ask how I arrived at the one in a billion figure, it is somewhat related to the earth’s actual population. If the population were a googleplex, a far larger number of mutually exclusive claims would be believed by a significant number of people, and so the lower bound would be much lower. Finally, I don’t understand why you say my positions are “less coherent”, when I denied the position that as you were about to point out, leads to mathematical inconsistency. This should make my position more coherent, not less.
It’s my map of your beliefs that became less coherent, not your actual beliefs. (Not necessarily!) As you know, I’ve thought your beliefs are mistaken from the beginning.
Note that I’m asking about a limit of ratios, not a ratio of limits. Actually, I’m not even asking you about the limits—I’d prefer some rough information about how those ratios change as n grows. (Are they bounded above? Do they grow linearly or logarithmically or what?) If you don’t know, why not?
So in order to discuss my claim at all, we need to exclude the fixed probability device and only consider real world claims.
This is bad form. Phrases like “unless you have actually computed the probability...”, “real world claim”, “natural language claim”, “significant number of people” are slippery. We can talk about real-world examples after you explain to me how your reasoning works in a more abstract setting. Otherwise you’re just reserving the right to dismiss arguments (and even numbers!) on the basis that they feel wrong to you on a gut level.
Edit: It’s not that I think it’s always illegitimate to refer to your gut. It’s just bad form to claim that such references are based on mathematics.
Edit 2: Can I sidestep this discussion by saying “Let CM(n) be any real world claim with P(CM(n)) = 1/n”?
Unless you’ve actually calculated the probability mathematically, a probability of one in a billion for a natural language claim that a significant number of people accept as likely true is always overconfident.
I nowhere stated that this was “based on mathematics.” It is naturally related to mathematics, and mathematics puts some constraints on it, as I have been trying to explain. But I didn’t come up with it in the first place in a purely mathematical way. So if this is bad form, it must be bad form to say what I mean instead of something else.
I could accept what you say in Edit 2 with these qualifications: first, since we are talking about “real world claims”, the probability 1/n does not necessarily remain fixed when someone brings up the possibility or asserts that the thing is so. This probability 1/n is only a prior, before the possibility has been raised or the thing asserted. Second, since it isn’t clear what “n” is doing, CM(5), CM(6), CM(7) and so on might be claims which are very different from one another.
I am not sure about the behavior of the ratios 1 and 2, especially given the second qualification here (in other words the ratios might not be well-behaved at all). And I don’t see how I need to say “why not?” What is there in my account which should tell me how these ratios behave? But my best guess for the moment, after thinking about it some more, would be the first ratio probably goes to infinity, but not as quickly as the fourth. What leads me to think this is something along the lines of this comment thread. For example, in my Scientology example, even if no one held that Scientology was true, but everyone admitted that it was just a story, the discovery of a real Xenu would greatly increase the probability that it was true anyway; although naturally not as much as given people’s belief in it, since without the belief, there would be a significantly greater probability that Scientology is still a mere story, but partly based on fact. So this suggests there may be a similar bound on things-which-have-been-raised-as-possibilities, even if much lower than the bound for things which are believed. Or if there isn’t a lower bound, such things are still likely to decrease in probability slowly enough to cause ratio 1 to go to infinity.
What is there in my account which should tell me how these ratios behave?
You responded positively to my suggestion that we could phrase this notion of “overconfidence” as “failure to update on other people’s beliefs,” indicating that you know how to update on other people’s beliefs. At the very least, this requires some rough quantitative understanding of the players in Bayes formula, which you don’t seem to have.
If overconfidence is not “failure to update on other people’s beliefs,” then what is it?
Here’s the abbreviated version of the conversation that led us here (right?).
S: God exists with very low probability, less that one in a zillion.
U: No, you are being overconfident. After all, billions of people believe in God, you need to take that into account somehow. Surely the probability is greater than one in a billion.
S: OK I agree that the fact that billions of people believing it constitutes evidence, but surely not evidence so strong as to get from 1-in-a-zillion to 1-in-a-billion.
Now what? Bayes theorem provides a mathematical formalism for relating evidence to probabilities, but you are saying that all four quantities in the relevant Bayes formula are too poorly understood for it to be of use. So what’s an alternative way to arrive at your one-in-a-billion figure? Or are you willing to withdraw your accusation that I’m being overconfident?
I did not say that “all four quantities in the relevant Bayes formula are too poorly understood for it to be of use.” Note that I explicitly asserted that your fourth ratio tends to infinity, and that your first one likely does as well.
If you read the linked comment thread and the Scientology example, that should make it clear why I think that the evidence might well be strong enough to go from 1 in a zillion to 1 in a billion. In fact, that should even be clear from my example of the random 80 digit binary number. Suppose instead of telling you that I chose the number randomly, I said, “I may or may not have chosen this number randomly.” This would be merely raising the possibility—the possibility of something which has a prior of 2^-80. But if I then went on to say that I had indeed chosen it randomly, you would not have therefore called me a liar, while you would do this, if I now chose another random 80 digit number and said that it was the same one. This shows that even raising the possibility provides almost all the evidence necessary—it brings the probability that I chose the number randomly all the way from 2^-80 up to some ordinary probability, or from “1 in a zillion” to something significantly above one in a billion.
More is involved in the case of belief, but I need to be sure that you get this point first.
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.)
I have been using “now you are saying” as short for “now I understand you to be saying.” I think this may be causing confusion, and I’ll try write more carefully.
I did misunderstand you, and it might change things; I will have to think. But now your positions seem less coherent to me, and I no longer have a model of how you came to believe them. Tell me more:
Let CM(n) be the assertion “one day I’ll play guitar on the moon, and then flip an n-sided coin and it will come up heads.” The point being that P(CM(n)) is proportional to 1/n. Consider the following ratios:
R1(n) = P(CM(n)|CM(n) is raised as a possibility)/P(CM(n))
R2(n) = P(CM(n)|CM(n) is raised as a possibility by a significant number of people)/P(CM(n))
R3(n) = P(CM(n)|CM(n) is believed by one person)/P(CM(n))
R4(n) = P(CM(n)|CM(n) is believed by a significant number of people)/P(CM(n))
How do you think these ratios change as n grows? Before I had assumed you thought that ratios 1. and 4. grew to infinity as n did. I still understand you to be saying that for 4. Are you now denying it for 1., or just saying that 1. grows more slowly? I can’t guess what you believe about 2. and 3.
First we need to decide on the meaning of “flip an n-sided coin and it will come up heads”. You might mean this as:
1) a real world claim; or 2) a fixed probability device
To illustrate: if I assert, “I happen to know that I will win the lottery tomorrow,” this greatly increases the chance that it will happen, among other reasons, because of the possibility that I am saying this because I happen to have cheated and fixed things so that I will win. This would be an example of a real world claim.
On the other hand, if it is given that I will play the lottery, and given that the chance of winning is one in a trillion, as a fixed fact, then if I say, “I will win,” the probability is precisely one in a trillion, by definition. This is a fixed probability device.
In the real world there are no fixed probability devices, but there are situations where things are close enough to that situation that I can mathematically calculate a probability, even one which will break the bound of one in a billion, and even when people believe it. This is why I qualified my original claim with “Unless you have actually calculated the probability...” So in order to discuss my claim at all, we need to exclude the fixed probability device and only consider real world claims. In this case, the probability of P(CM(n)) is not exactly proportional to 1/n. However, it is true that this probability goes to zero as n goes to infinity.
In fact, all of these probabilities go to zero as n goes to infinity:
P(CM(n))
P(CM(n) is raised as a possibility)
P(CM(n) is believed, by one or many persons)
The reason these probabilities go to zero can be found in my post on Occam’s razor.
Given this fact (that all the probabilities go to zero), I am unsure about the behavior of your cases 1 & 2. I’ll leave 3 for another time, and say that case 4, again remembering that we take it as a real world claim, does go to infinity, since the numerator remains at no less than 1 in a billion, while the denominator goes to zero.
One more note about my original claim: if you ask how I arrived at the one in a billion figure, it is somewhat related to the earth’s actual population. If the population were a googleplex, a far larger number of mutually exclusive claims would be believed by a significant number of people, and so the lower bound would be much lower. Finally, I don’t understand why you say my positions are “less coherent”, when I denied the position that as you were about to point out, leads to mathematical inconsistency. This should make my position more coherent, not less.
It’s my map of your beliefs that became less coherent, not your actual beliefs. (Not necessarily!) As you know, I’ve thought your beliefs are mistaken from the beginning.
Note that I’m asking about a limit of ratios, not a ratio of limits. Actually, I’m not even asking you about the limits—I’d prefer some rough information about how those ratios change as n grows. (Are they bounded above? Do they grow linearly or logarithmically or what?) If you don’t know, why not?
This is bad form. Phrases like “unless you have actually computed the probability...”, “real world claim”, “natural language claim”, “significant number of people” are slippery. We can talk about real-world examples after you explain to me how your reasoning works in a more abstract setting. Otherwise you’re just reserving the right to dismiss arguments (and even numbers!) on the basis that they feel wrong to you on a gut level.
Edit: It’s not that I think it’s always illegitimate to refer to your gut. It’s just bad form to claim that such references are based on mathematics.
Edit 2: Can I sidestep this discussion by saying “Let CM(n) be any real world claim with P(CM(n)) = 1/n”?
My original claim was
I nowhere stated that this was “based on mathematics.” It is naturally related to mathematics, and mathematics puts some constraints on it, as I have been trying to explain. But I didn’t come up with it in the first place in a purely mathematical way. So if this is bad form, it must be bad form to say what I mean instead of something else.
I could accept what you say in Edit 2 with these qualifications: first, since we are talking about “real world claims”, the probability 1/n does not necessarily remain fixed when someone brings up the possibility or asserts that the thing is so. This probability 1/n is only a prior, before the possibility has been raised or the thing asserted. Second, since it isn’t clear what “n” is doing, CM(5), CM(6), CM(7) and so on might be claims which are very different from one another.
I am not sure about the behavior of the ratios 1 and 2, especially given the second qualification here (in other words the ratios might not be well-behaved at all). And I don’t see how I need to say “why not?” What is there in my account which should tell me how these ratios behave? But my best guess for the moment, after thinking about it some more, would be the first ratio probably goes to infinity, but not as quickly as the fourth. What leads me to think this is something along the lines of this comment thread. For example, in my Scientology example, even if no one held that Scientology was true, but everyone admitted that it was just a story, the discovery of a real Xenu would greatly increase the probability that it was true anyway; although naturally not as much as given people’s belief in it, since without the belief, there would be a significantly greater probability that Scientology is still a mere story, but partly based on fact. So this suggests there may be a similar bound on things-which-have-been-raised-as-possibilities, even if much lower than the bound for things which are believed. Or if there isn’t a lower bound, such things are still likely to decrease in probability slowly enough to cause ratio 1 to go to infinity.
Ugly and condescending of me, beg your pardon.
You responded positively to my suggestion that we could phrase this notion of “overconfidence” as “failure to update on other people’s beliefs,” indicating that you know how to update on other people’s beliefs. At the very least, this requires some rough quantitative understanding of the players in Bayes formula, which you don’t seem to have.
If overconfidence is not “failure to update on other people’s beliefs,” then what is it?
Here’s the abbreviated version of the conversation that led us here (right?).
S: God exists with very low probability, less that one in a zillion.
U: No, you are being overconfident. After all, billions of people believe in God, you need to take that into account somehow. Surely the probability is greater than one in a billion.
S: OK I agree that the fact that billions of people believing it constitutes evidence, but surely not evidence so strong as to get from 1-in-a-zillion to 1-in-a-billion.
Now what? Bayes theorem provides a mathematical formalism for relating evidence to probabilities, but you are saying that all four quantities in the relevant Bayes formula are too poorly understood for it to be of use. So what’s an alternative way to arrive at your one-in-a-billion figure? Or are you willing to withdraw your accusation that I’m being overconfident?
I did not say that “all four quantities in the relevant Bayes formula are too poorly understood for it to be of use.” Note that I explicitly asserted that your fourth ratio tends to infinity, and that your first one likely does as well.
If you read the linked comment thread and the Scientology example, that should make it clear why I think that the evidence might well be strong enough to go from 1 in a zillion to 1 in a billion. In fact, that should even be clear from my example of the random 80 digit binary number. Suppose instead of telling you that I chose the number randomly, I said, “I may or may not have chosen this number randomly.” This would be merely raising the possibility—the possibility of something which has a prior of 2^-80. But if I then went on to say that I had indeed chosen it randomly, you would not have therefore called me a liar, while you would do this, if I now chose another random 80 digit number and said that it was the same one. This shows that even raising the possibility provides almost all the evidence necessary—it brings the probability that I chose the number randomly all the way from 2^-80 up to some ordinary probability, or from “1 in a zillion” to something significantly above one in a billion.
More is involved in the case of belief, but I need to be sure that you get this point first.
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.)
I have been using “now you are saying” as short for “now I understand you to be saying.” I think this may be causing confusion, and I’ll try write more carefully.
More soon.