Suppose I bet that a 30-digit random number generator will deliver the number 938726493810487327500934872645. And, lo and behold, the generator comes up with 938726493810487327500934872645 on the first try. If I am magically certain that I am actually dealing with a random number generator, I ought to conclude that I am hallucinating, because p(me hallucinating) > p(guessing a 30-digit string correctly).
Note that this is true even though p(me hallucinating the number 938726493810487327500934872645) is quite low. I am certainly more likely, for example, to hallucinate the number 123456789012345678901234567890 than I am to hallucinate the number 938726493810487327500934872645. But since I am trying to find the minimum meaningful probability, I don’t care too much about the upper bounds on the odds that I’m hallucinating—I want the lower bound on the odds that I’m hallucinating, and the lower bound would correspond to a mentally arbitrary number like 938726493810487327500934872645.
In other words, if you agree with me that p(I correctly guess the number 938726493810487327500934872645) < p(I’m hallucinating the number 938726493810487327500934872645), then you should certainly agree with me that p(I correctly guess the number 123456789012345678901234567890) < p(I’m hallucinating the number 123456789012345678901234567890). The probability of guessing the correct number is always 10^-30; the probability of hallucinating varies, but I suspect that the probability of hallucinating is more than 10^-30 for either number.
Choosing a number and betting that you will see it increases the probability that you will wrongly believe that you have seen that number in the future to a value that does not depend on how long that number is. P(hallucinate number N|placed a bet on N) >> P(hallucinate number N).
Yes, I completely agree. To show that I understand your point, I will suggest possible numbers for each of these variables. I would guess, with very low confidence, that on a daily basis, P(hallucinate a number) might be something like 10^-7, that P(hallucinate a 30-digit number N) might be something like 10^-37, and that P(hallucinate a 30-digit number N | placed a bet on N) might be something like 10^-9. Obviously, p(correctly guess a 30-digit number) is still 10^-30.
Even given all of these values, I still claim that we should be interested in P(hallucinate a 30-digit number N | placed a bet on N). This number is probably roughly constant across ostensibly sane people, and I claim that it marks a lower bound below which we should not care about the difference in probabilities for a non-replicable event.
I am not certain of these claims, and I would greatly appreciate your analysis of them.
Note that there are explanations other than “I correctly guessed”, and “I’m hallucinating”. “This generator is broken and always comes up 938726493810487327500934872645, but I’ve forgotten that it’s broken consciously, but remember that number”, or “The generator is really remotely controlled, and it has a microphone that heard me guess, and transmitted that to the controller, who wants to mess with my head.”
Oh, I completely agree. I’m using “hallucinating” as shorthand for all kinds of conspiracy theories, and assuming away the chance that the generator is broken.
Obviously the first thing you should do if you appear to guess right is check the generator.
I respectfully disagree.
Suppose I bet that a 30-digit random number generator will deliver the number 938726493810487327500934872645. And, lo and behold, the generator comes up with 938726493810487327500934872645 on the first try. If I am magically certain that I am actually dealing with a random number generator, I ought to conclude that I am hallucinating, because p(me hallucinating) > p(guessing a 30-digit string correctly).
Note that this is true even though p(me hallucinating the number 938726493810487327500934872645) is quite low. I am certainly more likely, for example, to hallucinate the number 123456789012345678901234567890 than I am to hallucinate the number 938726493810487327500934872645. But since I am trying to find the minimum meaningful probability, I don’t care too much about the upper bounds on the odds that I’m hallucinating—I want the lower bound on the odds that I’m hallucinating, and the lower bound would correspond to a mentally arbitrary number like 938726493810487327500934872645.
In other words, if you agree with me that p(I correctly guess the number 938726493810487327500934872645) < p(I’m hallucinating the number 938726493810487327500934872645), then you should certainly agree with me that p(I correctly guess the number 123456789012345678901234567890) < p(I’m hallucinating the number 123456789012345678901234567890). The probability of guessing the correct number is always 10^-30; the probability of hallucinating varies, but I suspect that the probability of hallucinating is more than 10^-30 for either number.
Choosing a number and betting that you will see it increases the probability that you will wrongly believe that you have seen that number in the future to a value that does not depend on how long that number is. P(hallucinate number N|placed a bet on N) >> P(hallucinate number N).
Yes, I completely agree. To show that I understand your point, I will suggest possible numbers for each of these variables. I would guess, with very low confidence, that on a daily basis, P(hallucinate a number) might be something like 10^-7, that P(hallucinate a 30-digit number N) might be something like 10^-37, and that P(hallucinate a 30-digit number N | placed a bet on N) might be something like 10^-9. Obviously, p(correctly guess a 30-digit number) is still 10^-30.
Even given all of these values, I still claim that we should be interested in P(hallucinate a 30-digit number N | placed a bet on N). This number is probably roughly constant across ostensibly sane people, and I claim that it marks a lower bound below which we should not care about the difference in probabilities for a non-replicable event.
I am not certain of these claims, and I would greatly appreciate your analysis of them.
Note that there are explanations other than “I correctly guessed”, and “I’m hallucinating”. “This generator is broken and always comes up 938726493810487327500934872645, but I’ve forgotten that it’s broken consciously, but remember that number”, or “The generator is really remotely controlled, and it has a microphone that heard me guess, and transmitted that to the controller, who wants to mess with my head.”
Oh, I completely agree. I’m using “hallucinating” as shorthand for all kinds of conspiracy theories, and assuming away the chance that the generator is broken.
Obviously the first thing you should do if you appear to guess right is check the generator.