Oops, it looks like I messed up. I misinterpreted you as talking about a distribution of when humanity would go extinct. But you weren’t talking about a distribution—you just said ” if humanity lives forever, that contradicts intuition about our chances of biting it in finite time.”
Maybe it’s because I think a distribution is the correct way to think about it. If the chances of humanity going extinct are a Poisson distribution normalized to 0.75, we have a 25% chance of living forever AND a finite chance of biting it in finite time—the apparent exclusion is resolved.
Of course, that still doesn’t answer the question “why were we born now and not infinitely later?” To do that, you have to invoke the axiom of choice, which is, in layman’s terms, “impossible things happen all the time, because it makes more sense than them not happening.”
-
“As for your last part, why is variance important?”
It’s because a prediction is only powerful if it excludes some possibilities. If any real measurement is within a standard deviation of the prediction, it’s a pretty boring prediction—it’s not enough to tell me that I will die in 50 years if the error bars are +/- 100.
Of course, the more I think about it the more I question how I got my result. It was pretty dependent on thinking of the mean as something drawn from a distribution rather than just plugging and chugging the obvious way.
It’s not about disproving certain possibilities; it’s about finding them increasingly unlikely. I never said that humanity will die out. The intuition at the beginning was that it might. Of course, if there’s any chance it will, the fact that we’re alive now instead of unimaginable far into the future shows that it almost definitely will.
“impossible things happen all the time, because it makes more sense than them not happening.”
But they’d be more likely to happen if there were fewer other possibilities. For example, if there was one dart thrower who was only guaranteed to hit the dartboard, and one who only ever hit the bullseye, if it hit in a given spot on the bullseye, it would be fairly good evidence that the second guy threw it.
Also, I don’t see what that has to with the Axiom of Choice. That’s an axiom of set theory, not statistics.
The axiom of choice was just to justify how it’s possible to be born now even if there are infinite people, which is an objection I have seen before, though you didn’t make it. Also it’s a fun reference to make.
It’s possible I’m wrong, but I’ll try to walk you through my logic as to why it isn’t informative that we’re born now. Imagine the “real” set of people who ever get born. When choosing from this bunch o’ people, the variance is - ^2 = N^2 /3-N^2 /4 = N^2 /12 (for large N).
So the standard deviation is proportional not to the square root of N, as you’d expect if you’re used to normal distributions, but is instead proportional to N. This means that no matter how big N is, the beginning is always the same number of standard deviations from the mean. Therefore, to a decent approximation, it’s not more surprising to be born now if there are 10^20 total people born than if there are 10^100.
Of course, writing this out highlights a problem with my logic (drat!) which is that standard deviations aren’t the most informative things when your distribution doesn’t look anything like a normal distribution—I really should have thought in two-tailed integrals or something. But then, once you start integrating things, you really do need a prior distribution.
So dang, I guess my argument was at least partly an artifact of applying standard distributions incorrectly.
Oops, it looks like I messed up. I misinterpreted you as talking about a distribution of when humanity would go extinct. But you weren’t talking about a distribution—you just said ” if humanity lives forever, that contradicts intuition about our chances of biting it in finite time.”
Maybe it’s because I think a distribution is the correct way to think about it. If the chances of humanity going extinct are a Poisson distribution normalized to 0.75, we have a 25% chance of living forever AND a finite chance of biting it in finite time—the apparent exclusion is resolved.
Of course, that still doesn’t answer the question “why were we born now and not infinitely later?” To do that, you have to invoke the axiom of choice, which is, in layman’s terms, “impossible things happen all the time, because it makes more sense than them not happening.”
-
“As for your last part, why is variance important?”
It’s because a prediction is only powerful if it excludes some possibilities. If any real measurement is within a standard deviation of the prediction, it’s a pretty boring prediction—it’s not enough to tell me that I will die in 50 years if the error bars are +/- 100.
Of course, the more I think about it the more I question how I got my result. It was pretty dependent on thinking of the mean as something drawn from a distribution rather than just plugging and chugging the obvious way.
It’s not about disproving certain possibilities; it’s about finding them increasingly unlikely. I never said that humanity will die out. The intuition at the beginning was that it might. Of course, if there’s any chance it will, the fact that we’re alive now instead of unimaginable far into the future shows that it almost definitely will.
But they’d be more likely to happen if there were fewer other possibilities. For example, if there was one dart thrower who was only guaranteed to hit the dartboard, and one who only ever hit the bullseye, if it hit in a given spot on the bullseye, it would be fairly good evidence that the second guy threw it.
Also, I don’t see what that has to with the Axiom of Choice. That’s an axiom of set theory, not statistics.
The axiom of choice was just to justify how it’s possible to be born now even if there are infinite people, which is an objection I have seen before, though you didn’t make it. Also it’s a fun reference to make.
It’s possible I’m wrong, but I’ll try to walk you through my logic as to why it isn’t informative that we’re born now. Imagine the “real” set of people who ever get born. When choosing from this bunch o’ people, the variance is - ^2 = N^2 /3-N^2 /4 = N^2 /12 (for large N).
So the standard deviation is proportional not to the square root of N, as you’d expect if you’re used to normal distributions, but is instead proportional to N. This means that no matter how big N is, the beginning is always the same number of standard deviations from the mean. Therefore, to a decent approximation, it’s not more surprising to be born now if there are 10^20 total people born than if there are 10^100.
Of course, writing this out highlights a problem with my logic (drat!) which is that standard deviations aren’t the most informative things when your distribution doesn’t look anything like a normal distribution—I really should have thought in two-tailed integrals or something. But then, once you start integrating things, you really do need a prior distribution.
So dang, I guess my argument was at least partly an artifact of applying standard distributions incorrectly.