This is probably the wrong place to ask, but I’m confused by one point in the DA.
For reference, here’s Wikipedia’s current version:
Denoting by N the total number of humans who were ever or will ever be born, the Copernican principle suggests that humans are equally likely (along with the other N − 1 humans) to find themselves at any position n of the total population N, so humans assume that our fractional position f = n/N is uniformly distributed on the interval [0, 1] prior to learning our absolute position.
f is uniformly distributed on (0, 1) even after learning of the absolute position n. That is, for example, there is a 95% chance that f is in the interval (0.05, 1), that is f > 0.05. In other words we could assume that we could be 95% certain that we would be within the last 95% of all the humans ever to be born. If we know our absolute position n, this implies[dubious – discuss] an upper bound for N obtained by rearranging n/N > 0.05 to give N < 20n.
My question is: What is supposed to be special about the interval (0.05, 1)?
If I instead choose the interval (0, 0.95), then I end up 95% certain that I’m within the first 95% of all humans ever to be born. If I choose (0.025, 0.975), then I end up 95% certain that I’m within the middle 95% of all humans ever to be born. If I choose the union of the intervals (0, 0.475) & (0.525, 1), then I end up 95% certain that I’m within the 95% of humans closer to either the beginning or the end.
As far as I can tell, I could have chosen any interval or any union of intervals containing X% of humanity and then reasonably declared myself X% likely to be in that set. And sure enough, I’ll be right X% of the time if I make all those claims or a representative sample of them.
I guess another way to put my question is: Is there some reason—other than drama—that makes it special for us to zero in on the final 95% as our hypothesis of interest? And if there isn’t a special-making reason, then shouldn’t we discount the evidential weight of the DA in proportion to how much we arbitrarily zero in on our hypothesis, thereby canceling out the DA?
Yes, yes, given that there’s so much literature on the topic, I’m probably missing some key insight into how the DA works. Please enlighten.
If I instead choose the interval (0, 0.95), then I end up 95% certain that I’m within the first 95% of all humans ever to be born.
What would this imply about the total number of humans? If you knew that you were the 50th percentile human, for example, that would give you the total number of humans, and the same is true for all percentiles.
I think the ‘continuous’ approach to the DA, which does not rely on the ‘naturalness’ of 1/20th, goes like the following:
Suppose across all of time there are a finite number of humans, and that we can order them by time of birth.
To normalize the birth orders, we can divide each person’s position in the ordering by the total number of people, meaning each person corresponds to a fractile between 0.0 and 1.0.
My prior should be that my fractile is uniformly distributed between 0.0 and 1.0.
Upon observing that I am human number 108 billion, I can now combine this with my prior on the fractile to compute the estimated human population.
There is a 1% chance that my fractile is between 0.99 and 1.0, which would imply there is a 1% chance the total number of humans is between 109B and 108B. (The larger number is earlier because it corresponds to being the 99th percentile human instead of the 100th percentile human.) We now add this up for all possible fractiles, to get a distribution and a mean.
This is a tool for integration. If I’m number N and my fractile is f, then the total number of humans is N/f. So we integrate ∫10Nfdf, which… does not converge. The expected number of future humans is infinite!
But while the expectation diverges, that doesn’t mean the most likely value is infinite humans. The median value is determined by f=0.5, where there are only 108B more humans. In fact, for any finite number of humans, one can calculate the probability that there will be that many or fewer humans—which is why the last 95% of humans is relevant. Those are the ones where the extinction numbers are soonest.
Any real resolution of the Doomsday Argument needs to replace the basic structure of “assume uniform prior on fractile distribution, combine with number of observed humans” with “assume uniform prior on fractile distribution, update fractile distribution based on modeled trajectory of history, combine with number of observed humans.” For example, one could look at human history and the future and say “look, extinction in the near future seems very likely, and growth to immense numbers seems very likely, but exactly 100B more humans looks very unlikely. We need to replace our Beta(1,1) distribution with something like a Beta(0.5,0.5) distribution.”
This is probably the wrong place to ask, but I’m confused by one point in the DA.
For reference, here’s Wikipedia’s current version:
My question is: What is supposed to be special about the interval (0.05, 1)?
If I instead choose the interval (0, 0.95), then I end up 95% certain that I’m within the first 95% of all humans ever to be born. If I choose (0.025, 0.975), then I end up 95% certain that I’m within the middle 95% of all humans ever to be born. If I choose the union of the intervals (0, 0.475) & (0.525, 1), then I end up 95% certain that I’m within the 95% of humans closer to either the beginning or the end.
As far as I can tell, I could have chosen any interval or any union of intervals containing X% of humanity and then reasonably declared myself X% likely to be in that set. And sure enough, I’ll be right X% of the time if I make all those claims or a representative sample of them.
I guess another way to put my question is: Is there some reason—other than drama—that makes it special for us to zero in on the final 95% as our hypothesis of interest? And if there isn’t a special-making reason, then shouldn’t we discount the evidential weight of the DA in proportion to how much we arbitrarily zero in on our hypothesis, thereby canceling out the DA?
Yes, yes, given that there’s so much literature on the topic, I’m probably missing some key insight into how the DA works. Please enlighten.
What would this imply about the total number of humans? If you knew that you were the 50th percentile human, for example, that would give you the total number of humans, and the same is true for all percentiles.
I think the ‘continuous’ approach to the DA, which does not rely on the ‘naturalness’ of 1/20th, goes like the following:
Suppose across all of time there are a finite number of humans, and that we can order them by time of birth.
To normalize the birth orders, we can divide each person’s position in the ordering by the total number of people, meaning each person corresponds to a fractile between 0.0 and 1.0.
My prior should be that my fractile is uniformly distributed between 0.0 and 1.0.
Upon observing that I am human number 108 billion, I can now combine this with my prior on the fractile to compute the estimated human population.
There is a 1% chance that my fractile is between 0.99 and 1.0, which would imply there is a 1% chance the total number of humans is between 109B and 108B. (The larger number is earlier because it corresponds to being the 99th percentile human instead of the 100th percentile human.) We now add this up for all possible fractiles, to get a distribution and a mean.
This is a tool for integration. If I’m number N and my fractile is f, then the total number of humans is N/f. So we integrate ∫10Nfdf, which… does not converge. The expected number of future humans is infinite!
But while the expectation diverges, that doesn’t mean the most likely value is infinite humans. The median value is determined by f=0.5, where there are only 108B more humans. In fact, for any finite number of humans, one can calculate the probability that there will be that many or fewer humans—which is why the last 95% of humans is relevant. Those are the ones where the extinction numbers are soonest.
Any real resolution of the Doomsday Argument needs to replace the basic structure of “assume uniform prior on fractile distribution, combine with number of observed humans” with “assume uniform prior on fractile distribution, update fractile distribution based on modeled trajectory of history, combine with number of observed humans.” For example, one could look at human history and the future and say “look, extinction in the near future seems very likely, and growth to immense numbers seems very likely, but exactly 100B more humans looks very unlikely. We need to replace our Beta(1,1) distribution with something like a Beta(0.5,0.5) distribution.”
As we are interested in future time of humanity existence we choose this interval. If we have different question, we may choose different intervals.