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Vaniver comments on Doomsday Argument Map - Less Wrong

6 Post author: turchin 14 September 2015 03:04PM

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Comment author: Vaniver 17 October 2015 07:07:28AM *  1 point [-]

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:

  1. Suppose across all of time there are a finite number of humans, and that we can order them by time of birth.

  2. 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.

  3. My prior should be that my fractile is uniformly distributed between 0.0 and 1.0.

  4. 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.

  5. 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.

  6. 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 , 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."