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By making the a priori assumption that we are equally likely to be anywhere along the chronological list of humans, the Doomsday argument implies that our position n is correlated with the future total number of humans, N.

But if the many-worlds interpretation of quantum theory is correct then all futures will exist with all values of N. In that case our position n is not correlated with any particular future with a particular value of N.

Thus, as it seems reasonable that we cannot predict the future of the human race, can we use the failure of the Doomsday argument as evidence for the many-worlds interpretation?

**More detail**

Using Bayes's Theorem the probability of the total population N given our position n, P(N|n) is P(N|n) = P(n|N)P(N) / P(n).

If we assume prior complete ignorance of n and N then we should use the improper priors P(N)=1/N and P(n)=1/n.

Assuming a uniform probability for our position n given a particular total population size N, P(n|N)=1/N, we find that Bayes's theorem says

P(N|n) = n / N^2.

This is the Doomsday argument prediction. It implicitly assumes only one future with some particular total population N with prior probability P(N)=1/N.

But in the many-worlds scenario we would have many futures with many values of total population N weighted by the function W(N)=1/N. The probability of our position n would then be given by the sum

P(n) = Sum[N=n to infinity] P(n|N) W(N)

P(n) = Sum[N=n to infinity] 1/N * 1/N = 1 / n

As mentioned above the probability P(n)=1/n and W(N)=1/N implies total ignorance. Therefore in the many-worlds scenario if we lived until the end of the human race our birth position n would not tell us anything about the final population size N that any particular version of ourselves will experience.

1My hunch is that when you run the maths for both together, you end up where you began... where your prediction of future population is purely contingent on how you evaluate the conditions of the current word you're in. I'm a little skeptical about the usefulness of knowledge gained by statistical tricks applied to variations of the anthropic principle. – Ask About Monica – 2017-11-27T17:33:47.037

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Suppose you have some phenomenon X which is a priori unlikely and a theory T that makes it more likely, can X be considered evidence for T? In a very weak sense, yes. Very weak because there could be plenty of other theories making X more likely, and the usual approach is to survey them to select the "best" one according to some criteria. In this case there is a long list of alternative explanations, so it does not really work as "evidence".

– Conifold – 2017-11-27T20:07:30.510