# How does the Doomsday argument make any sense?

I just don't understand the Doomsday argument at all. As Wikipedia tells it, you say that "I'm 95% likely to not be one of the first 5% people to be born", and then because of that multiply the number of people born so far by 20 and claim a 95% probability of that being the upper bound on total humans born with 95% certainty.

I just don't understand how this is supposed to work.

• The 100th person ever born could make the exact same calculation and find that with 95% confidence that humanity will go extinct after 1900 more people are born.
• Every single person has a unique $n$, therefore everyone who makes this calculation will find different results.
• If you draw the first $n$ elements from a set of size $N$, all that tells you is that $N$ is at least $n$. How could it possibly tell you how many more are left in $N$, regardless of what you do with the numbers?
• Isn't it just equivalent to saying that lifetimes of intelligent species (in terms of # individuals born) follow a simple power law distribution? How can we claim such a distribution, when we've only ever observed one such species, and even that not to conclusion?

Unless I'm missing something, the conclusions fail some pretty basic sanity checks, so the argument must be wrong. But where is the problem?

Wikipedia also has a non-mathematical analogy, which seems to me equally insane: It reminds me of people who say you have 50% chance of succeeding any challenge ("either you succeed or you fail"). I would say that if you know absolutely nothing a priori, estimating the probability to be 0.5 is no better than estimating it as 0.1 or any other number.

• Slightly related is the German tank problem. – Arthur Aug 11 '16 at 12:50
• Have you read the entire article? There is a lot of criticism and rebuttal around this concept. – naslundx Aug 11 '16 at 12:54
• @Arthur I think it's not related at all, because in the Tank problem samples are randomly drawn from the set. Of course if you pick a bunch of numbers from $[1, 1000]$ it's very unlikely that all are, say, less than 10. Meanwhile in the DA, you have interval $[1, N]$ from which you draw the first $n$ - how does that let you guess $N$? – Superbest Aug 11 '16 at 12:54
• One problem is treating that figure of 95% as if it were a probability. Which it is not. If it has any meaning at all, it is more like a "confidence level". – Lee Mosher Aug 11 '16 at 12:54
• @naslundx Yes, the rebuttals all seem to be ignoring the issues I've pointed out. Some of it was difficult to understand, so I may have missed it, though. – Superbest Aug 11 '16 at 12:55

None of your critiques look to me like problems for the argument.

1. Yes, the 100th person could have reasoned in that way. This reasoning would, in that scenario, have led to a bad estimate. So? For a probabilistic argument to be a good one, we don't demand that it lead to good estimates regardless of what evidence you happen to have. If a die which I know may or may not have come from a prank novelty company comes up six purely by chance on the first fifty rolls, I will guess that the die is not fair. My reasoning is good even though my conclusion is false.
2. Yes, every person has different evidence -- especially when we're constraining the evidence in the narrow what that this argument requires. Yes, different evidence leads to different estimates. So?
3. You ask "How could it possibly tell you how many more are left in $N$, regardless of what you do with the numbers?" The answer is easy if we add the following assumption: $P(n\leq 0.05\cdot N)\leq 0.05$. The Doomsday Argument includes this assumption. Now, we can disagree about whether, in the context of the Doomsday Argument, that assumption is reasonable. (FWIW, it seems reasonable to me.) But what is clear is that if you grant that assumption, then your question ("How could $n$ tell you...") is answered.
4. The Doomsday Argument need not commit itself to this at all. It would suffice to have only the assumptions that: (a) There exists some number $N$ which is (an upper bound on) the total number of humans who will live; and (b) my own birth number is assigned to me on a uniform distribution (i.e., I could equally likely have had any birth order number $1\leq n \leq N$). Quite possibly you could salvage a version of the DA with even weaker versions of (b).

It seems to me that the crux of your disagreement with the Argument is point 3. You do not accept the crucial assumption that you $P(n\leq 0.05 \cdot N)\leq 0.05$. I'm not sure how we should adjudicate whether that statement is true. Maybe, in the end, this is the kind of issue on which the kind of frequentist intuitions that drive all of us (even us committed Bayesians) cannot thrive. But if you do accept that assumption, then it seems difficult to escape the conclusion of the Doomsday Argument.

No assumption is being made about the distribution of lifetimes. The only assumption being made is that you can consider yourself to be a uniformly randomly chosen individual from the collective population of humanity over all time. Yes, it is true that the 100th human born would conclude that humanity would end by the time 1900 more people are born, but the chance of any individual following this logic and reaching that conclusion is only $100/N$ - a vanishingly low probability.

To your second bullet point, note that this logic gives a predicted upper bound on $N$, not a predicted value for $N$.

• There is still the issue that you would say $N=N_1$ and I would say $N=N_2 \ne N_1$. Since we're both part of the same humanity, how can we arrive at different predictions? – Superbest Aug 11 '16 at 13:00
• We have an upper bound on $N$, not a predicted value for $N$. – Dustan Levenstein Aug 11 '16 at 13:01
• Well, the DA claims to produce probability distributions for possible values of $N$ with only $n$ as input. I was saying "predict" as a shorthand, but the probability of $N$ being a certain number shouldn't depend on who makes the calculation either. – Superbest Aug 11 '16 at 13:07
• I haven't read the wikipedia article, but sure; everyone will have their own probability distribution for $N$ (à la Bayes). There's nothing contradictory there, as long as each distribution assigns a nonzero probability to the true value of $N$. – Dustan Levenstein Aug 11 '16 at 13:13

You're right: the Doomsday Argument is total nonsense. The primary fallacy is the notion that "you" are randomly assigned a spot among all the modern humans that have ever lived or will live. This is just as illogical as saying "I'm lucky I was born in New York", or anything like that. It just doesn't work that way. The argument implicitly assumes that first the world decides how many people there will be, and then "you" get randomly assigned to be one of them. And, for some reason, "you" get to definitely exist no matter how many people there will end up having been. All of these assumptions are necessary for the argument to work, and they have no connection to reality.