Confused about the doomsday argument, please help

by cousin_it1 min read14th Apr 201128 comments

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The doomsday argument says I have only a 10% chance of being within the first 10% of humans ever born, which gives nonzero information about when humanity will end. The argument has some problems with the choice of reference class; my favorite formulation (invented by me, I'm not sure if it's well-known) is to use the recursive reference class of "all people who are considering the doomsday argument with regard to humanity". But this is not the issue I want to discuss right now.

Imagine your prior says the universe can contain 10, 1000 or 1000000 humans, with probability arbitrarily assigned to these three options. Then you learn that you're the 50th human ever born. As far as I can understand, after receiving this information you're certain to be among the first 10% of humans ever born, because it's true in every possible universe where you receive such information. Also learning your index doesn't seem to tell you very much about the date of the doomsday: it doesn't change the relative probabilities of doomsday dates that are consistent with your existence. (This last sentence is true for any prior, not just the one I gave.) Is there something I'm missing?

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That wikipedia article is pretty bad. Read this and this instead. It looks like your confusion is caused by there being multiple versions of the doomsday argument; see the second question in the linked FAQ.

Assuming we're using Leslie's version:

As far as I can understand, after receiving this information you're certain to be among the first 10% of humans ever born, because it's true in every possible universe where you receive such information.

Yes, that's right: it's only before learning your birth rank that you assign a 10% probability of being in the first 10% of humans ever born.

Also learning your index doesn't seem to tell you very much about the date of the doomsday: it doesn't change the relative probabilities of doomsday dates that are consistent with your existence.

But it does: finding you're the 50th human is much more likely under the hypothesis that there will be 1000 humans than under the hypothesis that there will be 1000000 humans.

But it does: finding you're the 50th human is much more likely under the hypothesis that there will be 1000 humans than under the hypothesis that there will be 1000000 humans.

I have no coherent theory of updating on information like "you're the 50th human" that gives the result you suggest. I'm using the only coherent theory that I fully understand: updating on "the 50th human exists". (Note that it doesn't privilege the observer, i.e. if you know that the 51st human exists somewhere, you can update on that too.) If you have a theory for the former kind of updating, and it gives a different result, then I'd really like to learn it.

P(you're 50th | 1000 people) = 1/1000

P(you're 50th | 1000000 people) = 1/1000000

So the 1000 people world has more of its probability mass on the fact that you're the 50th person.

If that's how you update, then you can never update on observations in Big Worlds: see Bostrom's paper.

Hmm. It seems to me that I update just fine. If I flip a quantum coin and it comes up heads, and afterwards I face a decision problem whose outcome depends on that coinflip, then UDT prescribes behavior that looks like I had updated.

Anyway, if my way of updating is wrong, then what way is right?

I'm sympathetic to that approach, but most of the folk supporting or opposing the SSA (or SIA) Doomsday Argument aren't. From the title of your post I thought you were trying to understand what people were talking about. The Doomsday argument comes from updating principles like SSA, as discussed here.

I am trying to understand what people are talking about, precisely, and I asked on LW because people here are more likely to have a precise understanding of the DA than most philosophers.

If my original example takes place in a Big World (e.g. the total population depends on a quantum event that happened long ago), then it seems to me that the SSA doesn't make the DA go through. Let's say an urn contains 1 red ball, 1000 yellow balls and 1000000 green balls. Balls of each color are numbered. You draw a ball at random and see that it says "50", but you're colorblind and cannot see the color. Then Bayes says you should assign 0 probability to red, 0.5 to yellow, and 0.5 to green, thus the relative probabilities of "worlds compatible with your existence" are unchanged.

So I'm still confused. Does the updating rule used by the DA rely on a fundamental difference between big worlds and small worlds? This looks suspicious, because human decisions shouldn't change depending on classicalness or quantumness of a coinflip, yet the SSA seems to say they should, by arbitrarily delineating parts of reality as "worlds". There's got to be a mistake somewhere.

The implied algorithm is that you first pick a world size s from some distribution, and then pick an index uniformly from 1..s. This corresponds to the case where there are three separate urns with 1 red, 1000 yellow and 10^6 green balls, and you pick from one of the urns without knowing which one it is.

(I find the second part, picking an index uniformly from 1..s, questionable; but there's only one sample of evidence with which to determine what the right distribution would be, so there's little point in speculating on it.)

Let's say an urn contains 1 red ball, 1000 yellow balls and 1000000 green balls. Balls of each color are numbered.

This is not equivalent to the original problem. In the original problem, if there are 1000 people you have a 1/1000 chance of being the 50th and if there are 1,000,000 people you have a 1/1,000,000 chance of being the 50th. In your formulation, you have a 1/1,001,001 chance of getting each of the balls marked '50'.

It might be equivalent to have the urn contain one million red balls marked '1', one million yellow balls divided into one thousand sets which are each numbered one through one thousand, and one million green balls numbered one through one million. In this case, if you draw a ball marked '50', it can be either the one green ball that's marked '50', or any of the thousand yellow balls marked '50', and the latter case is one thousand times more likely than the former.

Thanks. Carl, jimrandomh and you have helped me understand what the original formulation says about probabilities, but I still can't understand why it says that. My grandparent comment and its sibling can be interpreted as arguments against the original formulation, what do you think about them?

In general I'm a lousy one to ask about probability; I only noticed this particular thing after a few days of contemplation. I was more hoping that someone else would see it and be able to use it to form a more coherent explanation.

I do think, regarding the sibling, that creating or destroying people is incompatible with assuming that a certain number of people will exist - I expect that a hypothesis that would generate that prediction would have an implicit assumption that nobody is going to be creating or destroying or failing to create people on the basis of the existence of the hypothesis. In other words, causation doesn't work like that.

Edit: It might help to note that the original point that led me to notice that your formulation was flawed was that the different worlds - represented by the different colors - were not equally likely. If you pick a ball out of your urn and don't look at the number, it's much more likely to be green than yellow and very very unlikely to be red. If you pick a ball out of my urn, there's an even chance of it being any of the three colors.

I thought about SSA some more and came up with a funny scenario. Imagine the world contains only one person and his name is Bob. At a specified time Omega will or won't create 100 additional people depending on a coinflip, none of whom will be named Bob.

  • Case 1: Bob knows that he's Bob before the coinflip. In this case we can all agree that Bob can get no information about the coinflip's outcome.

  • Case 2: Bob takes an amnesia drug, goes to sleep, the coinflip happens and people are possibly created, Bob wakes up thinking he might be one of them, then takes a memory restoration drug. In this case SSA leads him to conclude that additional people probably didn't get created, even though he has the same information as in case 1.

  • Case 3: coinflip happens, Bob takes amnesia drug, then immediately takes memory restoration drug. SSA says this operation isn't neutral and Bob should switch from case 1 to case 2. Moreover, Bob can anticipate changing his beliefs this way, but that doesn't affect his current beliefs. Haha.

  • Bonus: what if Omega is spacelike separated from Bob?

The only way to rescue SSA is to bite the bullet in case 1 and say that Bob's prior beliefs about the coinflip's outcome are not 50/50; they are "shifted" by the fact that the coinflip can create additional people. So SSA allows Bob to predict with high confidence the outcome of a fair coinflip, which sounds very weird (though it can still be right). Note that using UDT or "big-world SSA" as in my other comment will lead to more obvious and "normal" answers.

ETA: my scenario suggests a hilarious way to test SSA experimentally. If many people use coinflips to decide whether to have kids, and SSA is true, then the results will be biased toward "don't have kids" because the doomsday wants to happen sooner and pushes probabilities accordingly :-)

ETA2: or you could kill or spare babies depending on coinflips, thus biasing the coins toward "kill". The more babies you kill, the stronger the bias.

ETA3: or you could win the lottery by precommitting to create many observers if you lose. All these scenarios make SSA and the DA look pretty bad.

[-][anonymous]10y 0

OK, what's the correct way for me to update?

First, update on "there exists a Less Wronger with the handle 'cousin it'", along with everything else you know about you.

Depending on your posterior, there's basically two ways this could go:

  • Uniform from 1 human to so many that of course there's one like the one you know.

You now find very strong evidence that cousin it is the 50th human. You update the probability that he is from about 10^-100 to 10^-80.

  • Log distribution, with slightly shorter tails.

You now find very strong evidence that cousin it is the 50th human. You update the probability that he is from about 10^-10 to about 1-10^-10. Now, for all intents and purposes, you know cousin it is the 50th human. This is more likely if there are only 1000 people.

Basically, either the doomsday argument is correct, or humanity has been around for a very, very long time.

Incidentally, this isn't quite the same as what atucker and I think about it. If I only experienced one bit, I'd still think it works (ignoring the fact that if I had thoughts that complex, it would be more than one bit). It's still pretty close though.

updating on "the 50th human exists"

That doesn't happen either. Observing something doesn't tell you it's a priori probable. (But you could update to considering only the worlds in which you "exist", that is the worlds that you potentially control.)

[-][anonymous]10y 0

I have no interest in defending the doomsday argument; I'm just explaining what it says.

[-][anonymous]10y 0

Sorry. I didn't intend to attack the argument. I just want to learn what it says, precisely. If it uses reasoning that is not precise, then I'd like to know if it can be made precise.

[-][anonymous]10y 0

Grandparent comment said something along the lines of "I'm not interested in defending the doomsday argument, I'm just explaining what it says", then I deleted it not knowing it had been replied to.

My understanding of the doomsday argument, perhaps mistaken, is that one assumes that population growth is exponential in the long term, in which case most observers live just before the end.

The doomsday argument makes no assumptions about the rate of population growth.

Really? If they assume a uniform distribution, wouldn't that be unrealistic? If the assumptions result in different predictions for how an observer should update, how do they avoid making any?

The DA assumes a prior probability distribution for the number of people that will ever exist; beyond that, it doesn't matter for updating purposes when these people come into existence.

[-][anonymous]10y 0

Are you sure? If the population of observers are uniformly distributed over a time interval, I calculate that they should each multiply their observation of the age of their civilization by root 2 in order minimize their collective error. However, I don't see this estimate on the Wikipedia page and assume they have more complex assumptions about the distribution of observers.

... I also assume that the civilization lifetime is uniformly distributed over an unknown interval. I suppose it would be more reasonable to assume it is normally distributed over an unknown interval or -- even more accurately -- extremely right tailed.

This is doubtless a frequentist approach, which perhaps isn't allowed, but I asked myself, how should an observer update on the information that civilization has existed up until that time, if observers collectively wanted to minimize their total error.

Suppose that a civilization lasts T years, and I assume that observations are uniformly distributed over the T years. (This would mean, for example, that the civilization at each time point in the interval (0,T) makes a collective, single vote.) Given only the information that civilization has lasted x years (clearly, 0<=x<=T), the observer would guess that civilization will actually last some multiple of x years: cx. Should they choose a c that is low (close to 1) or high, etc?

The optimal value of c can be calculated as exactly sqrt(2).

By taking the minimum of the function that measures the total error, the integral of the error (error=|T-cx|) integrated from T to 0.

You would get a different c if you assume that growth is exponential and weight by the number of observers. It would be closer to 1.

Also, implicit is the assumption that T is uniformly distributed over an unknown range. Instead, T might be normally distributed with an unknown mean or extremely tight-tailed. These would also affect c, but by moving c further from 1 I think.

tldr;

If you observe that civilization is age X units, you should update that it will last another 0.4 X units.

I've been meaning to post about the Doomsday Argument for awhile. I have a strong sense that it's wrong, but I've had a hell of a time trying to put my finger on how it fails. The best that I can come up with is as follows: Aumann's agreement theorem says that two rational agents cannot disagree, in the long run. In particular, two rational agents presented with the same evidence should update their probability distribution in the same direction. Suppose I learn that I am the 50th human, and I am led to conclude that it is far more likely that only 1000 humans should ever live, than 100,000. But suppose I go tell Bob that I'm the 50th human; it would be senseless for him to come to the same conclusion that I have. Formally, it looks something like this:

P(1000 humans|I am human #50)>P(1000 humans)

but

P(1000 humans|Skatche is human #50)=P(1000 humans)

where the right hand sides are the prior probabilities. The same information has been conveyed in each case, yet very different conclusions have been reached. Since this cannot be, I conclude that the Doomsday Argument is mistaken. This could perhaps be adapted as an argument against anthropic reasoning more generally.

it would be senseless for him to come to the same conclusion that I have.

Why do you say that?

Suppose you have an urn containing consecutively numbered balls. But you don't know how many. Draw one ball from the urn and update your probabilities regarding the number of balls. Draw a second ball, and update again.

Two friends each draw one ball and then share information. I don't see why the ball you drew yourself should be privileged.

Two variants of this urn problem that may offer some insight into the Doomsday Argument:

  1. The balls are not numbered 1,2,3,... Instead they are labeled "1st generation", "2nd generation", ... After sampling, estimate the label on the last ball.

  2. The balls are labeled "1 digit", "2 digits", "3 digits" ... Again, after sampling, estimate the label on the last ball.

I see what you're saying, but I'm not sure if the analogy applies, since it depends a great deal on the selection process. When I learn that Julius Caesar lived from 100-44BCE, or that Stephen Harper lives in the present day, that certainly doesn't increase my estimated probability of humans dying out within the next hundred years; and if I lack information about humans yet to be born, that's not surprising in the slightest, whether or not we go extinct soon.

Really it's the selection process that's the issue here; I don't know how to make sense of the question "Which human should I consider myself most likely to be?" I've just never been able to nail down precisely what bothers me about the question.