Avoiding doomsday: a "proof" of the self-indication assumption

I upvoted this and I think you proved SIA in a very clever way, but I still don't quite understand why SIA counters the Doomsday argument.

Imagine two universes identical to our own up to the present day. One universe is destined to end in 2010 after a hundred billion humans have existed, the other in 3010 after a hundred trillion humans have existed. I agree that knowing nothing, we would expect a random observer to have a thousand times greater chance of living in the long-lasting universe.

But given that we know this particular random observer is alive in 2009, I would think there's an equal chance of them being in both universes, because both universes contain an equal number of people living in 2009. So my knowledge that I'm living in 2009 screens off any information I should be able to get from the SIA about whether the universe ends in 2010 or 3010. Why can you still use the SIA to prevent Doomsday?

[analogy: you have two sets of numbered balls. One is green and numbered from 1 to 10. The other is red and numbered from 1 to 1000. Both sets are mixed together. What's the probability a randomly chosen ball is red? 1000/1010. Now I tell you the ball has number "6" on it. What's the probability it's red? 1/2. In this case, Doomsday argument still applies (any red or green ball will correctly give information about the number of red or green balls) but SIA doesn't (any red or green ball, given that it's a number shared by both red and green, gives no information on whether red or green is larger.)]

My paper, Past Longevity as Evidence for the Future, in the January 2009 issue of Philosophy of Science, contains a new refutation to the Doomsday Argument, without resort to SIA.

The paper argues that the Carter-Leslie Doomsday Argument conflates future longevity and total longevity. For example, the Doomsday Argument’s Bayesian formalism is stated in terms of total longevity, but plugs in prior probabilities for future longevity. My argument has some similarities to that in Dieks 2007, but does not rely on the Self-Sampling Assumption.

I'm relatively green on the Doomsday debate, but:

The non-intuitive form of SIA simply says that universes with many observers are more likely than those with few; the more intuitive formulation is that you should consider yourself as a random observer drawn from the space of possible observers (weighted according to the probability of that observer existing).

Isn't this inserting a hidden assumption about what kind of observers we're talking about? What definition of "observer" do you get to use, and why? In order to "observe", all that's necessary is that you form mutual information with another part of the universe, and conscious entities are a tiny sliver of this set in the observed universe. So the SIA already puts a low probability on the data.

I made a similar point before, but apparenlty there's a flaw in the logic somewhere.

It seems understressed that the doomsday argument is as an argument about max entropy priors, and that any evidence can change this significantly.

Yes, you should expect with p = 2/3 to be in the last 2/3 of people alive. Yes, if you wake up and learn that there have only been tens of billions of people alive but expect most people to live in universes that have more people, you can update again and feel a bit relieved.

However, once you know how to think straight about the subject, you need to be able to update on the rest of the evidence.

If we've never seen an existential threat and would expect to see several before getting wiped out, then we can expect to last longer. However, if we have evidence that there are some big ones coming up, and that we don't know how to handle them, it's time to do worry more than the doomsday argument tells you to.

What bugs me about the doomsday argument is this: it's a stopped clock. In other words, it always gives the same answer regardless of who applies it.

Consider a bacterial colony that starts with a single individual, is going to live for N doublings, and then will die out completely. Each generation, applying the doomsday argument, will conclude that it has a better than 50% chance of being the final generation, because, at any given time, slightly more than half of all colony bacteria that have ever existed currently exist. The doomsday argument tells the bacteria absolutely nothing about the value of N.

The reason all these problems are so tricky is that they assume there's a "you" (or a "that guy") who has a view of both possible outcomes. But since there aren't the same number of people for both outcomes, it isn't possible to match up each person on one side with one on the other to make such a "you".
Compensating for this should be easy enough, and will make the people-counting parts of the problems explicit, rather than mysterious.

I suspect this is also why the doomsday argument fails. Since it's not possible to define a set of people who "might have had" either outcome, the argument can't be constructed in the first place.

As usual, apologies if this is already known, obvious or discredited.

At case D, your probability changes from 99% to 50%, because only people who survive are ever in the situation of knowing about the situation; in other words there is a 50% chance that only red doored people know, and a 50% chance that only blue doored people know.

After that, the probability remains at 50% all the way through.

The fact that no one has mentioned this in 44 comments is a sign of an incredibly strong wishful thinking, simply "wanting" the Doomsday argument to be incorrect.

weighted according to the probability of that observer existing

Existence is relative: there is a fact of the matter (or rather: procedure to find out) about which things exist where relative to me, for example in the same room, or in the same world, but this concept breaks down when you ask about "absolute" existence. Absolute existence is inconsistent, as everything goes. Relative existence of yourself is a trivial question with a trivial answer.

(I just wanted to state it simply, even though this argument is a part of a huge standard narrative. Of course, a global probability distribution can try to represent this relativity in its conditional forms, but it's a rather contrived thing to do.)

The wikipedia on the SIA points out that it is not an assumption, but a theorem or corollary. You have simply shown this fact again. Bostrom probably first named it an assumption, but it is neither an axiom or an assumption. You can derive it from these assumptions:

1. I am a random sample
2. I may never have been born
3. The pdf for the number of humans is idependent of the pdf for my birth order number

I don't see how the SIA refutes the complete DA (Doomsday Argument).

The SIA shows that a universe with more observers in your reference class is more likely. This is the set used when "considering myself as a random observer drawn from the space of all possible observers" - it's not really all possible observers.

How small is this set? Well, if we rely on just the argument given here for SIA, it's very small indeed. Suppose the experimenter stipulates an additional rule: he flips a second coin; if it comes up heads, he creates 10^10 extrea copies of you; if tails, he does nothing. However, these extra copies are not created inside rooms at all. You know you're not one of them, because you're in one of the rooms. The outcome of the second coin flip is made known to you. But it clearly doesn't influence your bet on their doors' colors, even when it increases the number of observers in your universe 10^8 times, and even though these extra observers are complete copies of your life up to this point, who are only placed in a different situation from you in the last second.

Now, the DA can be reformulated: instead of the set of all humans ever to live, consider the set of all humans (or groups of humans) who would never confuse themselves with one another. In this set the SIA doesn't apply (we don't predict that a bigger set is more likely). The DA does apply, because humans from different eras are dissimilar and can be indexed as the DA requires. To illustrate, I expect that if I were taken at any point in my life and instantly placed at some point of Leonardo da Vinci's life, I would very quickly realize something was wrong.

Presumed conclusion: if humanity does not become extinct totally, expect other humans to be more and more similar to yourself as time passes, until you survive only in a universe inhabited by a Huge Number of Clones

It also appears that I should assign very high probability to the chance that a non-Friendly super-intelligent AI destroys the rest of humanity to tile the universe with copies of myself in tiny life-support bubbles. Or with simulators running my life up to then in a loop forever.

As we are discussing SIA, I'd like to remind about counterfactual zombie thought experiment:

Omega comes to you and offers $1, explaining that it decided to do so if and only if it predicts that you won't take the money. What do you do? It looks neutral, since expected gain in both cases is zero. But the decision to take the $1 sounds rather bizarre: if you take the $1, then you don't exist!

Agents self-consistent under reflection are counterfactual zombies, indifferent to whether they are real or not.

This shows that inference "I think therefore I exist" is, in general, invalid. You can't update on your own existence (although you can use more specific info as parameters in your strategy).

Rather, you should look at yourself as an implication: "If I exist in this situation, then my actions are as I now decide".

The doomsday assumption makes the assumptions that:

1. We are randomly selected from all the observers who will ever exist.
2. The observers increase expoentially, such that there are 2/3 of those who have ever lived at any particular generation
3. They are wiped out by a catastrophic event, rather than slowly dwindling or other

(Now those assumptions are a bit dubious - things change if for instance, we develop life extension tech or otherwise increase rate of growth, and a higher than 2/3 proportion will live in future generations (eg if the next generation is immortal, they're guaranteed to be the last, and we're much less likely depending on how long people are likely to survive after that. Alternatively growth could plateau or fluctuate around the carrying capacity of a planet if most potential observers never expand beyond this) However, assuming they hold, I think the argument is valid.

I don't think your situation alters the argument, it just changes some of the assumptions. At point D, it reverts back to the original doomsday scenario, and the odds switch back.

At D, the point you're made aware, you know that you're in the proportion of people who live. Only 50% of the people who ever existed in this scenario learn this, and 99% of them are blue-doors. Only looking at the people at this point is changing the selection criteria - you're only picking from survivors, never from those who are now dead despite the fact that they are real people we could have been. If those could be included in the selection (as they are if you give them the information and ask them before they would have died), the situation would remain as in A-C.

Making creating the losing potential people makes this more explicit. If we're randomly selecting from people who ever exist, we'll only ever pick those who get created, who will be predominantly blue-doors if we run the experiment multiple times.

99% odd of being blue-doored at F is precisely the SIA: you are saying that a universe with 99 people in it is 99 times more probable than a universe with a single person in it.

Might it make a difference that in scenario F, there is an actual process (namely, the coin toss) which could have given rise to the alternative outcome? Note the lack of any analogous mechanism for "bringing into existence" one out of all the possible worlds. One might maintain that this metaphysical disanalogy also makes an epistemic difference. (Compare cousin_it's questioning of a uniform prior across possible worlds.)

In other words, it seems that one could consistently maintain that self-indication principles only hold with respect to possibilities that were "historically possible", in the sense of being counterfactually dependent on some actual "chancy" event. Not all possible worlds are historically possible in this sense, so some further argument is required to yield the SIA in full generality.

(You may well be able to provide such an argument. I mean this comment more as an invitation than a criticism.)

Final edit: I now understand that the argument in the article is correct (and p=.99 in all scenarios). The formulation of the scenarios caused me some kind of cognitive dissonance but now I no longer see a problem with the correct reading of the argument. Please ignore my comments below. (Should I delete in such cases?)

I don't understand what precisely is wrong with the following intuitive argument, which contradicts the p=.99 result of SIA:

In scenarios E and F, I first wake up after the other people are killed (or not created) based on the coin flip. No-one ever wakes up and is killed later. So I am in a blue room if and only if the coin came up heads (and no observer was created in the red room). Therefore P(blue)=P(heads)=0.5, and P(red)=P(tails)=0.5.

Edit: I'm having problems wrapping my head around this logic... Which prevents me from understanding all the LW discussion in recent months about decision theories, since it often considers such scenarios. Could someone give me a pointer please?

Before the coin is flipped and I am placed in a room, clearly I should predict P(heads)=0.5. Afterwards, to shift to P(heads)=0.99 would require updating on the evidence that I am alive. How exactly can I do this if I can't ever update on the evidence that I am dead? (This is the scenario where no-one is ever killed.)

I feel like I need to go back and spell out formally what constitutes legal Bayesian evidence. Is this written out somewhere in a way that permits SIA (my own existence as evidence)? I'm used to considering only evidence to which there could possibly be alternative evidence that I did not in fact observe. Please excuse a rookie as these must be well understood issues.

I'm not sure about the transition from A to B; it implies that, given that you're alive, the probability of the coin having come up heads was 99%. (I'm not saying it's wrong, just that it's not immediately obvious to me.)

The rest of the steps seem fine, though.

Essentially the only consistent low-level rebuttal to the doomsday argument is to use the self indication assumption (SIA).

What about rejecting the assumption that there will be finitely many humans? In the infinite case, the argument doesn't hold.

Your justification of the SIA requires a uniform prior over possible universes. (If the coin is biased, the odds are no longer 99:1.) I don't see why the real-world SIA can assume uniformity, or what it even means. Otherwise, good post.

If continuity of consciousness immortality arguments also hold, then it simply doesn't matter whether doomsdays are close - your future will avoid those scenarios.

SIA self rebuttal.

If many different universes exist, and one of them has infinite number of all possible observers, SIA imply that I must be in it. But if infinite number of all possible observers exist, the condition that I may not be born is not working in this universe and I can't apply SIA to the Earth's fate. Doomsday argument is on.

Just taking a wild shot at this one, but I suspect that the mistake is between C and D. In C, you start with an even distribution over all the people in the experiment, and then condition on surviving. In D, your uncertainty gets allocated among the people who have survived the experiment. Once you know the rules, in C, the filter is in your future, and in D, the filter is in your past.

Actually, if we consider that you could have been an observer-moment either before or after the killing, finding yourself to be after it does increase your subjective probability that fewer observers were killed. However, this effect goes away if the amount of time before the killing was very short compared to the time afterwards, since you'd probably find yourself afterwards in either case; and the case we're really interested in, the SIA, is the limit when the time before goes to 0.

I just wanted to follow up on this remark I made. There is a suble anthropic selection effect that I didn't include in my original analysis. As we will see, the result I derived applies if the time after is long enough, as in the SIA limit.

Let the amount of time before the killing be T1, and after (until all observers die), T2. So if there were no killing, P(after) = T2/(T2+T1). It is the ratio of the total measure of observer-moments after the killing divided by the total (after + before).

If the 1 red observer is killed (heads), then P(after|heads) = 99 T2 / (99 T2 + 100 T1)

If the 99 blue observers are killed (tails), then P(after|tails) = 1 T2 / (1 T2 + 100 T1)

P(after) = P(after|heads) P(heads) + P(after|tails) P(tails)

For example, if T1 = T2, we get P(after|heads) = 0.497, P(after|tails) = 0.0099, and P(after) = 0.497 (0.5) + 0.0099 (0.5) = 0.254

So here P(tails|after) = P(after|tails) P(tails) / P(after) = 0.0099 (.5) / (0.254) = 0.0195, or about 2%. So here we can be 98% confident to be blue observers if we are after the killing. Note, it is not 99%.

Now, in the relevant-to-SIA limit T2 >> T1, we get P(after|heads) ~ 1, P(after|tails) ~1, and P(after) ~1.

In this limit P(tails|after) = P(after|tails) P(tails) / P(after) ~ P(tails) = 0.5

So the SIA is false.

The crucial step in your argumentation is from A to B. Here you are changing your a-priori probabilities. Counterintuitively, the probability of dying is not 1/2.

This paradox is known as the Monty Hall Problem: http://en.wikipedia.org/wiki/Monty_Hall_problem

The doomsday example, as phrased, simply doesn't work.

Only about 5-10% of the ever-lived population is alive now. Thus, if doomsday happened, only about that percentage would see it within our generation. Not 66%. 5-10%. Maybe 20%, if it happened in 50 years or so. The argument fails on its own merits: it assumes that because 2/3 of the ever-human population will see doomsday, we should expect with 2/3 probability to see doomsday, except that means we should also expect (with p=.67) that only 10% of the ever-human population will see doomsday. This doesn't work. Indeed, if we think it's very likely that 2/3 of the ever-lived will be alive on doomsday, we should be almost certain that we are not among that 2/3.

More generally, the 2/3 conclusion requires generational population tripling over many generations. This has not happened, and does not appear to be likely to happen. If 2/3 of the ever-lived were alive today, and there were reason to believe that the population would continue to triple generationally, then this argument would begin to make sense. As it is, it simply doesn't work, even if it sounds really cool.

Incidentally, the wikipedia summary of the doomsday argument does not sound anything like this. It says (basically) that we're probably around the halfway point of ever-lived population. Thus, there probably won't be too many more people, though such is certainly possible. It does not follow from this that 2/3 of the ever-lived will be alive for doomsday; it only says that doomsday ought to happen relatively soon, but still probably several generations off.

I don't object to the rest of the reasoning and the following argument, but the paraphrasing of the Doomsday argument is a complete straw man and should be dismissed with a mere googling of "world population growth." I'm not sure that the logic employed does anything against the actual DA.

A - A hundred people are created in a hundred rooms. Room 1 has a red door (on the outside), the outsides of all other doors are blue. You wake up in a room, fully aware of these facts; what probability should you put on being inside a room with a blue door?

Here, the probability is certainly 99%. But now consider the situation:

B - same as before, but an hour after you wake up, it is announced that a coin will be flipped, and if it comes up heads, the guy behind the red door will be killed, and if it comes up tails, everyone behind a blue door will be killed. A few minutes later, it is announced that whoever was to be killed has been killed. What are your odds of being blue-doored now?

There should be no difference from A; since your odds of dying are exactly fifty-fifty whether you are blue-doored or red-doored, your probability estimate should not change upon being updated. The further modifications are then:

SIA: Given the fact that you exist, you should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.

"Other things equal" is a huge obstacle for me. Without formalizing "other things equal", this is a piece of advice, not a theorem to be proved. I accept moving from A->F, but I don't see how you've proved SIA in general.

How do I go about obtaining a probability distribution over all possible universes conditioned on nothing?

How do I get a distribution over universes conditioned on "my" existence? And what do I mean by "me" in universes other than this one?

SIA makes perfect sense to me, but I don't see how it negates the doomsday argument at all. Can you explain further?

I don't feel like reading through 166 comments, so sorry if this has already been posted.

I did get far enough to find that brianm posted this: "The doomsday assumption makes the assumptions that:

1. We are randomly selected from all the observers who will ever exist..."

Since we're randomly selecting, let's not look at individual people. Let's look at it like taking marbles from a bag. One marble is red. 99 are blue. A guy flips a coin. If it comes up heads, he takes out the red marble. If it comes up tails, he takes out the blue marbles. You then take one of the remaining marbles out at random. Do I even need to say what the probability of getting a blue marble is?

Edit again: OK, I get it. That was kind of dumb.

I read "2/3 of humans will be in the final 2/3 of humans" combined with the term "doomsday" as meaning that there would be 2/3 of humanity around to actually witness/experience whatever ended humanity. Thus, we should expect to see whatever event does this. This obviously makes no sense. The actual meaning is simply that if you made a line of all the people who will ever live, we're probably in the latter 2/3 of it. Thus, there will likely only be so many more people. Thus, some "doomsday" type event will occur before too many more people have existed; it need not affect any particular number of those people, and it need not occur at any particular time.

The primary reason SIA is wrong is because it counts you as special only after seeing that you exist (i.e., after peeking at the data)

My detailed explanation is here.

A - A hundred people are created in a hundred rooms. Room 1 has a red door (on the outside), the outsides of all other doors are blue. You wake up in a room, fully aware of these facts; what probability should you put on being inside a room with a blue door?

Here, the probability is certainly 99%.

Sure.

B - same as before, but an hour after you wake up, it is announced that a coin will be flipped, and if it comes up heads, the guy behind the red door will be killed, and if it comes up tails, everyone behind a blue door will be killed. A few minutes later, it is announced that whoever was to be killed has been killed. What are your odds of being blue-doored now?

There should be no difference from A; since your odds of dying are exactly fifty-fifty whether you are blue-doored or red-doored, your probability estimate should not change upon being updated.

Wrong. Your epistemic situation is no longer the same after the announcement.

In a single-run (one-small-world) scenario, the coin has a 50% to come up tails or heads. (In a MWI or large universe with similar situations, it would come up both, which changes the results. The MWI predictions match yours but don't back the SIA). Here I assume the single-run case.

The prior for the coin result is 0.5 for heads, 0.5 for tails.

Before the killing, P(red|heads) = P(red|tails) = 0.01 and P(blue|heads) = P(blue|tails) = 0.99. So far we agree.

P(red|before) = 0.5 (0.01) + 0.5 (0.01) = 0.01

Afterwards, P'(red|heads) = 0, P'(red|tails) = 1, P'(blue|heads) = 1, P'(blue|tails) = 0.

P(red|after) = 0.5 (0) + 0.5 (1) = 0.5

So after the killing, you should expect either color door to be 50% likely.

This, of course, is exactly what the SIA denies. The SIA is obviously false.

So why does the result seem counterintuitive? Because in practice, and certainly when we evolved and were trained, single-shot situations didn't occur.

So let's look at the MWI case. Heads and tails both occur, but each with 50% of the original measure.

Before the killing, we again have P(heads) =P(tails) = 0.5

and P(red|heads) = P(red|tails) = 0.01 and P(blue|heads) = P(blue|tails) = 0.99.

Afterwards, P'(red|heads) = 0, P'(red|tails) = 1, P'(blue|heads) = 1, P'(blue|tails) = 0.

Huh? Didn't I say it was different? It sure is, because afterwards, we no longer have P(heads) = P(tails) = 0.5. On the contrary, most of the conscious measure (# of people) now resides behind the blue doors. We now have for the effective probabilities P(heads) = 0.99, P(tails) = 0.01.

P(red|after) = 0.99 (0) + 0.01 (1) = 0.01

P(heads) = (2*1 + 3*1/3 + 3*1/3 + 4*0)/(2+3+3+4) = (2+1+1+0)/12 = 4/12 = 1/3

p(tails|survival) = p(tails) * p(survival|tails) / p(survival) =  .5 * .01 / .5 = .01

"P(red|after)" = p(heads|survival) * "p(red|heads)" + p(tails|survival) * "p(red|tails)" 
= .99 * 0 + .01 * 1
= .01 

p(red|survival) = p(red) * p(survival|red) / p(survival) 

BTW, whoever is knocking down my karma, knock it off. I don't downvote anything I disagree with, just ones I judge to be of low quality. By chasing me off you are degrading the less wrong site as well as hiding below threshold the comments of those arguing with me who you presumably agree with. If you have something to say than say it, don't downvote.