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6y40

These are valid points, but we have wandered a bit away from the initial argument, and we're now talking about numbers that can't be compared (my money is on TREE(3) being smaller in this example, but that's irrelevant to your general point), or ways of truncating in the infinite case.

But we seem to have solved the finite-and-comparable case.

Now, back to the infinite case. First of all, there may be a correct decision even if probabilities cannot be computed.

If we have a suitable utility function, we may decide simply not to care about what happens in universes that are of the type 5, which would rule them out completely.

Or maybe the truncation can be improved slightly. For example, we could give each observer a bubble of radius 20 mega-light years, which is defined according to their own subjective experience: how many individuals do they expect to encounter within that radius, if they were made immortal and allowed to explore it fully.

Then we truncate by this subjective bubble, or something similar.

But yeah, in general, the infinite case is not solved.

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14 Solving the Doomsday argument

by Stuart_Armstrong

17th Jan 2019

1 min read

14

The Doomsday argument gives an anthropic argument for why we might expect doom to come reasonably soon. It's known that the Doomsday argument works under SSA, but not under SIA.

Ok, but since different anthropic probability theories are correct answers to different questions, what are the question versions of the Doomsday argument, and is the original claim correct?

No Doomsday on birth rank

Simplify the model into assuming there is a large universe (no Doomsday any time soon) with many, many future humans, and a small one (a Doomsday reasonably soon - within the next 200 billion people, say), with equal probability. In order to think in terms of frequencies, which comes more naturally to humans, we can imagine running the universe many, many times, each with the $50 %$ Doomsday chance.

There are roughly a 108.5 billion humans who have ever lived. So, asking:

What proportion of people with birth rank 108.5 billion, live in a small universe (with a Doomsday reasonably soon)?

The answer to that question converges to $1 / 2$ , the SIA probability. Half of the people with that birth rank live in small universes, half in large universes.

Doomsday for time travellers

To get an SSA version of the problem, we can ask:

What proportion of universes, where a randomly selected human has a birthrank of 108.5 billion, will be small (with a Doomsday reasonably soon)?

This will give an answer close to $1$ as it converges on the SSA probability.

But note that this is generally not the question that the Doomsday argument is posing. If there is a time traveller who is choosing people at random from amongst all of space and time - then if they happen to choose you, that is a bad sign for the future (and yet another reason you should go with them). Note that this is consistent with conservation of expected evidence: if the time traveller is out there but doesn't choose you, then this a (very mild) update towards no Doomsday.

But for the classical non-time-travel situation, the Doomsday argument fails.

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Solving the Doomsday argument

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[-]drnickbone

6y100

Hi Stuart. It’s a while since I’ve posted.

Here’s one way of asking the question which does lead naturally to the Doomsday answer.

Consider two universes. They’re both infinite (or if you don’t like actual infinities, are very very large, so they both have a really huge number of civilisations).

In universe 1, almost all the civilisations die off before spreading through space, so that the average population of a civilisation through time is less than a trillion.

In universe 2, a fair proportion of the civilisations survive and grow to galaxy-size or bigger, so that the average population of a civilisation through time is much more than a trillion trillion.

Now consider two more universes. Universe 3 is like Universe 1 except that the microwave background radiation 14 billion years after Big Bang is 30K rather than 3K. Universe 4 is like Universe 2 again except for the difference in microwave background radiation. Both Universe 3 and Universe 4 are so big (or infinite) that they contain civilisations that believe the background radiation has temperature 3K because every measurement they’ve ever made of it has accidentally given the same wrong answer.

Here’s the question to think about.

Is there a sensible way of doing anthropics (or indeed science in general) that would lead us to conclude we are probably in Universe 1 or 2 (rather than Universe 3 or 4) without also concluding that we are probably in Universe 1 (rather than Universe 2)?

[-]Stuart_Armstrong

6y40

"How many copies of people like me are there in each universe?"

Then as long as your copies know that 3K has been observed, and excluding simulations and such, the answers are "(a lot, a lot, not many, not many)" in the four universes (I'm interpreting "die off before spreading through space" as "die off just before spreading through space").

This is the SIA answer, since I asked the SIA question.

[-]drnickbone

6y20

Thanks Stuart.

The difficulty is that, by construction, there are infinitely many copies of me in each universe (if the universes are all infinite) or there are a colossally huge number of copies of me in each universe, so big that it saturates my utility bounds (assuming that my utilities are finite and bounded, because if they’re not, the decision theory leads to chaotic results anyway).

So SIA is not an approach to anthropics (or science in general) which allows us to conclude we are probably in universe 1 or 2 (rather than 3 or 4). All SIA really says is “You are in some sort of really big or infinite universe, but beyond that I can’t help you work out which”. That’s not helpful for decision making, and doesn’t allow for science in general to work.

Incidentally, when you say there are “not many” copies of me in universes 3 and 4, then you presumably mean “not a high proportion, compared to the vast total of observers”. That’s implicitly the SSA reasoning being used for to discriminate against universes 3 and 4... but then of course it also discriminates against universe 2.

I’ve worked through pretty much all the anthropic approaches over the years, and they all seem to stumble on this question. All the approaches which confidently separate universes 3 and 4 also separate 1 from 2.

[-]Stuart_Armstrong

6y40

If we set aside infinity, which I don't know how to deal with, then the SIA answer does not depend on utility bounds - unlike my anthropic decision theory post.

Q1: "How many copies of people (currently) like me are there in each universe?" is well-defined in all finite settings, even huge ones.

Incidentally, when you say there are “not many” copies of me in universes 3 and 4, then you presumably mean “not a high proportion, compared to the vast total of observers”

No, I mean not many, as compared with how many there are in universes 1 and 2. Other observers are not relevant to Q1.

I'll reiterate my claim that different anthropic probability theories are "correct answers to different questions": https://www.lesswrong.com/posts/nxRjC93AmsFkfDYQj/anthropic-probabilities-answering-different-questions

[-]drnickbone

6y20

I get that this is a consistent way of asking and answering questions, but I’m not sure this is actually helpful with doing science.

If, say, universes 1 and 2 contain TREE(3) copies of me while universes 3 and 4 contain BusyBeaver(1000) then I still don’t know which I’m more likely to be in, unless I can somehow work out which of these vast numbers is vaster. Regular scientific inference is just going to completely ignore questions as odd as this, because it simply has to. It’s going to tell me that if measurements of background radiation keep coming out at 3K, then that’s what I should assume the temperature actually is. And I don’t need to know anything about the universe’s size to conclude that.

Returning to SIA, to conclude there are more copies of me in universe 1 and 2 (versus 3 or 4), SIA will have to know their relative sizes. The larger, the better, but not infinite please. And this is a major problem, because then SIA’s conclusion it dominated by how finite truncation is applied to avoid the infinite case.

Suppose we truncate all universes at the same large physical volume (or 4d volume) then there are strictly more copies of me in universe 1 and 2 than 3 and 4 (but about the same number in universes 1 and 2). That works so far - it is in line with what we probably wanted. But unfortunately this volume based truncation also favours universe 5-1:

5-1. Physics is nothing like it appears. Rather the universe is full of an extremely dense solid, performing a colossal number of really fast computations; a high fraction of which simulate observers in universe 1.

It’s not difficult to see that 5-1 is more favoured than 5-2, 5-3 or 5-4 (since the density of observers like me is highest in 5-1).

If we instead truncate universes at the same large total number of observers (or the same large total utility), then universe 1 now has more copies of me (because it has more civilisations in total). Universe 1 is favoured.

Or if I truncate universes at the same large number of total copies of me (because perhaps I don’t care very much about people who aren’t copies of me) then I can no longer distinguish between universes 1 to 4, or indeed 5-1 to 5-4.

So either way we’re back to the same depressing conclusion. However the truncation is done, universe 1 is going to end up preferred over the others (or perhaps universe 5-1 is preferred over the others), or there is no preference among any of the universes.

[-]Stuart_Armstrong

6y40

But we seem to have solved the finite-and-comparable case.

Now, back to the infinite case. First of all, there may be a correct decision even if probabilities cannot be computed.

If we have a suitable utility function, we may decide simply not to care about what happens in universes that are of the type 5, which would rule them out completely.

Then we truncate by this subjective bubble, or something similar.

But yeah, in general, the infinite case is not solved.

[-]drnickbone

6y40

Thanks again for the useful response.

My initial argument was really a question “Is there any approach to anthropic reasoning that allows us to do basic scientific inference, but does not lead to Doomsday conclusions?” So far I’m skeptical.

The best response you’ve got is I think twofold.

Use SIA but please ignore the infinite case (even though the internal logic of SIA forces the infinite case) because we don’t know how to handle it. When applying SIA to large finite cases, truncate universes by a large volume cutoff (4d volume) rather than by a large population cutoff or large utility cutoff. Oh and ignore simulations because if you take those into account it leads to odd conclusions as well.

That might perhaps work, but it does look horribly convoluted. To me it does seem like determining the conclusion in advance (you want SIA to favour universes 1 and 2 over 3 and 4, but not favour 1 over 2) and then hacking around with SIA until it gives that result.

Incidentally, I think you’re still not out of the woods with a volume cutoff. If it is very large in the time dimension then SIA is start going to favour universes which have Boltzmann Brains in the very far future over universes whose physics don’t ever allow Boltzmann Brains. And then SIA is going to suggest that not only are we probably in a universe with lots of BBs, we most likely are BBs ourselves (because almost all observers with exactly our experiences are BBs). So SIA calls for further surgery either to remove BBs from consideration or to apply the 4volume cutoff in a way that doesn’t lead to lots of Boltzmann Brains.

Forget about both SIA and SSA and revert to an underlying decision theory: viz your ADT. Let the utility function take the strain.

The problem with this is that ADT with unbounded utility functions doesn’t lead to stable conclusions. So you have to bound or truncate the utility function.

But then ADT is going to pay the most attention to universes whose utility is close to the cutoff ... namely versions of universe 1,2,3,4 which have utility at or near the maximum. For the reasons I’ve already discussed above, that’s not in general going to give the same results as applying a volume cutoff. If the utility scales with the total number of observers (or observers like me), then ADT is not going to say “Make decisions as if you were in universe 1 or 2 ... but with no preference between these ... rather than as if you were in universe 3 or 4”

I think the most workable utility function you’ve come up with is the one based on subjective bubbles of order galactic volume or thereabouts i.e. the utility function scales roughly linearly with the number of observers in the volume surrounding you, but doesn’t care about what happens outside that region (or in any simulations, if they are of different regions). Using that is roughly equivalent to applying a volume truncation using regular astronomical volumes (rather than much larger volumes).

However the hack to avoid simulations looks a bit unnatural to me (why wouldn’t I care about simulations which happen to be in the same local volume?) Also, I think this utility function might then tend to favour “zoo” hypotheses or “planetarium” hypotheses (I.e. decisions are made as if in a universe densely packed with planetaria containing human level civilisations, rather than simulations of said simulations).

More worryingly, I doubt if anyone really has a utility function that looks like this ie. one that cares about observers 1 million light years away just as much as it cares about observers here on Earth, but then stops caring if they happen to be 1 trillion light years away...

So again I think this looks rather like assuming the right answer, and then hacking around with ADT until it gives the answer you were looking for.

[-]Shmi

6y100

The Doomsday argument is utter BS because one cannot reliably evaluate probabilities without fixing a probability distribution first. Without knowing more than just the number of humans existing so far, the argument devolves into arguing which probability distribution to pick out of uncountable number of possibilities. An honest attempt to address this question would start with modeling human population fluctuations including various extinction events. In such a model there are multiple free parameters, such as rate of growth, distribution of odds of various extinction-level events, distribution of odds of surviving each type of events, event clustering and so on. The the minimum number of humans does not constrain the models in any interesting way, i.e. to privilege a certain class of models over others, or a certain set of free parameters over others to the degree where we could evaluate a model-independent upper bound for the total number of humans with any degree of confidence.

If you want to productively talk about Doomsday, you have to get your hands dirty and deal with specific x-risks and their effects, not armchair-theorize based on a single number and a few so-called selection/indication principles that have nothing to do with the actual human population dynamics.

[-]Stuart_Armstrong

6y20

The DA, in it's SSA form (where it is rigorous) comes as a posterior adjustment to all probabilities computed in the way above - it's not an argument that doom is likely, just that doom is more likely than objective odds would imply, in a precise way that depends on future (and past) population size.

However my post shows that the SSA form does not apply to the question that people generally ask, so the DA is wrong.

[-]Lookingforyourlogic

6y10

Interesting post. Could the same argument not be used against the Simulation argument?

Simplify the model into assuming there is a universe in which I, the observer, are one of many many observers in an ancestor simulation run by some future civilization, and a universe in which I am a biological human naturally created by evolution on earth, with equal probability. Again, we can imagine running the universe many, many times. But no matter how many people are in the considered universe, I can only have the experience of being one at a time. So, asking:

What proportion of people whose experiences I have live in a simulated world?

The answer to that question converges to 1/2, as well. But if every observer reasoned like this when asked whether they are in a simulation, most would get the wrong answer (assuming there are more simulated than real observers)! How can we deal with this apparent inconsistency? Of course, as you say, different answers to different questions. But which one should we consider to be valid, when both seem intuitively to make sense?

[-]Dr. Jamchie

6y10

Lets say you do not know your birth rank at first. Then someone asks you to guess whether the universe is around 200 billion or some very large number. Without any additional data you should estimate 50% for either one. Then you get to know that your birth rank is around 100 billion. Do you not then update, that smaller universe have bigger than 50% chance estimated previously?

[-]Stuart_Armstrong

6y30

Again, we have to be clear about the question. But if it's "what proportions of versions of me are likely to be in a large universe", then the answer is close to 1 (which is the SIA odds). Then you update on your birthrank, notice, to your great surprise, that it is sufficiently low to exist in both large and small universes, so update towards small and end up at 50:50.

[-]Dr. Jamchie

6y10

So what you are saying, is, before one knows his birth rank, one should assume infinite universe? This does actually corresponds to evidence about universe size, but not about human population size.

[-]Stuart_Armstrong

6y20

Again, it's what question you're asking. "If a copy of me existed, would it be more likely to exist in small universe or in an infinite one" has a pretty clear answer :-)

[-]avturchin

6y10

It is probably wrong to interpret DA as "doom is imminent". DA just say that we are likely in the middle of total population of all humans (or other relevant observers) ever born.

For some emotional reasons we are not satisfied to be in the middle and interprets it as "doom" - but there are 100 billion more people in the future according to DA. It becomes look like a doom, if we account for expected population growth, as in that case, the next 100 billion people will appear in a few hundreds years.

More over, DA tells that doom very soon is very unlikely, which I call "reverse DA".

[-]Stuart_Armstrong

6y30

There are two versions of the DA; the first is "we should roughly be in the middle", and the second is "our birth rank is less likely if there were many more humans in the future".

I was more thinking of the second case, but I've changed the post slightly to make it more compatible with the first.

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