TL;DR: SIA works only in a finite universe, as an argument for a larger universe. In the infinite universe, we can compare concentrations of observers in different regions, but it is SSA. 

 

SIA: All other things equal, an observer should reason as if they are randomly selected from the set of all possible observers. Note that "randomly selected" is weighted by the probability of the observers existing: under SIA you are still unlikely to be an unlikely observer, unless there are a lot of them. 

SSA: All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class. 

 

Main thesis: SIA becomes SSA in the infinite universe,  and the reference class in that case is the class of my exact copies.

Proof: In the infinite multiverse, there are no “possible observers”, as all possible observers actually exist somewhere, but maybe in small concentrations. (UPDATE: I discuss how infinite size of the universe generates all possible observers in comments.) Therefore, the idea of selecting from possible observers is meaningless. The selection is happening from actually existing observers. Thus, SIA turns into SSA in the multiverse.

The main difference between SIA and SSA is what they are looking at: SIA looks only at the fact that I exist at all, and SSA looks at some observables like a colour of the room in which I am located. In the infinite universe, the fact of my existence becomes uninformative: everybody exists. 

Thought experiments like “God toss a coin and created one observer” are meaningless, as there is no God and there are all possible outcomes of tossing a coin. In the infinite universe, there are many of my exact copies, so we can discuss how they are distributed. 

 

Now examples.

According to SIA, if there are two alternatives: (1) there are trillion observers in the universe, and (2) there are trillions of trillions of observers in the universe, – then the second hypothesis is overwhelmingly more probable. This is known as a Presumptuous Philosopher (PP) thought experiment and it is often used as a counterargument to SIA, as such predictive capability seems counterintuitive. Why does it look counterintuitive? Because it looks like a free lunch into the nature of the universe. (It is trickier with Sleeping Beauty, but let’s assume that she lives in MWI, and there are two branches, for head and for tails. In that case, probability calculations becomes straightforward.) 

But the real problem of the Presumptuous Philosopher experiment is that it assumes that the universe is finite, even though that there are serious arguments that it is either infinite or a very-very large (eternal inflation). The main feature of very-very large universe is that it includes all possible observers at least once. Tegmark showed that my copy is at the distance like 10^(10^29)m.  

In the very-very large universe the fact of my existence becomes non-informative, as we already know that all possible observers exist in it. Because of this, there is an alternative formulation of SIA: I am more likely to be in the part of the multiverse where most of my copies are located. There are many examples of the attempts to implement it: I am more likely to be in a simulation, in the world with MWI, in the real world vs being a Boltzmann Brain, in the world with panspermia etc. But it is not the real SIA: it is just SSA, applied to the whole multiverse where the reference class is my copies. 

This infinite-world SIA gives us almost the same prediction power as the finite-world-SIA, but there are a few exceptions. Finite-world-SIA favours the hypothesis of easy abiogenesis, so life should be everywhere. Infinite-world-SIA claims that I am the universe with the easiest abiogenesis from all other universes (with some caveats) but abiogenesis still could be an extremely small probability event like 1 for 10100 planets. 

Finite-SIA favours crazy hypotheses, but infinite-SIA just put me into the place where most observers-like-me live. In other words, Finite-SIA is Bayesian as it gives priors for hypothesis, and infinite-SIA is frequentist, as it is used to calculate probabilities based on the known world model. 

 

Reference class problem.

But what about the “reference class” which is mentioned in the SSA definition, but not in the SIA definition? 

Actually, there are even two reference classes in the SIA definition. The first one is the class of “all possible observers”. The second one is used to weight probabilities (as said in the definition: “Note that "randomly selected" is weighted by the probability of the observers existing”) and consists of my exact copies located in different situations. If it is zero, I do not exist. If there are many, I am located where most of my copies are located. 

This probably explains why it is sometimes called “SIA+SSA”: SIA tells why I am in the tails branch in Sleeping beauty, and SSA tells where exactly I am in that branch. 

Note also that “observers” in SIA still have to be “qualified observers”, who, in my opinion, are only those who already thinking about anthropics. 

In other words, SIA is a SSA with a very specific choice of the reference class. 

 

Attempt to save SIA. 

The fact of being born is described in SIA as if some “empty observer” falls into the position of a particular observer, which could be in two states: existing and non-existing. Only an existing observer could think. Therefore, as I think, I am real. 

SIA tries to use the opposite logic: if there is a concrete possible observer with the random name ZAQWSX, he has more chances to exist if there were more attempts to make him exist. Therefore, from the fact he exists, he could conclude that there were many attempts. 

The problem with this reasoning is that it doesn’t use his name ZAQWSX and other his unique information, so any other observer will reason the same way. 

Even if there was just one random observer in the universe, he can’t extract any information from his random name. Or is he? 

Imagine that there are 99 red rooms and 1 green room. I am a mind which was randomly put into rooms, I am interested not in the probability of being in some of the rooms, but in the question, of how many minds were put in such rooms at all. If I am red room, I got no new information. However, if I am in the green room, I can suggest that there were many other attempts to put minds in all rooms, so at least one got into the green room. 

Therefore, SIA could be saved only if I find myself in an un-typical location. Most observers can’t use this type of logic as they are typical. The presumptuous philosopher therefore should look not at the fact of his existence, but on some random coincidences in the observed physical laws. 

But here there is one case where PP works: from the fact that evolution succeeds on Earth despite many risks and improbable evolutionary transitions, follows that there were billions and billions of other planets where evolution fails. 

So even if we could not see stars, we will be able to guess their existence by this modified SIA.  

 

Unexpected conclusion or a second attempt to save SIA. 

We could try to save SIA by saying that the fact that “I exist at all” is an argument for an infinite universe

(I assume that there is a way to calculate observers’ measure even in an infinite universe and an interesting idea was suggested in the article “Watchers of the Multiverse” , in which observers’ measure is calculated via eternal timelines; another idea is UDASSA.) 

So, SIA proves that universe is infinite and stops here. For everything else, we use SSA with reference class of exact copies.    

 

My previous posts on the topic: 

Each reference class has its own end 

Anthropic effects imply that we are more likely to live in the universe with interstellar panspermia

Meta-Doomsday Argument: Uncertainty About the Validity of the Probabilistic Prediction of the End of the World

Could declining interest to the Doomsday Argument explain the Doomsday Argument?

Reverse Doomsday Argument is hitting preppers hard

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Instead of examining how SIA behaves given multiverse theories are true, a better approach is to examine what SIA says about the validity of multiverse theories.

And the result is simple, SIA heavily favours multiverse theories, as they greatly inflate the total number of observers in existence. It does not matter what kind of multiverse theories they are. It could be a very-very large universe (thus many casually independent regions),  it could be a plethora of universes with different physical parameters, it could be the many-worlds interpretations of quantum mechanics, it could also be the simulation argument where a super majority of observers are computer-generated. 

In my experience, most people are unwilling to bite this bullet and just say those theories are true simply because I exist. Two common ways I have seen people attempting to save SIA. 1: play with the reference class by arguing the reference class should not include observers from other universes. e.g. "I could not have been an observer from another universe." or "I reject the assumption that I could be a computer programme completetly". 2: play with infinity. e.g. "It is difficult to apply probabilistic judgments when infinity is part of the problem, and many multiverse theories imply infinity." But neither is very convincing. 

SIA favours many universes, but only of the sort that could produce you (not ones with different physical parameters such that they can only produce intelligent octopuses, not bipeds, since you are not an octopus).  

I don't see why multiverse theories would necessarily involve infinities.  There could be a finite number of universes, each of finite size.

True, SIA favours theory more likely to produce "me". However in the context of SIA, it does not define "me" by the specific physical parameters. SIA does not concern about the physical parameter of the observer, but the subjective state. Take the simulation argument as an example. My existence favours there are many computer-run simulations of human civilization. The physical parameters of me is unknown. I could very well be a programme living in a simulation instead of a real human.  Or use your example of intelligent octopuses. If a theory says there are many intelligent octopuses who each think they are a human being (maybe by octopuses-in-a-vet kind of experiment) , then SIA would still favour such a theory. And I could very well be such an octopus instead of a human physically. 

In my past experience, SIA supporters not liking this often resolves to limit the reference class. Something like "I reject the possibility that I could be a programme" outright. 

I agree with you completely on the infinity argument. I don't think it is a valid defense of SIA. Yet I have seen it used from time to time by its supporters. 

Yes, it's "more likely to produce me" in terms of subjective experience that counts, but if one ignores simulation-style scenarios, octopuses and bipeds will of course have distinct subjective experiences of seeing their own arms.

In simulation scenarios, the simulated world needn't have the same physical laws as the actual one (limited only by the imagination of the programmers), so people who think they're intelligent bipeds could exist in an actual universe where only intelligent octopuses can evolve.  But there are so many unresolved issues in such scenarios that I'm at a loss of how to think about them.  (For example, once the programmer has written the simulation program, for a deterministic computer, is it necessary to actually run the program in order for the simulated people to exist?)

I don't get the problem with octopuses here. What I am writing here is not causally connected with the number of my hands. Octopus-world-LW can have similar conversations. But as I find myself a biped, it a an argument that biped-world-LW are more often.

Sure, octopuses could write too.  But you are not, in fact, an octopus (assuming reality is what it seems).  So the evidence you have for evaluating cosmological theories does not favour universes/multiverses with large numbers of intelligent octopuses, since you have no evidence that intelligent octopuses exist.  But you do know that you exist.

If you like, you can bump up the probability of cosmological theories that posit a universe with a large number of intelligent observers, whether octopuses or bipeds (SIA), but then you have to push down the probability of those theories in which most of these observers are octopuses, since you aren't one (SSA).  The net effect is to just favour cosmological theories that make it more likely that you exist.

See my paper at http://www.cs.utoronto.ca/~radford/anth.abstract.html for a more extended exposition.

Thanks for the link. I saw your article before, but this explanation helps me to understand FNC better. 

Yes, I agree there are unresolved issues. There simply is no widely accepted way of reasoning about subjective experience. Given this, it seems more unreasonable to assert that simulation of subjective experience is absolutely impossible (prior probability=0). Yet even if we give a small prior to it, due to the overwhelmingly great number of human-like experience simulation theory suggests, SIA would push its probability to near unity. So many SIA supporters would make the above-mentioned assertion. 

Yes, agree and actually I said in the end of the post that "So, SIA proves that universe is infinite and stops here". 

The problem here is that after we use SIA to show that the world is infinite, we can't use it for anything else. For example, for any world with heads in Sleeping Beauty, there is a world where all the same except that there coin is tails. What do you think about this?

Yes I saw it and I agree. I was suggesting SIA=SSA given multiverse is not surprising because their difference lies in assessing whether or not multiverse is true. If we look past that part then they behave the same. 

For the sleeping beauty problem, accepting a multiverse theory does not change the conclusion for SIA supporters. It is still 1/3. Since there are 2 awakenings in the tails world and 1 in heads world, which all exists. Now applying an SSA type of updating would give 1/3. 

Of course one can argue if the world is infinite then there are infinite awakenings for both heads and tails. How to do a Bayesian update in this case becomes unclear. However, I just don't think that is a very powerful counter to SIA. Infinity is always a difficult concept, anthropics paradoxes already exist without it. 

My opinion is that while SSA and SIA are both wrong, SIA does so consistently. It generates less paradox than SSA. The "free lunch" of confirming theories with more observers simply because "I exist" is the exception. 

I agree with all you said except conclusion :)

For me, both are right. SIA works by showing that all possible observers exists.

SSA works now by looking on the distribution of observables. 

Understandable. Most if not all thinkers treat anthropic reasoning as some sort of observation selection effect. It has been the case since Nick Bostrom laid the foundation of the subject. I might be the only one thinking treating first-person concepts (like "myslef" "I" "now" and "here")as random sample is where all the mistake is. 

There is a theory called "full non-indexical conditioning". 

Do you know about it? Is it close to your view? I am not yet very good in it, but I saw some papers on arXiv.

Yes, I know about it. In fact, it is originally purposed by Prof. Radford Neal from university of Toronto. And he is replying right below you. :)

No, it is very different from my view. In my opinion, FNC is similar to SIA and SSA. It still assumes a particular sampling procedure and considers indexical (like "I" or "now") as the outcome of this sampling process. Though FNC is more subtle as it does not explicitly state anything in the like of "consider oneself as randomly selected from such and such". Nonetheless, it performs probability calculations as if someone with all the specific experience of "me" (first person) is being sampled and successfully found among existing observers. 

My opinion is indexicals cannot be regarded as a sampling outcome and they cannot be removed from anthropic problems. They should be treated as concepts inherently understood. e.g. I don't need anything to differentiate myself from all people. I just inherently know who "I" is. Anthropic problems are set up using indexicals as such. They should be solved as such. 

The octopus example helped me to grok the FNC, but I still don't have a clear example which will help me to better understand your point of view.

There is something lost in the discussion of the octopus example between me and Prof Neal. What I meant is that if a theory suggests there are many intelligent octopuses whose subjective experience is human-like. (maybe like matrix-style octopus-in-a-vets experiment). i.e. the octopus thinks they have are biped humans, even though they are physically not. Then no matter how crazy that sounds, as long as the theory greatly inflates the total number of observers with human-like experience, SIA will endorse it with a high degree of confidence. FNC does so too. 

For my position, see this post for a start. 

Thanks for the link.

Proof: In the infinite multiverse, there are no “possible observers”, as all possible observers actually exist somewhere, but maybe in small concentrations. Therefore, the idea of selecting from possible observers is meaningless. The selection is happening from actually existing observers. Thus, SIA turns into SSA in the multiverse.

I think this is incorrect. An infinite multiverse does not imply all possible observers exists. For example consider the set of all possible universes, and pick any infinite subset of that. That is a valid infinite multiverse which does not contain all possible observers.

I answered on the similar question above.  Mathematically what you said is true. If we speak about physical system, it is chaotic, in a sense that it has non-zero chance of generating any possible outcome. 

It depends what you mean by possible. SIA is about all possible multiverses I might be in. If you assign a probability to whether or not the multiverse allows X to exist, SIA will say you should consider yourself randomly drawn from all possible observer instances across both possible multiverses.

Even indeterministic physics has forbidden states. Chaos is defined as critical dependence on initial conditions, not as the ability to reach any state. Passing through all possible at states means passing through all states that are not forbidden by the physics in question, not all states that are armchair conceivable by you.

Here I don't discuss passing through all possible physical states, but through all possible observer states.

Anyway, I should have discussed the thesis "all possible observers actually exist in multiverse" as a lemma for the central argument about SIA. Maybe will make a post about it later. 

You still have the issue that "possible" has multiple meanings, and that physical possibility is more restrictive than conceivability.

I think that the issue could be tracked back to the SIA definition. It says “possible observers”, and I think that physically possible are meant.

For example, it is conceivable that each planet is inhabited by billions trolls, but there is no physical basis for this. (though folk mythology claims exactly this - any stone has its spirit)

You still have the issue that "possible" has multiple meanings, and that physical possibility is more restrictive than conceivability.

Different possible multiverses can have different densities of particular observers.

The SIA predicts that you're in those multiverses which have a higher density of observers exactly like you.

Sorry, long comment. 

TL;DR: SIA favours infinities of highest order, but spends all its power on it and thus not very useful in distinguishing multiverses with different densities of observers.

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Imagine that there are two hypothetical variants of multiverse, both have countable infinite number of all possible observers, but in one variant there is a 1000 times higher concentration of observers-like-me than another. (“Hypothetical” here means not “physically possible”, but two different hypotheses to which we want to give priors; if they were physically possible, they both will co-exist, and I will be in region with higher density, assuming that the size of regions is the same. But it is SSA.)

The fact of my existence does not provide any new information which would help to choose between to hypothesis, as I exist in both variants. 

But bona fide SIA seems to insists on the variant which with the higher density of observers anyway. However, two countable infinites are equal, so the fact that the second multiverse has a higher concentration of observers is also not an argument which favours the denser multiverse even under SIA. (I am not sure here, may return to this point later; anyway the next paragraph will overwrite this  uncertainty.)

But if there is a third hypothetical multiverse which has a higher order of infinity of observers, say, uncountable number of them, it will win according to SIA. Therefore, SIA favours multiverse with the highest order of infinity of observers. 

In practice, it means that SIA favors “many bubbles multiverse” over enteral inflation multiverse, as bubble are uncountable. It also favours mathematical multiverse over bubbles one, as mathematical multiverse includes all types of infinites. In short, higher orders of infinity by Tegmark are more likely according to SIA. 

But SIA is one-time gun. Even if it works, it distinguishes different orders of infinity in Tegmark’s model, but after that it becomes completely useless practically: it doesn’t help us with any smaller task, like deciding what are “a priory” chances of abiogenesis or what is the density of observers. 

What is a "possible" observer? Why would an "infinite" universe contain all of them?

By analogy, there are infinitely many real numbers in the interval [0,1]. But not all real numbers are in that interval.

If the universe is infinite and chaotic, it will pass through all possible states, which includes all possible combinations of atoms and all possible observers. It follows from Poincaré recurrence theorem

Even very improbable observers can appear as Boltzmann brains. Basically, it follows from the law of large numbers: if there are infinitely many tries, then any finite number will appear. Therefore, we need to add a constrain that observer's mind is finite.

I'm not sure that follows. It could depend on the structure of the universe. While many or most states are reached there could be unreached areas due to being outside of attractors. Even a random walk in three dimensions doesn't reach all points.

This objection may works for some strange astrophysical object, like cubic-size stars. If we limit our idea of observers to the finite (and not too large) size Turing machines, it seems unlikely. Any such observer could appear via random generating of a file.

I think by adding in the constraint of turning machines, you increase the complexity and thus reduce the likelihood, not increase it. Now you require Turing machines in the first place.

Turing machines are rather universal thing which will appear many times everywhere, so I don't see how it reduces the likelihood. 

I don't see a problem here, I will win in another branch of MWI. Or I miss something?