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FQXi BLOGS
CATEGORY: Blog
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TOPIC: Should we worry if we are weird?
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I've just finished reading an interesting paper by Hartle and Srednicki critiquing the assumption that 'we are typical', used in various cosmological model-testing arguments.
Here is the basic issue. There is an open problem in cosmology as to how to test a theory that entails a 'multiverse', which is to say an ensemble of regions, each member of which appears as a 'universe' to its inhabitants (i.e. appears larger than their horizon), but across which the observable 'cosmological' observables may vary. As we can measure only one set of such values, how do we test the model that gave rise to the ensemble? There are a number of rather different ways people have proposed to look at this.
My basic take on this is outlined here and here . In brief, I like to think of asking the question: "Suppose I were a randomly chosen X. What would I observe?" Here, 'X' might stand for 'universe' or 'point in space' or 'observer', etc. For each such 'X', we might try to calculate the answer in the theory. Then we would have to make a basic "philosophical" assumption that the probability distribution for a randomly chosen 'X' is actually closely related to the probability that we will in fact see some particular thing when we go out and look. After making this assumption, if we measure something that would be very, very unusual for a typical X, we then conclude that either (a) our observations are not closely related to those of a typical X, or (b) the theory is incorrect.
I've found that most methods people advocate fall somewhere on what might be seen as a 'spectrum' of conditionalization. On one end of the spectrum -- least conditioning -- we might 'count vacua' (in the string-theory landscape). On the other end we might 'condition' on all possible facts at our disposal, and try to predict something we have not yet seen. So a lot of the disagreement, as I see it, centers around what 'X' to pick, and in my book there are arguments in favor (and against) just about every choice.
For example, assuming that we are 'typical humans' (out of all that ever have and will exist) leads to the 'doomsday paradox' , and even stranger variations on it. Or, we might choose X='Observers just like me who know everything that I know' (which I've seen referred to as 'top-down' reasoning, or 'Full non-indexical conditioning'). This sounds compelling: it's what we do in lots of experimental physics -- that is, we do not worry about *how* a particular experimental setup came to be, but rather what will happen *given* everything about that setup. But in a cosmological context, I contend that using this sort of reasoning, we can never rule a theory out, which is bad. I present my argument simply in the form of a dialogue, then in more detail in the attachments below.
But back to Hartle and Srednicki, they are not quite advocating any particular 'X', but rather trying to see exactly what Bayesian reasoning tells us to do. In particular, given two theories A and B with 'prior' probabilities P_A and P_B, we should take all of the data D at our disposal, compute the 'probability of D given A' P(D|A), and the 'probability of D given B', P(D_B), then use Bayes theorem to find that the (relative) probabilities of theories A and B will be P(D|A)P_A/P(D_B)P_B.
Reasoning this way has some nice consequences. For example (and this seems to be a strong motivation behind their paper) if we don't assume that we are a 'typical' anything, then I think we do indeed remove the 'Boltzmann's brain' problem , along with the closely related 'doomsday' problem. A 'side effect' is that we should not in any way prefer a theory in which we are 'typical': if the data D occurs in two theories, we should not accord any preference to one that creates many more instances of D.
To me, this sort of reasoning does have some compelling qualities, but it also seems to me that it would be extremely weak in discriminating between cosmological theories. For example, suppose A is 'the big bang theory', and B is 'a 500 kg ball of gas in a box that exists forever.' It seems as though theory A does, indeed, give rise to our observations D. But the box, if it lasts long enough, would *also* give rise to them, in the form of, yes, a 'Boltzmann brain' that fluctuates thermally into existence in just such a way as to think it has measured all of the data D. On the basis of Bayesian inference, if we were to accord equal prior probability to these, we could not any further discriminate between them, since P(D|A) ~ P(D|B) ~ 1. This seems rather displeasing, as the data D arises very naturally in A, but in an exceedingly strange way in theory B. (I wonder if Hartle & Srednicki may have been grappling with this, as late in the paper they mention 'counting' and similar ideas as things that we might want to take into account in our prior probabilities. But then this is just burying the crucial point in a different place.)
So in my mind, the question of how we can reason in 'multiverse' cosmology in a way that (a) actually allows us to effectively discriminate between models, but (b) does not lead to any weird paradoxes, is still very much open.
| | attachments:
td_fnac_worries.txt,
simp_topdown.jpg |
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Interestingly, I have been reading the Hartle-Srednicki paper for a number of weeks, there appears some logical method that I have not quite grasped, as I am a big fan of Hartle, I keep it handy.
The bayesian reasoning is something I am only recently familiar with, but for example I was always taught that the:
http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
did not reveal any prime number pattern, so I used a bit of logic to "move the goalposts", forgive the basic condition of this link as it has been somethign I want to eventually tidy up:
http://homepage.ntlworld.com/paul.valletta/PRIME%20GRIDS.htm
but the basic picture is that given certain facts, one can bend the rules?
The fact is that if one alters the "fixed" Eratosthenes Sieve (from fixed columns and rows), to one that is dynamic, then the random prime numbers contained within a fixed grid as per Eratosthenes, fall into sections that reveal fractal/prime properties co-incedence?
Model predictions have many variables, strintheory for instance has a lot of problems because the Universe has not yet devulged it's hidden dimensions, this I believe is because the Universe is still evolving, and dimensionally it happens to be in a 3+1 phase at this moment in time.
The Universe of the future will definately have more dimensions than it has currently, thus stringtheory is a
" not yet correct " theory!..but I do not believe there will be any evidence for extra-dimensions, other than the maths, the maths are certainly correct, it just has to be time-stamped into the evolving Universe model.
Even in the far off future, Einstein's Theory of Relativity
will be correct, it will just have to be classified as a dimensional phase dependant theory, relevant to a specific time-slot within the Universe.
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About the attached text file: "Worries about 'Top down' or 'Full non-indexical conditioning'"
Why not take into account prior probabilities of theories A and B? Then everything becomes well defined...
Of course, the scientists in that example may not know what values to assign to the a priori probabilities, but they must have some rough idea. How else did they come up with theories A and B in the first place if they didn't think they had some reasonable chance of being correct?
About the 500 kg ball of gas in a box :
"This seems rather displeasing, as the data D arises very naturally in A, but in an exceedingly strange way in theory B"
Some time ago I was thinking about this problem in terms of algorithmic complexity and artificial intelligence, which allows one to formalize the notion of "exceedingly strange". Suppose we simulate both models in a computer (assumed to be powerful enough to simulate every relevant detail). We want to talk to Anthony who, we know, "lives" in both these worlds.
We can try to locate Anthony by using a search algorithm. Compared to the "gas in a box" universe, the search algorithm for the big bang universe isn't all that complicated. Also the run time to locate Anthony is much less. And after locating him, it's much easier to talk to him.
In fact, if you analyze a conversation between Anthony in the gas in the box universe and us, then you see that you mus constantly seach for Anthony using the search algorithm, because he disintegrates all the time. The search algorithm must be capable of locating Anthony at each time over and over again. i.e. it must be able to predict the state Anthony will be in given his state at a slightly earlier time.
So, the search algorithm does the bulk of the computation that generates Anthony's consciousness. Can we then really say that Anthony exists at all in that universe? It's a bit like how Strong AI proponents refute Searl's Chinese Room Argument...
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Count Iblis:
I'm not sure what you mean. Sure, we could give these theories different prior probabilities, but on what basis? That is, I think the relevant question is how we are supposed to use the data to distinguish theories, when it is not clear a-priori which is correct (i.e. fairly simialr prior probabilities). I think perhaps I am missing what you are getting at.
Your second point is interesting. Indeed part of what is so troubling about the 'gasball' theory is that along with producing the proper Anthony, you produce every other possible macrostate that you can compose (via course-graining) out of you ergodically-sampled microstates. So all of Anthony's particular qualities (for better or worse) are given no more 'credit' than any random blob of gas -- in fact Anthony is stupendously more rare. Your take, using computability, is an interesting perspective on this -- I wonder if some sort of 'search algorithm measure' could make sense?
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I am wondering if there is an example where typicality can be used as a quantitative tool, in context better explored than multiverse scenarios. For example for observables that are clearly anthropically determined such as the distance of the earth to the sun, is there a way to estimate that number based on appropriately chosen measure? we probably know more about planetary systems than about branches of the wave functions or pocket universes.
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Actually, what I was suggesting is just the approach by Hartle and Srednicki you wrote about. If there is no prior preference for theory A or B, then you can just take the prior probabilites equal to each other. These will then be updated using Bayes theorem. Any change in the updated probabilities is then due to the data, so there is nothing ambiguous/strange about that...
About the Boltzmann brain, perhaps one can also use the algorithmic complexity of the unitary transformation you need to generate the observer from the intitial state. The probability that Anthony can be found in some state |psi> is sum over k of ||^2 where the Anthony_k> form a complete set of states containing Anthony. The state vector of the universe is obtained from some intitial state: |psi> = U(t)|psi(0)>
Now, we already assume that universes with simple laws of physics are more likely than universes specified by complex laws. E.g. who believes that the laws of physics will turn out to be specified by trillions of arbitrary parameters :) So, we already have a notion a complexity measure for the unitary transformation U(t) = Exp[-i H t], i.e. we don't think that a very complicated H is very likely.
So, perhaps we need to multiply:
sum over k of ||^2
by a probability which depends on the complexity of U(t). Here we insert the value of t in U and then look at the comlexity of that transformation. The larger we make t, the moe bytes you need to specify U(t). So, the Boltzmann brain contributions we get by integrating over t till infinity get supressed.
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Moshe:
I'm glad you brought up the planet example, which I think is illustrative in several ways:
(1) Indeed, I don't think you'll find many people who will argue that 'anthropic' selection effects are unimportant in answering (say) "why is the earth-sun distance 1.5e11m?" I think this should give pause to those who say inclusion of such selection effects is "not science." Alternatively, we can define a set of questions (such as that one) to which we simply cannot give scientific answers. I prefer leaving these within the purview of science.
(2) BUT contemplating performing such a calculation (easy compared to the much harder task of applying this reasoning in cosmology) does cause one to despair a bit. An analogy I contemplate sometimes is the program of testing the the big-bang cosmology with just a pair of binoculars. In principle, sufficiently clever cosmological (and planet/star formation and exobiology, etc.) theorists could run the calculation through, generating the probability distributions for galaxies, stars, planets, planets 1.5e11m from the stars, planets with "life", etc. But would we succeed? Would we really be able to pin down even one cosmological parameter this way? And would we ever have come up with dark matter or dark energy?
But I think what you were asking is one of methodology, and the relevant analog might be convincingly and satisfactorily explaining (since it is already measured) the earth-sun distance given the standard cosmological model. What would this mean? Well, suppose I had a planet-formation theory that entailed that the incredibly vast majority of earth-mass planets form in rich galaxy clusters, because the only way (in my theory) planet formation works is for planets to condense out of 10^8 K gas of > 0.1 solar metallicity, then be captured by stars. Since we have very little observational evidence regarding the distribution of earth-mass planets, this might be hard to rule out on that count. But I might then be surprised that *we* are not in a rich cluster. This surprise could lead me to (a) accept that we are very wierd, or (b) theorize that there is some additional selection effect that forces the probability distribution over to spiral galaxies in small groups, or (c) figure that my theory is wrong.
My reading of Hartle and Srednicki, BTW, would be that the 'planet condensation' theory is just as good as a more conventional planet-formation theory, insofar as both give at least one instance of an earthlike-planet in a large spiral galaxy.
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Thanks Anthony.
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