Revising priors and anthropic reasoning

by PhilGoetz1 min read6th Feb 201127 comments

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I believe that life on Earth arose spontaneously.  I also believe the galaxy around me is largely devoid of life.  I reconcile these things using the anthropic principle.

I also believe that fundamental cosmological constants have values convenient for the development of life.  I don't know if it makes sense to pretend that those constants could have had other values - it seems to me like arguing that e could have been 2.716.  But it's certainly done.  And again, the anthropic principle is sometimes invoked, as an alternative to, say, God.

Suppose somebody came up with a new theory of cosmological constants, that claimed that only certain values are allowable, and that a large percentage of the allowable sets would make life possible.  Then you wouldn't have to use the anthropic principle.  Wouldn't you be more comfortable with that?

But if that's so, doesn't it mean that you really attach a low prior to the anthropic principle?  And that you don't truly accept the anthropic principle?

How do you do Bayesian belief revision when one of your alternative hypotheses uses the anthropic principle?  Can you give a strong preference to the hypothesis that does not require it?  Because I know that I would.

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How do you do Bayesian belief revision when one of your alternative hypotheses uses the anthropic principle? Can you give a strong preference to the hypothesis that does not require it? Because I know that I would.

Your existence can never be Bayesian evidence for you either way due to conservation of expected evidence. Your non-existence can never be Bayesian evidence the other way for you.

In a wider sense it seems far from obvious that a claim like "most observers are instances of probable observers" is true, and even if it is true it's not clear that you are allowed to draw inferences from that as though being a particular instance of you was a random sample, that may be just a malfunction of human reasoning caused by the particular way we evolved. Updateless reasoning suggests that you should not distinguish between yourself and other instances of you in that way.

That's not the question I'm asking. If you have two hypotheses to explain something, and one of them uses anthropic reasoning, and the other does not, how much can you favor the one that does not?

If anthropic reasoning is used to get around a prior of 10^-11, can I favor the hypothesis not requiring anthropic reasoning by a factor of 100? If I favor the latter hypothesis at all, shouldn't that show up in the priors; and shouldn't anthropic reasoning therefore lose out to, eg., the God hypothesis?

My other reply to the post about the updateless perspective probably fits better.

I'm not sure what a hypothesis without anthropic reasoning would be. Anthropic reasoning is just that you are certain to observe a world where your existence is possible. if two different hypotheses both allow your existence they both benefit from anthropic reasoning, if any hypothesis ever does. If your existence is drastically more likely conditional on one hypothesis being true than it is conditional on the other being true then the first benefits more from anthropic reasoning itself, while the latter implies more anthropic reasoning for the observation that you exist. Which of those hypotheses would you describe as using anthropic reasoning?

If hypothesis A has a prior of 10^-10, hypothesis B a prior of 10^-15, P(PhilGoetz|A)=10^-20 and P(PhilGoetz|B)=10^-5, then I think it's permissible for PhilGoetz to drastically favor hypothesis B over A whenever the importance of the decision directly scales with P(PhilGoetz).

You don't have to have a low prior for the anthropic principle; the anthropic principle might tell you that there should be a low prior. In some formulations of the anthropic principle, such as the self-sampling assumption, a universe that makes it "more likely for you to exist" is favoured, depending on the specifics of what it means for someone to be more likely to exist in one theory than in another.

Sounds like this is about using the anthropic principle in situations of logical uncertainty. You say, "what if there's a new theory where most possible values of basic physical properties favor life", but you don't know whether there's any such theory, and whether there is such a theory is a logical (mathematical) fact.

This debate exists in string theory. People used to expect that string theory would predict a single unique ground state for the universe, which would lead to predictions about particle masses and so forth, and this would make string theory falsifiable. Now very large numbers of apparently stable or metastable string vacua have been constructed, and there's a view that some combination of anthropic and statistical thinking will be required to get predictions.

However, the real story is simply that string theory is still a work in progress, and we still don't know the right way to think about cosmology in string theory, and cosmology is half the necessary context for this problem. The other half is the dynamics of the "geometric moduli", the parameters describing the Calabi-Yau spaces and so forth, and that isn't solved either. String theorists still haven't even constructed a vacuum which provably gives you back the standard model - because the particle masses depend on the moduli dynamics, and the moduli dynamics are very difficult, though progress is occurring.

The next level of progress beyond moduli dynamics is the cosmological level, and that's where we'll find out whether eternal inflation or Hawking's no-boundary proposal or something else is the right way to think about the very big picture. And incidentally, the whole history of string theory suggests that there will be a specific right way to approach the problem.

We aren't at that level yet, and so meanwhile people speculate about the degree to which the anthropic principle might be relevant in string theory. But perhaps you can see that these speculations are not like empirical speculations; they are speculations about a logical fact. Whether or not the anthropic principle plays a role in getting predictions out of string theory is a question which "platonically" already has an answer. We only have an anthropic debate in string theory because the theory is unfinished and we don't know its answer to that question yet. So it's exactly an example of what you're talking about.

Assuming string theory is actually correct. As opposed to say loop quantum gravity or something we haven't even thought of.

Here is an attempt at an updateless answer, because the problem is too confusing for me from an individual perspective. I'm not sure in how far this contradicts my earlier answer.

Assume multiverse branches A and B with equal mesure/prior probability. A has 99 times as many instances of the agent as B. If the agent weights the consequences of the actions of each instance equally the instances of the agent will in most cases behave like individualist single world agents believing "A" to be true with 0.99 confidence. Most human level problems probably are of that type. There may be problems where the majority answer in A and B is equally important, or some other weighting where the answer in A is not 99 times as important as the one in B. In those cases the agent won't behave like agents believing in "A" with 0.99 confidence.

Suppose somebody came up with a new theory of cosmological constants, that claimed that only certain values are allowable, and that a large percentage of the allowable sets would make life possible. Then you wouldn't have to use the anthropic principle. Wouldn't you be more comfortable with that?

No, I'd be quite surprised and see that supposed relationship in much more of a need for an explanation than us happening to observe (somewhat?) intelligent life on Earth.

Suppose somebody came up with a new theory of cosmological constants, that claimed that only certain values are allowable, and that a large percentage of the allowable sets would make life possible. Then you wouldn't have to use the anthropic principle. Wouldn't you be more comfortable with that?

Another way of putting your question, is "how much would you expect a discovery of a new theory that expands the allowable values of cosmological constants?" - fair? It's less trap-like, for better or worse :)

But if that's so, doesn't it mean that you really attach a low prior to the anthropic principle? And that you don't truly accept the anthropic principle?

Your link defines "the anthropic principle" as "the philosophical argument that observations of the physical Universe must be compatible with the conscious life that observes it."

How do you unpack this statement (especially the modal "must") so that it has any implications at all? That particular formulation looks tautological. It seems to say no more than that an observed universe contains observers.

People have tried to make anthropic intuitions precise in different ways. Insofar as these efforts succeed, they have different implications for how one ought to do Bayesian updating. Which one are you talking about?

Very sharp question, thanks you.

Suppose somebody came up with a new theory of cosmological constants, that claimed that only certain values are allowable, and that a large percentage of the allowable sets would make life possible. Then you wouldn't have to use the anthropic principle. Wouldn't you be more comfortable with that?

It seems a bit hypothetical. We can easily create cellular automata universes with different physical laws. By most obvious ways of enumerating them, very few would support life.

By most obvious ways of enumerating them, very few would support life.

Can you expand on this claim? The most obvious enumerations to me involve something like a restricted Solomonoff prior. And given that, I don't see how one gets that very few would support life. We know very little about the behavior of cellular automata on average in any useful sense.

The territory has been surveyed to some degree. For example, consider Wolfram's Elementary 1D Cellular Automata. Most are not even computationally universal.

On the other hand, David Eppstein - who has looked into the issue quite a bit - claims that gliders are "commonplace" - contradicting the spirit of Woflram's: "Except for a few simple variants on the Game of Life, no other definite class-4 two-dimensional cellular automata were found in a random sample of several thousand outer totalistic rules."

Eppstein's glider work is a bit of a caveat - but I generally approve of the idea that "life exists at the edge of chaos". Looking at Langton's "lambda" parameter, most automata are either off in the chaotic realm, or exhibit trivial, degenerate behaviour - with only a relatively small number in between.

First off, Wolfram's attempt to classify automata into four categories is flawed: it isn't obvious that every cellular automata falls into one of those categories and not all the categories are even rigorously defined. Also the claim that not many other "definite class-4" automata were found is unhelpful. The ways that things can turn out to be Turing complete are often surprising and non-obvious (cf for example the word problem or Hilbert's 10th problem.) So the claim that an automaton rule isn't obviously Turing complete is extremely weak evidence that it isn't.

The fact that 1D cellular automata are weak is both unsurprising and not good evidence for anything. From a Solomonoff perspective, adding in extra dimensions is pretty cheap (Tangent: one issue that I don't fully understand is why we shouldn't be deeply surprised by how few dimensions our universe seems to have. One of the strongest arguments for string theory that I'm aware of is that it answers this by saying that there really are more dimensions. However, string theory doesn't really do a satisfactory job in that regard because the behavior of the individual dimensions is likely to vary, while the Occamian approach predicts that dimensions that are all treated the same way should be cheap.) But there are all the 2D automata and no one has even done almost any investigation of low dimensionsal automata for higher than 2. 3 and 4 dimensions would be the obvious ones where interesting things might happen. And there's also been very little work at automata that uses a basic tiling form other than squares (e.g. equilateral triangles, or hexagons, or pairs of octagons and squares with possibly different rules for the two types.)

Incidentally, I suspect that most of the so-called "chaotic" cellular automata are Turing complete but simply feindishly difficult to establish as such.

There's also an issue that's relevant to all of this- whether or not an automata is Turing complete is not the same as the question as to whether or not it can support "life." There are decent arguments for this claim, but it is far from obvious.

First off, Wolfram's attempt to classify automata into four categories is flawed: it isn't obvious that every cellular automata falls into one of those categories and not all the categories are even rigorously defined.

Yes, David has a page about that topic.

Without studying it in detail, I'm skeptical of the importance its claim. David says he has found class 1 and 3 systems that produce gliders. Perhaps David is poor at classifying systems. The existence of gliders strongly implies a system is class 2 or 4.

But perhaps he really has found class 1 and 3 systems in which a glider can exist. Is that important? Well, since the classification is statistical, being "class 1" really means "behavior is more class-1-like than any other class". So particular structures can still exist in a class 1 or class 3 CA that exhibit a particular behavior. They're just less-common.

I think Wolfram's classifications are very well-defined as these things go (you can probably state them mathematically in terms of Lyapunov exponents, or number of attractors for finite worlds, or number of state transitions over time in Monte Carlo simulations, for instance). The problem is that the classification is statistical, so you can't say "this is class 1 and therefore behavior X is impossible".

The more important point, popping back up to the grandparent of this comment, is that I'm not convinced that the distribution of possible CAs resembles the distribution of possible physics.

Without studying it in detail, I'm skeptical of the importance its claim. David says he has found class 1 and 3 systems that produce gliders. Perhaps David is poor at classifying systems. The existence of gliders strongly implies a system is class 2 or 4.

But perhaps he really has found class 1 and 3 systems in which a glider can exist. Is that important? Well, since the classification is statistical, being "class 1" really means "behavior is more class-1-like than any other class". So particular structures can still exist in a class 1 or class 3 CA that exhibit a particular behavior. They're just less-common.

It's the second paragraph that's right here, I believe.

I generally agree with Epstein's comments. Except that in his classification scheme, I tend to regard the "Contraction impossible." case as pretty similar to the "Expansion impossible" case.

I think a major point of interest of Wolfram's classification is that is suggested that universal automata were in class 4. If that is more false than true, the classification scheme loses much of its interest.

Incidentally, I suspect that most of the so-called "chaotic" cellular automata are Turing complete but simply feindishly difficult to establish as such.

Maybe. It is a fun speculation - but I don't think anyone really knows. Rule 30 is an obvious test case. If that ever gets proved to be universal, the conjecture would become quite a bit more plausible.

If so, more automata would be lively. My speculation is partly based on "most" automata exhibiting chaotic dynamics, which don't obviously support life or universal computation - because everything explodes.

There's also an issue that's relevant to all of this- whether or not an automata is Turing complete is not the same as the question as to whether or not it can support "life." There are decent arguments for this claim, but it is far from obvious.

If you are Turing complete you can simulate anything - including arbitrary living systems.

...and if you aren't Turing complete, the dynamics are usually pretty limited. Surely these two things are really the same thing.

If you are Turing complete you can simulate anything - including arbitrary living systems.

...and if you aren't Turing complete, the dynamics are usually pretty limited. Surely these two things are really the same thing.

No. They aren't. First at the most basic level you have that many would argue that simulating isn't the same thing as actually being (many on LW would consider this to be wrong or so incoherent to not even be wrong and I'm inclined to sympathize with that view but it is something that needs to be considered). More seriously, just because the rule system is capable of simulating Turing machines when one can control the initial configuration doesn't mean that any given universe actually will have that happen for a fixed configuration. For example, consider Conway's Life. This is Turing complete in the strong sense that we have a computable map that takes Turing machines to Life configurations such that 1) the corresponding Life configuration will will form a stable non-empty configuration iff the Turing machine halts in an accepting state and 2) the corresponding Life configuration will die-off iff the Turing machine halts in an a rejecting state. But, and this is an important but, if we start with some Life configuration it won't necessarily run through all or any interesting Turing machines. For example, the universe might obey Conway's laws but have all squares start empty, or one that starts with a handful of live cells that quickly die.

Another problem is that one can conceive of (although I'm not aware of any known examples) of a set of cellular automata rules that allow something looks like "life" to occur (growing, dieing, reproducing, competing) but is too weak to be Turing complete.

Note also that being Turing complete doesn't mean you can simulate anything- it means you can simulate anything in our universe. Universes that obey drastically different laws of physics are not necessarily Turing computable (so for example the HPMR world would be difficult for a Turing machine to handle due to the time-travel issues.) One can give even more straightforward examples, such as a universe identical to ours except that there's a little black box that when fed a description of a Turing machine in some simple method will output a certain signal iff that Turing machine runs on the blank tape. The fact that our universe doesn't do anything like that is essentially an empirical statement, not a statement about all universes (although there are plausibility arguments to think that all universes might behave this way. This is one issue that comes up when discussing weakened versions of the Tegmark ensemble.).

The upshot is that asking at whether or not a given cellular automata rule is Turing complete is not necessarily the same as asking whether or not that rule can support life.

You make several points:

"many would argue that simulating isn't the same thing as actually being"

A useless and pointless argument, IMO. Patternism is a better philosophy. Artificial life is really alive.

"More seriously, just because the rule system is capable of simulating Turing machines when one can control the initial configuration doesn't mean that any given universe actually will have that happen for a fixed configuration."

The claim would be something like: living systems are Turing complete and Turing complete systems are capable of supporting life. I don't think what you said impacts on that claim.

Another problem is that one can conceive of (although I'm not aware of any known examples) of a set of cellular automata rules that allow something looks like "life" to occur (growing, dieing, reproducing, competing) but is too weak to be Turing complete.

Yes, maybe. This seems like a pretty rare and esoteric possibility to me - but I wouldn't say it was impossible.

If you define life as having anything to do with constructive evolution and adaptations, though, then I am going to have to ask for an example.

Note also that being Turing complete doesn't mean you can simulate anything- it means you can simulate anything in our universe.

Noted - but that doesn't seem to have much to do with life.

A useless and pointless argument, IMO. Patternism is a better philosophy. Artificial life is really alive.

I'd be inclined to agree that patternism is more coherent. But that doesn't make the argument useless or pointless (indeed it is a pretty pointed issue since it matters a lot in this context).

Yes, maybe. This seems like a pretty rare and esoteric possibility to me - but I wouldn't say it was impossible.

If you define life as having anything to do with constructive evolution and adaptations, though, then I am going to have to ask for an example.

Well, Sniffnoy has already pointed out the finite computation limit. But the following is an almost example. (Note that this example is not completely deterministic and allows arbitrarily many states in cells although any fixed configuration has only finitely many such in use.) Example: Cells can have any non-negative integer in them (with zero representing the empty cell): In any given iterarion, for any cell , there is a (1-1/(k+1))/2 chance that the cell is filled with k, where k is the largest number in an adjacent cell, and a ( (1-1/(k+1))/2) chance that it will fill with k+1. Now, start this with a single cell with k=1 and all others empty. The population will rapidly expand, and "evolve" towards higher and higher integers, expanding rapidly from the initial point, with different populations of integers expanding outwards and competing. One could make more complicated versions of this to allow more complicated adaptions (say giving a slight benefit to primes to preserve their current value rather than be overwritten). This would deal with the fact that this model doesn't allow any interesting evolution (and indeed the ladder nature of the evolution in question makes it a very poor model, almost reflecting certain common misconceptions about evolution somehow being progressive.)

Note also that being Turing complete doesn't mean you can simulate anything- it means you can simulate anything in our universe. Noted - but that doesn't seem to have much to do with life.

Right, I was bringing this up because you made a comment about Turing machines simulating anything.

Also note that a universe based on real numbers could reasonably have life and could not be simulated by a Turing machine but only a Blum-Shub-Smale machine.

Cells can have any non-negative integer in them (with zero representing the empty cell): In any given iterarion, for any cell , there is a (1-1/(k+1))/2 chance that the cell is filled with k, where k is the largest number in an adjacent cell, and a ( (1-1/(k+1))/2) chance that it will fill with k+1. Now, start this with a single cell with k=1 and all others empty. The population will rapidly expand, and "evolve" towards higher and higher integers, expanding rapidly from the initial point, with different populations of integers expanding outwards and competing.

Hmm. Yes. That meets my criteria for adaptation.

It's pretty simple - I shoulda thought of that myself, and saved you some time.

Another problem is that one can conceive of (although I'm not aware of any known examples) of a set of cellular automata rules that allow something looks like "life" to occur (growing, dieing, reproducing, competing) but is too weak to be Turing complete.

Yes, maybe. This seems like a pretty rare and esoteric possibility to me - but I wouldn't say it was impossible.

How about a really large finite state machine? While important theoretically, for the purposes we're discussing, I don't think we should need the infinitude of a Turing machine.

Right - I don't literally mean to imply infinity. I just mean in the way that partial recursive functions can perform arbitrary computations (memory permitting) - or lambda calculus, or cellular automata.

So: a substrate capable of universal computation if it were extended to infinity.

The anthropic principle is a logical necessity, given its basic requirements, but it only makes things more likely, especially when there's uncertainty about whether the basic requirements are fulfilled or not. Even probable things can be wrong.