Robin Hanson writes:
> Nick B. writes:
> >Well, suppose there were two possible worlds, A and B, that are a
> >priori equally probable. In A there are a hundred humans and nothing
> >else. In B there are a hundred humans and a million stones. If you
> >know this and nothing more except that you exist and are a human,
> >what would you say the posterior probabilities are for the two
> >worlds?
> >
> >I would say 1/2. Finding myself being a human would give me no
> >information as to the number of stones.
> >
> >But in order to get this result we need to assume that there was a
> >zero chance that you would have been a stone.
>
> I wasn't considering universes of different sizes, but on reflection
> my prior gives exactly the result you prefer. As you recall, my
> approach was to divide universes into space-time regions, define
> states as universe-space-time combinations, and put equal priors
> across such states.
But in the present example, A and B were postulated to be a priori equally probable. Do you deny that this is a coherent assumption? It would seem that even on your approach you could agree that A and B were a priori equally probable if, e. g. B:s advantage in terms of number of states were compensated by an advantage of A in terms of simplicity of laws.
Or suppose that one of A or B had been created from a smaller starter-universe C through some random indeterministic process that statistically had a fifty-fifty per cent chance of churning out an A or a B. This would seem to give you equal priors for A and B.
> Consistent perhaps, but it doesn't address the model I detailed.
> Perhaps it is my fault for not more clearly distinguishing states
> from universes. A state says which is the true universe *and* which
> space-time spot I occupy. I was preferring equal priors over
> *states*.
What you are doing is you define a state to be:
Si = "u is the real universe and I occupy slot x in u."
where u is a member of U, the set of possible universes, and x is a spacetime slot in u.
Then you argue for the prior p(Si)=1/ |{i: Si is a state}|
One thing that I find problematic is your use of the indexical "I" in the definitions of the states. Are you using this term as a rigid designator that denotes the same individual in all possible worlds where that individual exists? What if anything does it refer to in those worlds where you do not exist? For example, in "* is the real universe and I occupy slot A1 [where there is a rock]" - here it seems more like you're using "I" as variable ranging over all slots in a given universe.
I would say that there are no "I"s associated with dead rocks. In order to be the sort of thing that could be referred to by the term "I" you have to be a person, and rocks are not persons. So the prior probability that I should be a rock must be zero. If you replace "I" with "this thing here-and-now" then it's really clear that this doesn't refer to anything in particular except if it is accompanied by an act of pointing. But if this is what you mean then in order to make sense of the state descriptions we have to understand "I" as a rigid designator. If we do that, then do you really think there is a fact of the matter, in a world containing ten similar rocks and nothing else, as to which one of these rocks is "really" you? Are there then ten physically identical possible worlds in each of which you are really a different rock? Sounds very metaphysical to me.
> >But when you are conditioning on your existing, what you do is
> >increase the probability of those worlds with many observers (SIA).
> >... And I say, if you look in
> >one place, and find an X there, then that gives you reason to believe
> >there are many Xs; but *only* if the reason why you looked where you
> >looked was not that there was an X there. Otherwise independence
> >fails. That's why you can't infer "There are many quick evolutionary
> >processes." (I assume you agree with that?)
>
> No, I don't follow this.
> I don't see what independence has to do
> with anything. To repeat myself again, the way to figure out what
> you can infer is to set up a state space, a prior, information sets,
> and turn the crank.
As I indicated, I'm not even convinced yet that you have coherently specified a state space. The specification seems to make use of rigid designators in a way that doesn't make any sense to me.