Nick Bostrom writes:
>> I'll assume A universes
>> have 10 humans + N stones, and B universes have just 10 humans.
>>
>> I don't want to work out the math for 1000 universes, but two should be
>> enough to see what works. In that case there are four possible worlds:
>> -- Two A "universes", with 2*(10+N) space-time slots, and "prior" of 1%.
>> -- Two B "universes", with 2*10 slots, and "prior" 81%.
>> -- An A and a B "universe," with 2*10 + N slots, and "prior" 9%.
>> -- Another possible world that looks just like the last one.
>> These "priors" are over worlds, but not necessarily over states.
>
>I'm not sure where you got these priors from.
You had previously assumed each baby of C had a 10% chance of being A and a 90% chance of B.
>It doesn't really matter but it may be easiest to see the objection
>that I'm trying to make if we assume the priors are:
>
>AA=1/4
>AB=1/2
>BB=1/4
>
>One can imagine this distribution arising from throwing a fair coin
>in the C universe twice. This is the probabilities of the different
>world combinations relative to an information set that only contains
>the information about the set-up.
>
>Then, when you conditionalize upon being human (but you don't yet
>have any other information), if you assume that you are a random
>sample from all possible states/space-time slots (as you say we
>should do), you get the posterior:
>
>AA=0
>AB=0
>BB=1
>
>in the limiting case where N is very large. (Reason: if there existed
>an A universe, you would almost certainly have been one of the
>stones.)
No! You are confusing priors on universes with priors on states, even though I tried to clearly distinguish these in my previous post:
>These "priors" are over worlds, but not necessarily over states.
If you're going to go with equal chance of A or B, then I'd say there
are four possible worlds: AA, AB, BA, BB, and 80 + 4N space-time slots
among these worlds. Giving equal probability to these *slots*, then
conditioning on being human, you get equal probability to be in A vs. B.
>If we extend these descriptions to include which slot "I" occupy,
>we get 80 + 4N states. If I make the relative priors between states
>equal to the relative "priors" between associated worlds, then, yes,
>very little state prior is associated with the second world with two
>B "universes." But conditioning on observing that I'm a human, I'm
>back to estimating a 81% chance that there are two B "universes."
Robin Hanson
hanson@econ.berkeley.edu http://hanson.berkeley.edu/
RWJF Health Policy Scholar, Sch. of Public Health 510-643-1884
140 Warren Hall, UC Berkeley, CA 94720-7360 FAX: 510-643-8614