Re: Doomsday Example

Robin Hanson (hanson@econ.berkeley.edu)
Sat, 22 Aug 1998 21:12:06 -0700 (PDT)

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.

It seems as if both of your worlds have one time slot, world A has 100 spatial slots, and world B has a million + 100 spatial slots. Equal priors on states implies unequal priors on worlds (where the prior of a world adds up all the priors of states associated with it.) But then conditioning on being a human, we get equal world posteriors back again.

>> >> >> What prior would you assign to world * in my example? ...
>> >> > ... I would say P(*) = 1/3. This would be the
>> >> >absolute prior. Then you take account of the fact that you exist,
>> >> >... you renormalize and get P(#) = P(@) = 1/2. ...
>> >> you seem to prefer p(*) = 1/3, p(#) = 7/20, p(@) = 3/20. ...
>> >> I instead prefer all three being 1/3. ...
>> >But I too would say all absolute priors three are 1/3. ...
>> We seem to disagree on the math here. I say that since the probability
>> of existing conditional on being in universes # and @ is different,
>which I deny! ...
>Let's take it slowly and write it out in full:
>Let * be "Universe * exists." And similarly for # and @. Let Me be "I
>exist." P(*)=1/3 P(#)=1/3 P(@)=1/3
>P(Me | #) = ... (3/2)P(Me)
>P(Me | @) = ... (3/2)P(Me)
>I don't see any inconsistency.

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*. If the prior of a universe it taken to be the sum of priors of associated states, then in that sense I also prefer equal priors over universes.

In my model "I exist" was interpreted first as "I'm an H or P". Since universe @ has more space-time slots filled by Hs and Ps than universe # has, then P(I'm H or P | @) > P(I'm H or P | #) if all the states associated with a universe have equal priors.

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. That tells you everything there is to say about what you can infer. No independence need be assumed for this to work.