Re: reasoning under computational limitations

Wei Dai (
Sat, 3 Apr 1999 19:50:35 -0800

On Sat, Apr 03, 1999 at 12:10:03PM +0000, Nick Bostrom wrote:
> We don't get an exact cancellation of the DA unless the SIA has the
> property that, other things equal, if there are ten times more people
> on hypothesis A than on hypothesis B, then A is ten times more
> probable than B after conditionalizing on your own existence. And
> this does of course mean that your existence should make it a priori
> certain (probability one) that there are infinitely many people,
> provided that had a non-zero probability to start with. And that
> seems wrong.

My version of SIA (let's call it SIA-1) does have that property when the measures of all observers are equal. But when there are an infinite number of people on one of the hypotheses, it is not possible for the measures to be equal. Suppose either (A) the universe contains an infinite number of people or (B) it contains one person, and before conditionalizing on your own existance you assign them equal probability. Also suppose the total measure of observer-instants for A is .5. Given A, it's not possible for every observer to have equal measure (otherwise their measures would either add up to zero or infinity), so the probability that you have birth rank 1 given A and you exist has to be some non-zero finite number, say .01. Therefore the measure of the first observer given A is .5*.01 = .005. Suppose the measure of the first observer given B is also .005.

If you apply SIA-1 now,

P(A | I exist) = P(I exist | A) * P(A) / P(I exist) = .5*.5 / (.5*.5+.5*.005) = 100/101.

If you then apply the DA,

P(A | I exist and my birth rank is 1) = P(my birth rank is 1 | I exist and A) * P(A | I exist) * P(I exist) / P(I exist and my birth rank is 1) = .01 * 100/101 * (.5*.5+.5*.005) / .005 = .5.

Everything cancels out nicely and you never assign probability one to A at any point.