From: Wei Dai (weidai@eskimo.com)
Date: Sat Apr 03 1999 - 20:50:35 MST
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.
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