# Re: predictions

From: Nick Bostrom <bostrom.domain.name.hidden>
Date: Sat, 28 Feb 1998 15:24:03 +0000

Wei Dai's first paradox was as follows:

> At time 0 the experiment starts. At time 1 a coin is flipped and the
> result observed by the experimenter. At time 2 a second coin is flipped
> and the result observed by the experimenter. At time 3 the experimenter is
> duplicated if and only if both coins show heads. Now suppose the coins are
> fair so that the experimenter would predict at time 0 that at time 1 he
> will observe heads with probability 1/2. Applying Tegmark's method, he
> would also predict that at time 3 he will observe two heads with
> probability 2/5. But suppose he has already observed heads at time 1, then
> he would predict that with probability 2/3 he will observe another head at
> time 3. So he is left with the following beliefs at time 0:
>
> 1. At time 1 I will observe heads with probability 1/2.
>
> 2. If I observe heads at time 1, I will observe another head at time 3
> with probability 2/3.
>
> 3. I will observe two heads at time 3 with probability 2/5.
>
> But these three statements contradict each other since 1 and 2 together
> implies
>
> 3a. I will observe two heads at time 3 with probability 1/3.
>
> I don't know what to make of this... anyone have any ideas?

1.
I think premiss 1is false. In order to see this, we have to consider
observer-moments, rather than persons. Let's assume the world is
empty except for the things refered to in the paradox. Then, finding
that your present observer-moment is at time 0 gives you reason
(because of Bayes' theorem) to prefer a hypothesis according to which
a larger fraction of all observer-moments are at time 0 to a
hypothesis according to which a smaller fraction of all
observer-moments are at that time. In the present example, that means
that finding yourself at t=0, you should conclude that the chance
that both coins will land heads is less than 1/4. This also means
that the chance of the first coin landing heads is less than 1/2.

If the world cointains a lot of other observes (outsiders), then the
very fact that your present observer-moment is in this experiment in
the first place indicates that the experiment contains many
observer-moments, i.e. that the chance of both coins landing heads is
greater than 1/4. If you then in addition find that your present
obserever-moment is at time 0, then that gives you reason (as
heads downwards again. (As the number of outsiders (observer-moments
outside the experiment) goes to infinity, I think you will end up
with a probability asymptotically approaching the one you have
assumed.

2.

3.
If all branches exist (if there is one real world in which the coins
both land heads, and other real worlds in which they land the other
ways) then I might say the following: If I find myself at time 1 and
observing tails, there is a greater than zero chance that I will find
myself later at time 3 observing two heads! For there will be future
observer-moments observing two heads and other future
observer-moments observing at least one tail, and there doesn't seem
to be any fact of the matter as to which one of these
observer-moments is "really" the future me. I think this might be the
solution to the paradox. It is, metaphorically speaking, possible for
"me" to jump from one branch to another, since there is no fact of
the matter as to which which of the several future me:s I should say
is the true continuation of my present me.

(If only one branch exists, then I would reject premiss 3.)

_____________________________________________________
Nick Bostrom
Department of Philosophy, Logic and Scientific Method
London School of Economics
n.bostrom.domain.name.hidden
http://www.hedweb.com/nickb
Received on Sat Feb 28 1998 - 07:30:02 PST

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