Re: predictions

From: Wei Dai <weidai.domain.name.hidden>
Date: Sat, 28 Feb 1998 14:31:27 -0800

On Sat, Feb 28, 1998 at 03:24:03PM +0000, Nick Bostrom wrote:
> Here are some comments:
> 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
> explained above) to adjust your probability estimate of getting two
> 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.
> Your paradox can be simplified to only one coin toss.
>
> 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.)

It's very difficult to follow your reasoning here. Let me try to simplify
things as much as possible. (You're right that the paradox can be reduced
to just one coin toss.) Suppose there are just two universes. In universe
A the experimenter flips a coin at time 1 and observes heads, and at time
2 he is duplicated. In universe B the experimenter flips a coin at time 1
and observes tails, and at time 2 he is not duplicated. Both universes
start at time 0 and end at time 3 and contain no other observers.

Now let's use this collection of universes to test our two definitions of
sensory probability P(X|Y). With definition A, the experimenter believes
at time 0:

A1. At time 1 I will observe heads with probability 1/2.

A2. If I observe heads at time 1, at time 2 I will observe heads with
probability 1.

A3. At time 2 I will observe heads with probability 2/3.

These probabilities are not consistent with each other. But with
definition B we have:

B1. At time 1 I will observe heads with probability 1/2.

B2. If I observe heads at time 1, at time 2 I will observe heads with
probability 2.

B3. At time 2 I will observe heads with probability 2/3.

These probabilities are consistent.
Received on Sat Feb 28 1998 - 14:32:01 PST