a baysian solution

From: Wei Dai <weidai.domain.name.hidden>
Date: Tue, 14 Apr 1998 11:26:41 -0700

This is an attempt to save baysianism in the face of the coin-toss copying
paradox. There are two parts to this solution:

1. Statements which use the word "I", "me", or "my" do not count as well
defined hypotheses. They are not assigned probabilities.
2. There is only one real universe. (The One Universe Hypothesis or 1UH)

The justification for 1 is that the word "I" is being used to refer to
different objects in different statements, so what appears to be
        P(AB|C) <> P(A|BC)P(B|C)
is actually
        P(AB|C) <> P(A|DC)P(D|C)
which is not interesting. Instead of "I", we need to be more specific. For
example, "I will perceive X at time t" should be replaced with "Everyone
at time t who remembers being in mind state M will perceive X" or "At
least one person at time t who remembers being in mind state M will
perceive X".

Now if we accept 1 and the AUH, then every hypothesis would have a
probability of either 1 or 0. That is why 2 is also needed.

This approach solves several related paradoxes. First recall the single
coin-toss copying paradox. In this experiment there are two possible
universes. In universe A the experimenter (t=1) tosses a coin, (t=2)
observes heads, and (t=3) is duplicated. In universe B the experimenter
(t=1) tosses a coin, (t=2) observes tails, (t=3) is not duplicated. The
paradox is that

P(I will observe heads at t=3|I am at t=1) = 2/3,
P(I will observe heads at t=2|I am at t=1) = 1/2, and
P(I will observe heads at t=3|I will observe heads at t=2) = 1

are not consistent with each other. Under the proposed solution, these
statements will be restated as well defined hypotheses. Let M be the
experimenter's mind state at t=1.

"I will observe heads at t=3" becomes R = "Everyone at t=3 who remembers
being in state M will observe heads."
"I will observe heads at t=2" becomes S = "Everyone at t=2 who remembers
being in state M will observe heads."
"I am at t=1" becomes T = "The real universe contains a person with mind
state M at t=1."

The relevant probabilities then become:

P(R|T) = 1/2
P(S|T) = 1/2
P(R|S) = 1

and the paradox is solved.

This solution takes care of the Doomsday Argument as well. (See Nick's
paper for a description of the DA.) Consider two possible universes.
Universe A ends after the 100th birth. Universe B ends after the 200th
birth. Suppose someone in one of these universes has not yet learned his
birth rank, and denote his mind state as M. His P(universe A is real) =
P(universe B is real) = 1/2. Now suppose someone tells him that his birth
rank is 40. Since "My birth rank is 40" is not a well defined hypothesis,
he restates that as E = "At least one person with mind state M has birth
rank 40." Since P(E | universe A is real) = P(E | universe B is real) = 1,
this knowledge does not change his beliefs about which universe is real.

Finally consider the infinite births paradox. In this paradox there is
only one possible universe. This universe is infinite in time and has an
infinite number of births. Each person is born conscious and is told his
birth rank after a day. The paradox is P(My birth rank is n|I was just
born) = 0 for all n. This can now be restated as P(At least one person
with mind state M has birth rank n|There exist a person with mind state M)
= 1 where M is the mind state of everyone who was just born (we assume
they all have identical mind states).

So a solution to the copying paradox and various other paradoxes is
available, but it requires a specific denial of the AUH. I believe solving
these paradoxes within AUH is possible, but more difficult, because it
would require a new decision theory.
Received on Tue Apr 14 1998 - 11:29:06 PDT