many minds interpretation of probability

Wei Dai (
Wed, 15 Oct 1997 13:37:33 -0700 (PDT)

On Mon, 13 Oct 1997, Hal Finney wrote:

> This is related to the many-minds school of QM, where an entity with
> a given brain state can be said to span many (perhaps infinitely many)
> parallel universes. My mind right now is shared among multiple instances
> in different universes. As events happen which distinguish the universes,
> my mind in effect splits into multiple, disjoint, non-interacting parts.
> Conceivably in some of those universes I could be uploads, being watched
> over by beings who are simulating me, while in others I could be living in
> the base reality. In yet others I could be in some kind of VR. And I
> suppose in some I could be an insane alien being who has fantasized
> the entire world of human existence, including an identity for himself
> which happens coincidentally to match in every detail my own identity.
> With an infinite number of universes, it's bound to happen somewhere.

It seems that the many-minds interpretation applys not just to quantum
mechanics, but to probability itself. Under this interpretation, the
probability that a statement is true to someone is the fraction of minds
in the same state as his that live in worlds where the statement is true.
So when I say a coin has probability 1/2 of landing heads, I mean that
half of me (or more precisely half of minds in the same state as mine)
live in worlds where the coin will land heads.

This interpretation offers a simple solution to the copying problem I gave
a couple of weeks earlier. The central question in that problem was: after
you wake up following a cloning operation that conditioned on a coin
landing heads, what is the probability that the coin landed heads? Before
you went to sleep, your probability of the coin landing heads was 1/2.
This means half of you live in worlds where the coin will land heads, but
since there is just one of you in each of those worlds, that implies the
number of worlds where the coin lands heads as is the same as the number
of worlds where the coin lands tails. After you wake up, the probability
is 2/3 since 2/3 of you (which include the originals and clones since they
have the same mind states) live in worlds where coin landed heads.

Of course the many-minds interpretation wouldn't be interesting if it
doesn't offer any advantages over the baysian interpretation, but the
baysian interpretation seems to fail when cloning is involved. After
waking up and also being told that you're not a clone, a simple Dutch book
argument says your probability for heads should be 1/2, which implies that
your probability should be 2/3 before knowing that you're not a clone. But
as Eliezer pointed out, the baysian interpretation requires probability
shifts (in this case the shift from 1/2 before sleep to 2/3 after sleep)
to be based on new information, which does not seem to be present.