possibilities for I and Q

From: Wei Dai <weidai.domain.name.hidden>
Date: Tue, 9 Jun 1998 15:12:03 -0700

I put off considering possibilities for the theory of identity, F.
Although it is philosophically interesting and necessary for computing
probabilities for "I will observer X", these probabilities are not
actually predictions since they are not testable. I don't have and can't
possibly get any empirical information about whether or not I actually
will observe X. I only have information about what I currently experience
and what I remember.
So here are some possibilities for I (the set of all observer-instants)
and Q (the probability measure over I).

1. I is the set of all finite binary strings that represent
observer-instants. Q(i) is proportional to UP(i) where UP is the universal
prior.

2. I is the set of all infinite binary inputs for some universal prefix
machine that produce encodings of observer-instants as outputs. Q(J) is
defined iff J is the set of all infinite binary strings that share a
common prefix, in which case Q(J) is proportional to 2^-l, where l is the
length of that prefix.

3. I is the set of all observer-instants in worlds that can be encoded as
binary strings. Q(i), where i=<w,j> for some world w and index j, is
proportional to UP(w).

4. I is the set of all observer-instants in worlds that can be encoded as
binary strings. Q(i), where i=<w,j> for some world w and index j, is
proportional to UP(w)/N(w) where N(w) is the number of observer-instants
in w.

Some observations: (1) and (2) produces the same probabilities for "I
currently observe X", but it might be easier to construct a theory of
identity for (2) that leads to intuitive probabilities for "I will observe
X, given I currently observe Y." Both (3) and (4) are ill-defined unless
we restrict to worlds that contain finite numbers of observer-instants.
(3) can be discarded immediately, because Q would be dominated by counting
worlds that contain very many observer-instants and have high universal
priors. (1) and (2) have the advantage over (4) that they are less
sensitive to how we define what counts as observers, but it might be
easier to find an intuitive interpretation for (4).
Received on Tue Jun 09 1998 - 15:27:01 PDT