the one universe

From: Wei Dai <weidai.domain.name.hidden>
Date: Thu, 16 Apr 1998 00:51:42 -0700

Assuming there is just one real universe, a natural question would be
which universe is real. The traditional bayesian way of answering this
question would be to use something like the principal of indifference to
come up with a prior probability for each possible universe being real,
and then take into account any knowledge one might have about the real
universe to compute a posterior probability. I'm going to give that a try.

Suppose the only thing you know about the real universe is that it is the
output of some program running on some particular universal prefix machine
U. The principal of indifference would tell you to assign a uniform
distribution for the input tape of this machine. That is, each cell of the
input tape has probability 1/2 of being 0 and 1/2 of being 1, independent
of all other cells. It's not clear where the knowledge that the universe
is the output of a computer program comes from, but let's use this
distribution as the prior and see what happens. This setup is very similar
to the one in Juergen Schmidhuber's paper, except we have only one prefix
machine running one program. It turns out to have counter-intuitive
consequences.

One piece of information about the real universe you have direct access to
is your own mind state. This is captured in the statement D = "The real
universe contains at least one person with mind state M" where M is your
current mind state. I'm going to assume this is the ONLY piece of
information about the real universe you have direct access to. Everything
else must be computed from the prior and this data. The justification for
this is that I can't think of any other information that is not part of or
derived from D.

Right away you know that any universe that does not contain at least one
person with mind state M cannot be real. It's also not hard to see that
for any two universes that both contain at least one person with mind
state M, the ratio of their posterior probabilities is the same as the
ratio of their priors. This means the universe most likely to be real
given D is the one that has the highest prior among the universes that
contain at least one person with mind state M.

I don't have a proof for this, but I'm fairly confident that for any
reasonable prefix machine U and normal human mind state M, this universe
would be the counting universe, i.e. the universe generated by the program
that enumerates all possible bit strings. This is because the counting
program is the shortest program whose output includes at least one
encoding of a person with mind state M.

Well the conclusion is really absurd, but is there anything wrong with the
argument? If not we'll have to either come up with another prior for the
1UH, or give it up. I think it may not be possible to find a non-contrived
prior for the 1UH that would lead to intuitive results, but I'll have to
work out the arguments for that.
Received on Thu Apr 16 1998 - 00:52:31 PDT