Wei Dai: the one universe
1998 Apr 16
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Wei Dai: the one universe @ Satoshi Nakamoto
- Author
-
Wei Dai
- Email
-
satoshinakamotonetwork@proton.me
- Site
-
https://satoshinakamoto.network
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.
Discussions: https://riceissa.github.io/everything-list-1998-2009/0096.html
Wei Dai: the one universe
1998 Apr 16 See all postsWei Dai
satoshinakamotonetwork@proton.me
https://satoshinakamoto.network
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.
Discussions: https://riceissa.github.io/everything-list-1998-2009/0096.html