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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

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

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