# Re: join post

From: Abram Demski <abramdemski.domain.name.hidden>
Date: Wed, 26 Nov 2008 14:55:08 -0500

Russel,

I do not see why some appropriately modified version of that theorem
couldn't be proven for other settings. As a concrete example let's
just use Schmidhuber's GTMs. There would be universal GTMs and a
constant cost for conversion and everything else needed for a version
of the theorem, wouldn't there be? (I am assuming things, I will look
up some details this afternoon... I have the book you refer to, I'll
look at the theorem... but I suppose I should also re-read the paper
about GTMs before making bold claims...)

--Abram

On Tue, Nov 25, 2008 at 5:41 PM, Russell Standish <lists.domain.name.hidden> wrote:
>
> On Tue, Nov 25, 2008 at 04:58:41PM -0500, Abram Demski wrote:
>>
>> Russel,
>>
>> Can you point me to any references? I am curious to hear why the
>> universality goes away, and what "crucially depends" means, et cetera.
>>
>> -Abram Demski
>>
>
> This is sort of discussed in my book "Theory of Nothing", but not in
> technical detail. Excuse the LaTeX notation below.
>
> Basically any mapping O(x) from the set of infinite binary strings
> {0,1}\infty (equivalently the set of reals [0,1) ) to the integers
> induces a probability distribution relative to the uniform measure dx
> over {0,1}\infty
>
> P(x) = \int_{y\in O^{-1}(x)} dx
>
> In the case where O(x) is a universal prefix machine, P(x) is just the
> usual Solomonoff-Levin universal prior, as discussed in chapter 3 of
> Li and Vitanyi. In the case where O(x) is not universal, or perhaps
> even not a machine at all, the important Coding theorem (Thm 4.3.3 in
> Li and Vitanyi) no longer holds, so the distribution is no longer
> universal, however it is still a probability distribution (provided
> O(x) is defined for all x in {0,1}\infty) that depends on the choice
> of observer map O(x).
>
> Hope this is clear.
>
> --
>
> ----------------------------------------------------------------------------
> A/Prof Russell Standish Phone 0425 253119 (mobile)
> Mathematics
> UNSW SYDNEY 2052 hpcoder.domain.name.hidden
> Australia http://www.hpcoders.com.au
> ----------------------------------------------------------------------------
>
> >
>

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Received on Wed Nov 26 2008 - 14:55:24 PST

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