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...)


On Tue, Nov 25, 2008 at 5:41 PM, Russell Standish <[EMAIL PROTECTED]> 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                         [EMAIL PROTECTED]
> Australia                      
> ----------------------------------------------------------------------------
> >

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