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...)

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--Abram 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 http://www.hpcoders.com.au > ---------------------------------------------------------------------------- > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---