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)
UNSW SYDNEY 2052                         [EMAIL PROTECTED]
Australia                                http://www.hpcoders.com.au

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