Re: Computing Randomness

From: Russell Standish <>
Date: Fri, 30 Mar 2001 09:50:37 +1000 (EST) wrote:
> Russell Standish wrote:
> > In my approach (which didn't have this technical nicety), the ensemble
> > of descriptions obeyed a uniform distribution.
> I grab the opportunity to point out that there is no such thing as a
> uniform distribution on infinitely many things. For example, there is
> no uniform distribution on the natural numbers.

Correct about uniform probability distributions, but I don't think I
called it a PDF. Uniform measure distributions do exist over sets of
infinite measure, but cannot be normalised, hence cannot be turned
into a PDF.

> Uniform distributions that seem so natural in the finite case do not
> even make sense in the general case. The natural distributions on the
> possible universes are all nonuniform!
> To deal with infinities one might think of using measures instead of
> distributions. Measures do not assign probabilities to individual elements
> but to sets of them. For example, assign measure 2^-n to all bitstrings
> with a common prefix of size n. As n grows this measure goes to zero
> very quickly - and renders unlikely any finite beginning or future of
> any universe history, including the regular and describable ones.

Except for the fact of equivalencing. More to the point, you have
fingered the reason why wabbits aren't observed.

> Luckily, there are more natural and more plausible measures
> that do not necessarily vanish as history size grows:
> Juergen Schmidhuber

Dr. Russell Standish Director
High Performance Computing Support Unit, Phone 9385 6967
UNSW SYDNEY 2052 Fax 9385 6965
Room 2075, Red Centre
Received on Thu Mar 29 2001 - 16:28:35 PST

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