# Re: Predictions & duplications

From: <juergen.domain.name.hidden>
Date: Mon, 22 Oct 2001 17:51:17 +0200

> From R.Standish.domain.name.hidden:
> juergen.domain.name.hidden wrote:
> > M measure:
> > M(empty string)=1
> > M(x) = M(x0)+M(x1) nonnegative for all finite x.
>
> This sounds more like a probability distribution than a measure. In
> the set of all descriptions, we only consider infinite length
> bitstrings. Finite length bitstrings are not members. However, we can
> identify finite length bitstrings with subsets of descriptions. The
> empty string corresponds to the full set of all descriptions, so the
> first line M(empty string)=1 implies that the measure is normalisable
> (ie a probability distribution).

Please check out definitions of measure and distribution!
Normalisability is not the critical issue.

Clearly: Sum_x M(x) is infinite. So M is not a probability
distribution. M(x) is just measure of all strings starting with x:
M(x) = M(x0)+M(x1) = M(x00)+M(x01)+M(x10)+M(x11) = ....

Neglecting finite universes means loss of generality though.
Hence measures mu(x) in the ATOE paper do not neglect finite x:

mu(empty string)=1
mu(x) = P(x)+mu(x0)+mu(x1) (all nonnegative).

And here P is a probability distribution indeed!
P(x)>0 possible only for x with finite description.

Juergen Schmidhuber

http://www.idsia.ch/~juergen/
http://www.idsia.ch/~juergen/everything/html.html
http://www.idsia.ch/~juergen/toesv2/
Received on Mon Oct 22 2001 - 08:52:08 PDT

This archive was generated by hypermail 2.3.0 : Fri Feb 16 2018 - 13:20:07 PST