> > 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


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