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From: <juergen.domain.name.hidden>

Date: Mon, 22 Oct 2001 17:51:17 +0200

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

Date: Mon, 22 Oct 2001 17:51:17 +0200

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

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