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From: Juergen Schmidhuber <juergen.domain.name.hidden>

Date: Fri, 26 Oct 2001 15:45:48 +0200

*> From: Juho Pennanen <juho.pennanen.domain.name.hidden>
*

*> So there may be no 'uniform probability distribution' on the set of all
*

*> strings, but there is the natural probability measure, that is in many
*

*> cases exactly as useful.
*

Sure, I agree, measures are useful; I'm using them all the time. But in

general they are _not_ the same thing as probability distributions.

*> From: Russell Standish <R.Standish.domain.name.hidden>
*

*> The only reason for not accepting the simplest thing is if it can be
*

*> shown to be logically inconsistent. This far, you have shown no such
*

*> thing, but rather demonstrated an enormous confusion between measure
*

*> and probability distribution.
*

Russell, this is really too much; please read any intro to measure

theory,

e.g., the one by Halmos. Measures in general are _not_ probability

distributions. They are more than that; you can view them as _sets_ of

probability distributions on sets that are related to each other in a

specific way. Probability density functions define measures and

therefore

in general aren't probability distributions either. The uniform PDF on

[0..1] yields the perfect trivial example; that's exactly why I used it

before. In the computability context, compare LI/Vitanyi 1997, chapters

4.3 (discrete sample space with (semi)distribution m(x)) vs chapter 4.5

(continuous sample space with (semi)measure M(x)).

*> 1) Halting theorem implies the set of halting programs is of measure
*

*> zero within the set of all programs
*

You mean, given a uniform measure? But why should you need the halting

theorem for this trivial statement? In fact, what exactly do you mean

by

halting theorem? (Schoenfield's book provides a good intro to

mathematical

logic and computability theory.)

*> The GP program (aka Universal Dovetailer) is not the simplest
*

*> thing. The simplest describable thing is the set of all descriptions,
*

*> that simply exist without being generated. That has precisely zero
*

*> information or complexity, whereas the UD has some complexity of the
*

*> order of a few 100 bits. The simplest prior is the uniform one, ie
*

*> there is no bias whatsoever in the selection.
*

Exist without being generated? Precisely zero information or

complexity? But with respect to what? Any traditional definition of

information/simplicity requires the specification of a formal

description

method, such as a Turing machine. You don't need one? Then how do you

define information or complexity? How exactly do you define "simple" ?

Received on Fri Oct 26 2001 - 06:46:43 PDT

Date: Fri, 26 Oct 2001 15:45:48 +0200

Sure, I agree, measures are useful; I'm using them all the time. But in

general they are _not_ the same thing as probability distributions.

Russell, this is really too much; please read any intro to measure

theory,

e.g., the one by Halmos. Measures in general are _not_ probability

distributions. They are more than that; you can view them as _sets_ of

probability distributions on sets that are related to each other in a

specific way. Probability density functions define measures and

therefore

in general aren't probability distributions either. The uniform PDF on

[0..1] yields the perfect trivial example; that's exactly why I used it

before. In the computability context, compare LI/Vitanyi 1997, chapters

4.3 (discrete sample space with (semi)distribution m(x)) vs chapter 4.5

(continuous sample space with (semi)measure M(x)).

You mean, given a uniform measure? But why should you need the halting

theorem for this trivial statement? In fact, what exactly do you mean

by

halting theorem? (Schoenfield's book provides a good intro to

mathematical

logic and computability theory.)

Exist without being generated? Precisely zero information or

complexity? But with respect to what? Any traditional definition of

information/simplicity requires the specification of a formal

description

method, such as a Turing machine. You don't need one? Then how do you

define information or complexity? How exactly do you define "simple" ?

Received on Fri Oct 26 2001 - 06:46:43 PDT

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