RE: [Fis] Re: miscellanea / temperature / symmetry

Dear Guy,

I can follow Michel, but I don't understand your intervention.
Probabilistic entropy is a measure of dividedness, not of symmetry.
Symmetry is a geomterical property, while probabilistic entropy provides
an information calculus. Perhaps, a numerical example would be helpful.
Can you provide one?

For example, if we take a symmetrical distribution of relative
frequencies like 3,2,1,1,2,3, the entropy in this distribution would be
similar to any other order of it. For example: 1,1,2,2,3,3. One can use
probabilistic entropy for optimization of the dividedness, for example,
in grouping and clustering. But I cannot easily think of an entropy
measure that would enable us to distinguish between these two series.

Can you -- or perhaps Shu-Kun -- explain why you nevertheless argue in
this way? It is not sufficient to tell us that one concept is "more
powerful" than another without a demonstration.

With kind regards,

Loet

> -----Original Message-----
> From: fis-bounces@listas.unizar.es
> [mailto:fis-bounces@listas.unizar.es] On Behalf Of hoelzer@unr.edu
> Sent: Thursday, April 22, 2004 6:08 PM
> To: fis@listas.unizar.es
> Subject: Re: [Fis] Re: miscellanea / temperature / symmetry
>
>
> Dear Michel et al.,
>
> On Apr 22, 2004, at 1:56 AM, Michel Petitjean wrote:
>
> > About symmetry and its relations to entropy: symmetry theory and
> > entropy (information theory) are both related to
> distributions. But,
> > symmetry theory has to work for distributions, discarding
> whether they
> > are finite discrete, or infinite, or continuous.
> Informational entropy
> > is mainly connected to finite discrete distributions, despite some
> > extensions in the continuous case. Much more problematic is
> that the
> > entropy associated to a discrete distribution does not care of the
> > numerical values to which non null probabilities are
> attached, and in
> > fact, entropy exists even if the probabilities are defined on a non
> > numerical space: e.g.: P(red)=1/3, P(green)=1/3, P(blue)=1/3 is a
> > distribution for which entropy can be calculated. This is false for
> > symmetry calculations, which deals mainly with euclidean
> multivariate
> > distributions. So, entropy and symmetry are quite different.
>
> It seems that you have staked out a position that is entirely at odds
> with the view advocated earlier by Shu-Kun. My own view is very much
> aligned with that of Shu-Kun, so I would like to consider how your
> rational argument leads to such a contradictory conclusion. It seems
> to me that the source of the difference lies in the
> distinction between
> the concept of entropy and our tools for estimating its value. You
> might argue that the concept of entropy was originally derived as a
> statistical calculation, so that subsequent development of
> the concept
> distorted its correct meaning; but I would argue that its conceptual
> development represents an ordinary and heuristically valuable
> aspect of
> science. This process has arguably led to a more useful concept of
> entropy (e.g., one that includes meaning related to
> symmetry), despite
> an increase in conceptual ambiguity and divergence between
> the concept
> and its original measure. I think that the relationship between
> concept and measure has converged on the kind of relationship that we
> often encounter when theory provides a concept first. In
> this case, a
> variety of measures may later be advocated to represent the concept,
> but it is recognized that each measure is only an error-prone
> approximation of the concept. As an evolutionary biologist I would
> point to the concept of fitness and its measures as an example.
>
> In sum, I think that the growth and development of the concept of
> entropy beyond the scope of its traditional measures to include
> concepts like symmetry has been heuristically positive,
> although I also
> recognize the difficulties that this introduces.
>
> Cheers,
>
> Guy Hoelzer
>
> Department of Biology
> University of Nevada Reno
> Reno, NV 89557
>
> Phone: 775-784-4860
> Fax: 775-784-1302
>
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>

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Received on Thu Apr 22 20:35:25 2004