[Fis] Re: Data, observations and distributions / was: Re: Probabilistic Entropy

From: Stanley N. Salthe <[email protected]>
Date: Tue 20 Apr 2004 - 23:54:13 CEST

Commementing upon the Michel's posting:
     It is interesting that theoretically one may morph one probability
density function into another by changing aspects of its logical form.
What you get is a suite of different functions which seem fairly complete
with respect to the possibilities of the data -- normal/poisson, more
dispersed, more compressed, skewed, multimodal, etc.
     Unless it is possible to collect data that do not fit any known
distribution, statistical theory must be complete. My point is that what
we deal with, even in empirical science, is our own construction, which I
call Nature, as opposed to the World, which is largely unknown (likely
unknowable).

STAN

>To: <fis@listas.unizar.es>
>Subj: Data, observations and distributions / was: Re: Probabilistic Entropy
>
>Dear Loet,
>
>I see what you mean when you write:
>> The data are
>> measurement results which can be considered as probability
>> distributions. Thus, the expected information content of these
>> distributions and the meaning which these are given in (highly codified)
>> discourses provide the basis of science.
>
>But I do not agree. Is there any probabilist confusing the empirical
>distribution defined from a sample, and the values themselves ?
>(are there probabilists among FISers ?).
>Would you confuse a distribution with its observations, for a continuous
>distribution (e.g. a gaussian) ? Surely no. Same thing for a Poisson
>distribution (infinite discrete). The confusion arises in the finite
>discrete case. In Probability, the distribution associated to a random
>variable X exists, discarding if observations are made or not. For a
>finite discrete r.v. taking equiprobable values such that P(X=xi)=1/N,
>we still have no observations, unless we perform the "experiment",
>and even if getting a sample of size N of X, we are not ensured
>to observe exactly one time each value xi (may be x1 appears twice,
>may be x2 is not observed,...). The empirical distribution based on
>data is a special case, in which the N observed values (I assume the
>usual euclidean case) are input in the definition of a probability
>law and its distribution, for which each value has probability 1/N.
>And even here, we cannot say that the N data "are" a sample of the
>empirical law, even if observing such a sample has a certain
>probability to occur. The data are measured, then the empirical
>distribution is built.
>The distributions exist, in general, outside any experiment (in the
>probabilistic sense). Once the experiment is done, we have observations
>(which merit together the name of "sample" under some conditions).
>Now returning to science, and having, say N data xi (e.g. points in R^d):
>they are not observations of any probability law: building a r.v. X
>with a distribution such that P(X=xi)=1/N, is just a step in assigning
>a mathematical model to the physical phenomenon from which the N data
>were measured. And most time, the empirical distribution has little
>interest by itself: the modeller will look if the N data, considered
>as if they are a sample of some parent population, let him to
>know something about the parent distribution (may be gaussian, may
>be anything) and its associated parameters.
>Information is attached to a distribution, with or without any
>probabilistic or physical experiment. Measures are just physical data.
>The relations between data and distributions exist in the spirit of the
>modeller. I would say that the probabilistic entropy H, when it exists,
>is just a parameter of a distribution, as the mean, the median or the
>extreme values.
>
>Michel Petitjean Email: petitjean@itodys.jussieu.fr
>Editor-in-Chief of Entropy entropy@mdpi.org
>ITODYS (CNRS, UMR 7086) ptitjean@ccr.jussieu.fr
>1 rue Guy de la Brosse Phone: +33 (0)1 44 27 48 57
>75005 Paris, France. FAX : +33 (0)1 44 27 68 14
>http://www.mdpi.net http://www.mdpi.org
>http://petitjeanmichel.free.fr/itoweb.petitjean.html
>http://petitjeanmichel.free.fr/itoweb.petitjean.freeware.html
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Received on Tue Apr 20 22:24:21 2004

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