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Old 07-21-2011, 12:13 AM
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Trivial Hijack!

Quote:
Originally Posted by Peter Kennedy
It is extremely convenient to assume that errors are distributed normally, but there exists little justification for this assumption... Poincare is said to have claimed that "everyone believes in the [Gaussian] law of errors, the experimenters because they think it is a mathematical theorem, the mathematicians because they think it is an empirical fact.
The provenance of this quote has been less than clear for a few decades, but google resolves the matter. Poincare quotes a conversation with Gabriel Lippmann in Calcul des probabilités (1912). Others had difficulty locating the precise citation.

Boring Historical Details

Here's the original french:
Quote:
Originally Posted by Poincaré in Calcul des probabilités
108. Cela ne nous apprendrait pas grand'chose si nous n'avions aucune donnée sur phi et psi. On a donc fait une hypothèse sur phi, et cette hypothèse a été appelée loi des erreurs.
Elle ne s'obtient pas par des déductions rigoureuses ; plus d'une démonstration qu'on a voulu en donner est grossière, entre autres celle qui s'appuie sur raffirmation que la probabilité des écarts est proportionnelle aux écarts. Tout le monde y croit cependant, me disait un jour M. Lippmann, car les expérimentateurs s'imaginent que c'est un théorème de mathématiques, et les mathématiciens que c'est un fait expérimental.
Voici comment Gauss y est arrivé.
Lorsque nous cherchons la meilleure valeur à donnera a z, nous n'avons pas d'autre ressource que de prendre la moyenne entre x1, x2, ...,xn en l'absence de toute considération qui justifierait un autre choix. Il faut donc que la loi des erreurs s'adapte à cette façon d'opérer. Gauss cherche quelle doit être phi pour que la valeur la plus probable soit la valeur moyenne.
Emphasis in original: p. 170-71. Here is one translation combining google translate, my dismal high school french and some poetic license:
This doesn't teach us much if we don't have data on phi and psi. So we form an hypothesis for phi and call it The Law of Errors.

That can't be deduced rigorously, though there are demonstrations such as the probability of the deviations is proportional to the differences.1 Everyone believes however, as Gabriel Lippmann told me one day, the experimenters because they imagine that it is a mathematical theorem and the mathematicians because it is an experimental fact.

Here's how Gauss did it.

When we seek the best value to give to z, we have no alternative but to take the average of x1, x2, ..., xn in the absence of any considerations that would justify a choice. Therefore the law of errors fits this mode of operation. Gauss looks for what should be the most likely value of phi, the mean value.
And here's my paraphrase: as the great physicist Gabriel Lippmann once told Poincare, "Everybody believes in the Gaussian Law of Errors, the experimenters because they imagine that it is a mathematical theorem and the mathematicians because they believe it is an empirical fact."


1Translators note: huh?
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