Bienayme

Ir�n�e-Jules Bienaym�

b. 28 August 1796, Paris, France
d. 19 October 1878, Paris

Sometime around 1834, Bienaym� became an active member of the Societ� Philomatique de Paris; many of his papers were published in its journal. In all, thirty-six published articles were written by Bienaym�. Those which appeared in L'Institute, Paris were printed first in the Extraits des Proc�s-Verbaux de la Societ� philomatique de Paris. Those papers published by the Philomatic Society typically had little mathematical detail and for this reason they are unsatisfactory.

The list of publications has been taken from that compiled by Heyde and Seneta: I.J. Bienaym�, Statistical Theory Anticipated, Springer-Verlag 1977. Since Heyde and Seneta fully summarize his researches, there is little annotation below and the reader is directed to that work.

Demography and Social Statistics

(1829) "Mortalit� des nouveaux-n�s." Moniteur Universel, 18 April 1829, p. 564.

(1835) "Sur les erreurs pr�sum�es des documents � l'aide desquels on a calcul�, en France, les tables de population." Comptes Rendus Acad. Sci., Paris, 1, p. 364.

(1837) "De la dur�e de la vie en France depuis le commencement du XIX^e si�cle." Annales d'hygi�ne publique et de m�decine l�gale, 18, pp. 177-218.

This particular paper examines the validity of the mortality table of Antoine Deparcieux and that of Emmanuel Etienne Duvillard in his time. The first table may be found in the Essai sur les probabilit�s de la duree de la vie humaine (1746). The second is in the Analyse et Tableaux de l'influence de la petite v�role sur la mortalit� � chaque age ( 1806). Bienaym� mentions in particular three other works as well. These are Recherches et consid�rations sur la population de La France by Moheau (1778), Recherches sur la population (1766) by Messance and especially Poisson's "M�moire sur la proportion des naissances des filles et des gar�ons".

The technique is most interesting in that the census counts of 20 year old males made for army recruitment and which reflect births twenty years earlier are used for this validation.

(1837) "Naissances des enfants dans le mariage. Soc. Philomat. Paris Extraits, Ser. 5, 153-155. L'Institut, 223, Vol. 6 (1838), 18.

(1838) "Probabilit� des jugements et des t�moignages" Soc. Philomat. Paris Extraits, Ser. 5, pp. 93-96. L'Institut, 235, Vol. 6, 207-208. This is connected to the 1st Supplement to Laplace's Th�orie Analytique des Probabilit�s. It concerns distinguishing the probability of speaking the truth when affirming a fact from that of speaking truly when denying a fact.

(1839) "Effets de l'int�r�t compos�." Soc. Philomat. Paris Extraits, Ser. 5, 60-65. L'Institut, 286, Vol. 7, pp. 208-209.

This paper, "The effects of compound interest," makes the observation that an assurance company incurs too much risk if insuring too few persons. We all know that a modest sum can grow to an enormous quantity if interest is compounded. But the forcasting of revenue and expenses depends upon events happening as expected. Companies with a small number of subscribers may have large losses early which cannot be recovered easily.

The following remarks and papers concern the viability of the Benefit Society of Metz. This society sold annuities that were, in the mind of Bienaym�, not fully funded. We find him engaged in an attempt to get the managers of the fund to make changes to remedy the situation.

(1857) [Untitled remarks on a report by M. Didion concerning the calculation of pensions by the Soci�t� de Metz, 1855-1859]. Comptes Rendus Acad. Sci., Paris, 44, 573.

(1862) "Remarques sur les Soci�t�s de Secours mutuels." Comptes Rendus Acad. Sci., Paris, 54, pp. 536-537.

(1862) [Untitled remarks following the paper Soci�t� de Metz, 1862]. Comptes Rendus Acad. Sci., Paris, 54, 796.

(1862) "Remarques sur les Soci�t�s de Secours mutuels." Comptes Rendus Acad. Sci., Paris, 54, pp. 889-891.

(1865) [Untitled remarks in connection with the presentation of a pamphlet entitled "Calcul des pensions dans les Soci�t�s de pr�voyance" by M. Didion] Comptes Rendus Acad. Sci., Paris, 60, p. 415.

Statistical Trends

(1838) "M�moire sur la probabilit� des r�sultats moyens des observations; d�monstation directe de la r�gle de Laplace." M�m. pres. Acad. Roy. Sci. Inst. France, 5, 513-558. "Th�or�me sur la probabilit� des r�sultats moyens des observations" Soc. Philomat. Paris Extraits, Ser. 5, 42-49. L'Institut, 284, Vol. 7, pp. 187-189. "Calcul des probabilit�s" Soc. Philomat. Paris Extraits, Ser. 5, 18-22. L'Institut, 326, Vol. 8, pp. 111-112.

The problem is this: What is the probability of the deviation of which a mean is able to be affected among any great number of different values of a phenomenon or of an event, given by so many observations, when one is not aware of the law of the probabilities of the partial deviations of each observation?

I have not attempted to correct the work of Bienaym� and it appears as published (subject to my own errors of translation and reproduction, of course).

(1839) "Th�or�me sur la probabilit� des r�sultats moyens des observations" Soc. Philomat. Paris Extraits, Ser. 5, 42-49. L'Institut, 284, Vol. 7, pp. 187-189. Here the concern is the situation where probability distribution associated with a sequence of observations will change during the course of collecting data.

(1840) "Calcul des probabilit�s" Soc. Philomat. Paris Extraits, Ser. 5, 18-22. L'Institut, 327, Vol. 8, pp. 119-120. This is on the understanding of the meaning of the phrase "the constancy of mean results."

(1840) "Probabilit�s. Principe nouveau du calcul des probabilit�s avec ses applications aux sciences d'observation" Soc. Philomat. Paris Extraits, Ser. 5, 37-43. L'Institut, 333, Vol. 8, pp. 167-169. A sequence of observations may be partitioned into a sequence of subsequences each of which (under certain circumstances) should behave as the entire sequence.

Miscellaneous papers on statistics

(1840) "Application � la statistique m�dicale" Soc. Philomat. Paris Extraits, Ser. 5, 10-13. L'Institut, 322, Vol. 8, pp. 78-79. This paper discusses in a very superficial way the problems associated with determining the effectiveness of a treatment.

(1840) "Calcul des probabilit�s. Solution d'une probl�me du calcul des probabilit�s se rettachant avec �lections" Soc. Philomat. Paris Extraits, Ser. 5, 23-26. L'Institut, 327, Vol. 8, pp. 119-120. One supposes the very great number of the electors of a great country divided between two opinions in a known ratio; one supposes moreover the electors apportion at random into numerous colleges, and one demands what is, with a great probability, the number of the colleges in which the majority will belong to the opinion which possesses a known plurality in the electoral body.

(1840) "Quelques propri�t�s des moyens arithm�tiques de puissances de quantit�s positives." Soc. Philomat. Paris Extraits, Ser. 5, 67-68. L'Institut, 342, Vol. 8, pp. 216-217.

(1841) "Notice sur un nouveau quanti�me perp�tuel." Comptes Rendus Acad. Sci. Paris, 13, 1103.

(1843) [Untitled communication in the area: Histoire des math�matiques, concerning Pascal] Soc. Philomat. Paris Extraits, Ser. 5, 49-50.

(1845) "De la loi de multiplication et de la dur�e des familles." Soci�t� philomatique de Paris. Extraits des proc�s-verbaux des s�ances. 5th ser., 10, pp. 37-39. Also in L'Institut, Paris. No. 589, Vol. 13. pp 131-132. On the extinction of families. This anticipates the theory of branching processes and gives the Criticality Theorem.

Least Squares

(1852) "M�moire sur la probabilit� des erreurs d'apr�s la m�thode des moindres carr�s," Journal des math�matiques pures et appliqu�es, 17, pp. 33-78; Reprinted in M�moires present�s par savants �trangers � l'Acad�mie des sciences. 15 (1868), pp. 615-663. See also Antoine Augustine Cournot's Exposition de la Th�orie des Chances which is cited by Bienaym� in this paper. The concern of Bienaym� here is the correct estimation of probabilites when there are several random variables involved. See Comptes Rendus Acad. Paris 34, pp. 90-92 or Liouville's Jour. Math. 17, pp. 31-32 for a review: "Sur un M�moire de M. Jules Bienaym�, concernant la probabilit� des erreurs d'après la m�thode des moindres carr�s.". Also see Meyer's Calcul des Probabilit�s pp. 377-408.

(1853) "Sur les diff�rences qui distinguent l'interpolation de M. Cauchy de la m�thode des moindres carr�s, et qui assurent la sup�riorit� de cette m�thode." Comptes Rendus Acad. Sci., Paris, 37, 5-13. Liouville's J. Math. Pures Appl. (1) 18, 299-308. Read 4 July 1853. This paper refers back to the 1835 paper of Cauchy, "M�moire sur l'interpolation." Bienaym� is very concerned that no one confuse Cauchy's method of solving systems of equations with the method of least squares. He argues that the method of elimination of Cauchy is no different from that used by Gauss. Moreover, he observes that Cauchy shows in the paper "M�moire sur l'�valuation d'inconnues d�termin�es par un grand nombre d'�quations approximatives du premier degr�" that his own method can be used to computed the least squares values.

(1853) "Remarques de M. Bienaym� � l'occasion des Notes inser�es par M. Cauchy dans les Comptes Rendus de deux des s�ances pr�c�dentes." Comptes Rendus Acad. Sci., Paris, 37, pp. 197-198. Read 8 August 1853.

(1853) "Consid�rations � l'appui de la d�couverte de Laplace sur la loi de probabilit� dans la m�thode des moindres carr�s," Comptes Rendus... de l'Acad�mie des sciences, 37, pp. 309-324. Reprinted in Journal des math�matiques pures et appliqu�es, 2nd ser., 12 (1867), pp. 158-176. Read 20 August 1853. Here Bienaym� defends Laplace against Cauchy. See on the page devoted to Cauchy a discussion of the controversy between the methods of interpolation and of least squares. Of interest also is a demonstration that least squares is superior to all other least positive integer powers. We should also note that in this paper he derives the Bienaym�-Chebyshev Inequality. See also "Des valeurs moyennes," by Chebyshev.

(1855) "Sur un principe que M. Poisson avait cru d�couvir et qu'il avait appel� Loi des grands nombres." Comptes Rendus Acad. Sci. Morales et Politiques (3) 11, pp. 379-389. Also J. Soc. Statist. Paris, (1876), pp. 199-204. Here Bienaym� denies the existence of the Law of Large Numbers. He refers first to the Poisson's two papers delivered to the Academy: "Recherches sur la probabilit� des jugements, principalement en mati�re criminelle." and "Note sur la loi des grands nombres" and also to his own paper "Th�or�me sur la probabilit� des r�sultats moyens des observations. Sur la probabilit� des r�sultats moyens lorsque les causes sont variables durant les observations."

Other papers

(1851) De la mise � alignement des maisons. Publications de la Revue municipale. Dondey-Dupr�, Paris (24 pp).

(1852) Notice sur les travaux scientifiques de M. I.-J. Bienaym�. Bachelier, Paris (4 pp.)

(1870) "Traduction de deux passages de Stob�e inexpliqu�s jusqu'ici." Comptes Rendus Acad. Sci., Paris, 71, pp. 460-462.

(1871) "Rectification de listes d'articles d�tach�s de M. Cauchy, publi�es dans deux Catalogues diff�rents, et restitution � M. Cournot de quelques-uns de ces articles." Comptes Rendus Acad. Sci., Paris, 72, pp. 25-29.

(1873) "Rapport sur le Concours pour le prix de Statistique, fondation Montyon." Comptes Rendus Acad. Sci., Paris, 75 , pp. 1306, 1349 - s�ance du 25 nov. 1872). Liouville's J. Math. Pures Appl., (2) 18 , pp. 164-179.

(1874) "Sur une question de probabilit�s." Bull. Soc. Math. France, 2, pp. 153-154. Bienaym� considers the distribution of extrema in a sequence of observations, essentially a test for randomness. This same problem is applied in the following.

(1875) "Application d'un th�or�me nouveau du calcul des probabilit�s." Comptes Rendus Acad. Sci., Paris, 81, pp. 417-423. Bull. Math. Astr., 9, pp. 219-225. Apparently first published in Bull. Soc. Math. France, 2 (1874). pp. 153-154.