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| This article ''Iren&eacute;e-Jules Bienaym&eacute;'' was adapted from an original article by E. Seneta, which appeared in ''StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies''. The original article ([<nowiki>http://statprob.com/encyclopedia/Irenee-JulesBienaymeBienayme.html</nowiki> StatProb Source], Local Files: [[Media:Bienayme.pdf|pdf]] | [[Media:Bienayme.tex|tex]]) is copyrighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the [[:Category:Statprob|Category StatProb]].
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'''Iren&eacute;e-Jules BIENAYM&Eacute;'''

b. 28 August 1796 - d. 19 October 1878

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'''Summary.''' Bienaym&eacute; was a Civil Servant. A disciple of Laplace, he
proved the Bienaym&eacute;-Chebyshev Inequality some years before Chebyshev, and
stated the Criticality Theorem of branching processes completely correctly in
1845. His work on corrrecting the use of the Duvillard life table is perhaps his
greatest achievement as statistician in the public domain.

Bienaym&eacute; was born in Paris but began his secondary education at the ''lyc&eacute;e''
in Bruges, then part of the French Empire.  His father held a senior administrative
position in this town before moving his family back to Paris.  Bienaym&eacute; entered the
&Eacute;cole Polytechnique in 1815, but this institution was closed in 1816 due to the
fall of the Empire and the return of the Bourbons.  With the death of his father in
1816, he entered the Ministry of Finances and rose to the rank of Inspector General in
1836.  While carrying out his responsibilities as public servant, he was to become a
self-made scientist, publishing first on demography and actuarial matters, and then on
mathematical statistics.  He was elected to the Société Philomatique de Paris
in January, 1838 and was active in its affairs.  His contributions to its meetings were
reported in the now-obscure newspaper-journal ''L'Institut, Paris'', being reprinted at
the end of the year in the collections ''Procès-Verbaux de la Société
Philomatique de Paris - Extraits''.  Most of his publications in the period 1837 to 1845
appear in this medium, and are characterized, to the frustration of the reader, by lack
of mathematical proofs for assertions sometimes far ahead of their time.  The most
startling of his contributions occurs in this way when he gives, in 1845, a completely
correct statement of the Criticality Theorem for simple branching processes, which
precedes the partly correct one of [[Galton, Francis|F. Galton]] and H.W. Watson by over 30 years and the
first subsequently correct one by over 80 (Heyde and Seneta, 1972; Bru, Jongmans and
Seneta, 1992). (This theorem describes how the probability, $ q $, of extinction of a surname
depends on the average number, $ m $, of male children per male parent. If $ m \leq 1 $
then $ q=1 $, but if  $ m>1 $ then $ q<1 $, and so there is a positive probability of
survival of surname.) In a letter to [[Quetelet, Adolphe|Quetelet]] of 21 April 1846, Bienaymé
confides that his
everyday work and the state of his health do not permit him complete preparation of his
writings for publication, and that he works seriously on applications which are of
interest to both of them.  His ill-health, especially his trembling hands, were to
plague him to the end of his life.  (Quetelet, born the same year as Bienaym&eacute;, had
shortly before the letter paid a visit.  Their contact was to
continue, with Bienaym&eacute;'s last letter to Quetelet dated September 1871).

In 1848 Bienaym&eacute; lost his job in the Ministry of Finances for political reasons
associated with the changes of regime.  Shortly afterwards he was asked to give some
lectures on probability at the Faculté des Sciences, Paris.  Again due largely to
politics the Chair for probabilities was finally given to Lam&eacute; who began
his course in November, 1850, and spoke thus on 26 April, 1851:

<blockquote>
It is my pleasure to count among my friends a savant (M. Bienaym&eacute;) who today,
almost alone in France, represents the theory of probabilities, which he has cultivated
with a kind of passion, and in which he has successively attacked and destroyed errors.
It is to his counsels that I owe a proper understanding...
</blockquote>

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Finally Bienaym&eacute; was reinstalled in August 1850 as "Inspecteur
g&eacute;n&eacute;ral des finances, charg&eacute; du service des retraites pour la vieilesse et
des soci&eacute;t&eacute;s des secours mutuels".  Although he finally resigned in April 1852,
his applied statistical interests were continued in the context of the Paris Academy of
Sciences (to which he was elected as "acad&eacute;micien libre'' in July, 1852), where
he was referee for 23 years for the Prize of Statistics of the Montyon Foundation, the
highest French award in the area.  His eminence for such a role was enhanced by the
fact that he had worked hard to correct the state of affairs where upto about 1837 many
insurance companies in France had used the Duvillard life table to considerable financial
advantage, and the correction is deemed by some as his greatest achievement in the
public domain.

The period 1851-1852 also contains Bienaym&eacute;'s early contacts with J.J. Sylvester
(1814-1897) and [[Chebyshev, Pafnutii Lvovich|Chebyshev]], and his contribution to the enhancement of their
international standing.  The contact with Chebyshev was to become particularly
significant.

For Bienaym&eacute;, Laplace's ''Th&eacute;orie analytique
des probabilit&eacute;s'' of 1812 was the guiding light, and much of his work is concerned
with elaborating, generalizing and defending Laplacian positions.  When the first
treatise on probability in Russian (Buniakovsky's ''Foundations of the Mathematical
Theory of Probabilities'', clearly modelled on Laplace) appeared in 1846, one
biographer of Buniakovsky claims that Bienaym&eacute; and Gauss both learned Russian in
order to be able to read it.  (Certainly the linguistically gifted Bienaym&eacute; knew
Russian.)  Bienaym&eacute; was passionate in the defence of scientific truth as he
perceived it and of his friends such as [[Cournot, Antoine Augustin|Cournot]], to the extent of attacking
Cauchy and [[Poisson, Simeon-Denis|Poisson]]. [[Bertrand, Joseph|J. Bertrand]] (1822-1900), author of ''Calcul des
Probabilités'', a powerful Macchiavellian figure, eventually helped `bury' Bienaymé's
reputation by unjustified criticism.  Contributing to his being largely forgotten till
the 1960's were the facts that Bienaymé was modest as regards his own achievements, made
no great efforts to assert his priority, and was ahead of his time in mathematical
statistics.  He left no disciples, not being in academia; and wrote no book.
However, more recently interest has revived, and on the 200th anniversary of
his year of birth, at a conference in Paris, some 12 papers on his life and work were
presented, in the presence of representatives of the still flourishing family
Bienaymé.

It is appropriate to say something of the famous and useful
Bienaym&eacute;-Chebyshev Inequality, more commonly known by Chebyshev's name
alone.  Both Bienaym&eacute; in 1853 and Chebyshev in 1867 proved it for sums of
independent random variables.  Bienaym&eacute;'s proof, the simple proof which we
use today, is for identically distributed random variables, treating the sample
mean $\bar X$ in its own right as a single random variable, and is within his 
best known paper ``Consid&eacute;rations \`{a} l'appui de la d&eacute;couverte de
Laplace sur la loi de probabilit&eacute; dans la m&eacute;thode des moindres carr&eacute;s." 
Chebyshev's proof is for discrete random variables and is rather more involved. 
Bienaym&eacute;'s paper of 1853 is reprinted in 1867 in Liouville's journal immediately
preceding the French version of Chebyshev's paper.  The aim of both authors was a general
form of the Law of Large Numbers.  Eventually, in a paper presented at a
conference in France and published in Liouville's journal in 1874, Chebyshev
acknowledges Bienaym&eacute;'s priority, and extracts from Bienaym&eacute;'s approach what is
the essence of the ``Method of Moments".  Chebyshev in 1887 used this method to give an
incomplete proof of the Central Limit Theorem for sums of independent but not
identically distributed summands, his final and great achievement in probability theory.
This proof was then taken up and generalized by his student [[Markov, Andrei Andreevich|Markov]].

In the context of one of the polemics between Markov and P.A. Nekrasov (1853-1924) 
in response to a statement by Nekrasov that the idea of Bienaymé is exhausted 
within the works of P.L. Chebyshev, Markov says:

<blockquote> 
quad The reference here to Chebyshev is misleading, and the statement of
P.A. Nekrasov that the idea of Bienaym&eacute; is exhausted is contradicted by a
sequence of my papers containing a generalization of the method of
Bienaym&eacute; to settings which are not even touched on in
the writings of P.A. Nekrasov.
</blockquote>

The first paper which Markov lists, published in Kazan, is that in which Markov chains
first appear in his writings as a stochastically dependent sequence for which the 
Weak Law of Large Numbers holds.  This paper was written to contradict an assertion 
of Nekrasov that independence was a necessary condition for this law.  Thus 
according to Markov, Bienaymé might well be regarded as playing a role in the 
evolution of Markov chain theory.

The Method of Moments, however, like the Inequality, has come to be ascribed to
Chebyshev.

To conclude, here is an extract from a letter written by Bienaymé on 5 April 1878,
just before his own death, to E.C. Catalan (1814-1894).  It is a testament, prophetic
and a guide for our own times, with a touch of the old fire so evident in his
controversies.

<blockquote>
You do not see then that everything in the world is only probabilities, or even just
conjectures; and that in days to come all questions, more or less scientific, will be
better understood, or even solved [in these terms] when sufficient education is 
given to minds capable of it by it good teaching of probability.  I don't say to 
all minds, as there are weak intellects, and a great number of fools...
</blockquote>


====References====
{|
|-
|valign="top"|{{Ref|1}}||valign="top"|   Bru, B., Jongmans, F. and Seneta, E. (1992). I.J. Bienaymé:  Family information and the proof of the criticality theorem. ''International Statistical Review'', '''60''', 177-183. 
|-
|valign="top"|{{Ref|2}}||valign="top"|   Centre d'Analyse et de Mathématique Sociales (1997).  ''Irenée-Jules Bienaymé, 1796-1878''. Actes de la journée organisée le 21 juin 1996. C.A.M.S.-138.  Série ``Histoire du Calcul des Probabilités" No.28, 124 pp. (54 Boulevard Raspail, 75270 PARIS Cedex 06).  
|-
|valign="top"|{{Ref|3}}||valign="top"|   Heyde, C.C. and Seneta, E. (1972). The simple branching process, a turning point test and a fundamental inequality: A historical note on I.J. Bienaymé.  ''Biometrika'' '''59''', 680-683. 
|-
|valign="top"|{{Ref|4}}||valign="top"|   Heyde, C.C. and Seneta, E. (1977). ''I.J. Bienaymé : Statistical  Theory Anticipated'', Springer, Berlin. 
|-
|valign="top"|{{Ref|5}}||valign="top"|   Jongmans, F. and Seneta, E. (1993). The Bienaymé family history from archival materials and background to the turning-point test.  ''Bulletin de la Société Royale des Sciences de Liège'', '''62''', 121-145. 
|-
|valign="top"|{{Ref|6}}||valign="top"|   Seneta, E. (1982). Bienaymé, Irenée-Jules. ''Encyclopedia of  Statistical Sciences'' (S. Kotz and N.L. Johnson, eds.) Wiley, New York. Vol. 1, pp.231-233.
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<references />

Reprinted with permission from
Christopher Charles Heyde and Eugene William Seneta (Editors),
Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.

[[Category:Statprob]]
[[Category:Biographical]]