Laplace, Pierre Simon De
Laplace, Pierre Simon De
Pierre Simon de Laplace (1749-1827), renowned French mathematician, was born in Beau-mont-en-Auge, a village 4 miles west of Pont 1’Eveque in Normandy. He was the second of two children of Pierre de Laplace, a syndic of the parish, who owned and farmed a small estate. When Laplace arrived in Paris, barely twenty years old, he had finished his studies and begun his own research. His ability soon impressed d’Alembert, whose disciple he was to become. D’Alembert’s patronage secured Laplace a position as a teacher at the Ecole Royale Militaire, where he remained until changes in the organization brought his teaching there to an end. In 1783 he became an artillery inspector (Duveen & Hahn 1957), and in this capacity he made the acquaintance of the young Bonaparte. A mutual respect developed, and from then on Laplace enjoyed Bonaparte’s increasingly powerful support. The two became colleagues at the Academie des Sciences; as first consul, Napo-leon appointed Laplace minister of the interior, a position he gave up shortly afterward to become chancellor of the Senat Conservateur. In 1814, however, he turned against Napoleon; he was eventually made a marquis by Louis xvm, a surprising but not uncommon turn of events in those troubled times.
Laplace achieved distinction not only in mathematics; it was his literary style that won him election to the Academie Franchise in 1816. His famous sentence on the hidden determinism of natural laws is a good example of his style:
If there were an intelligence that for a given instant could comprehend all the forces that animate nature and the condition of each being that composes it; if, moreover, this intelligence were sufficiently great to submit these data to analysis, it would create a single formula that would embrace both the movements of the vastest bodies of the universe and those of the smallest atoms: to this intelligence nothing would be uncertain, and the future, as the past, would be present to its eyes. ([1814] 1951, p. 4)
Although Laplace’s work has literary elegance, his demonstrations are not always rigorous; often they are even obscure. He frequently wrote “it is clear that … ’ in place of long and difficult calculations.
Laplace presided over the famous Société d’Arcueil, one of the informal scientific societies that flourished in the nineteenth century. It took its name from Laplace’s estate, where he and the great chemist Claude Louis Berthollet periodically received their students to discuss scientific questions in an informal atmosphere.
Although our primary concern here is with Laplace’s contributions to probability and statistics, it is worth noting that the full scope of his work includes physics (the theory of capillary phenomena, the exact formula for the speed of sound), pure mathematics, and celestial mechanics (he was dubbed a “second Newton”). In probability and its applications Laplace systematized and further ex-tended the scattered researches of his predecessors, bringing the subject to full flower in the third edition (1820) of his great treatise, Theorie analytique des probabilites (1812). This is why Tod-hunter wrote: “On the whole, the Theory of Prob-ability is more indebted to him than to any other mathematician” ([1865] 1949, p. 464).
Laplace was not content to make important discoveries; he also thought it necessary to communicate them to a wide public. To this end he wrote two popular works addressed to the intelligent and educated general reader, Exposition du systeme du monde (1796) and Essai philosophique sur les probabilites (1814). Other French mathematicians have continued this practice; thus, Emile Borel thought it useful to write an analogous work, Le hasard (1914), which took into consideration the progress made since Laplace.
Generating functions and characteristic functions . Motivated by problems arising, for instance, in the mathematical treatment of games of chance, Laplace, in the epochal Memoire sur les suites (1782), developed the general theory of a powerful power-series technique for solving finite-difference equations, or recurrence relations, which he termed the “method of generating functions” (calcul des fonctions génératrices; ibid., p. 1). In Book 1, Part 1, of his Theorie analytique des probabilités (1812) he reproduced this Mémoire almost entirely, and in Book 2 he made repeated use of generating functions in solving a great variety of probability problems arising in the mathematical treatment of games of chance. The probability-generating function of a random variable, X—that is, the mean (or expected) value of tx, where t is a “dummy” real variable—had already been used (without being named) by De Moivre, in studies of games of chance (1730, pp. 191–197; [1718] 1756, pp. 41–43), and by Simpson and Lagrange, in their studies of the distribution of the arithmetic mean of independent observations under various laws of error (Simpson 1756; 1757; Lagrange 1770–1773). However, it is Laplace’s extensive discussion of generating functions (1782)
and the applications of them in his Théorie analytique des probabilités that is the actual source of their widespread use in probability theory, combinatory analysis, and the solution of finite-difference equations and recurrence formulas (see David & Barton 1962; Feller 1950; Frechet 1940–1943; Jordan 1939; Riordan 1958; Uspensky 1937). The invention of the characteristic function (fonction caracteristique) of the distribution of a random variable X (that is, the expected value of eitx, where i =?—1) and the associated inversion formula for deducing the distribution function of a random variable from its characteristic function are often attributed to later writers, such as A. L. Cauchy, 1789–1857, and Henri Poincaré, 1854–1912. Laplace, however, introduced this very same (characteristic) function, without assigning it a name, and gave the associated inversion formula for the case of a discrete random variable, in article 21 of Book 1 of his Théorie analytique des probabilités (it also appears in Oeuvres completes, vol. 7, pp. 83–84) and then employed such functions systematically in Book 2, Chapter 4 (cf. Molina 1930, arts. 3–4). The technical advantages possessed by the characteristic function permit simplification of many manipulations and proofs, although when X takes only integral values the generating function is usually adequate. [seeProb-ability, article onFormal Probability.]
Bayesian inference . Laplace was apparently the first to have stated in a general form what is now called Bayes’ theorem (Theorie analytique [1812] 1820, book 2, art. 1; Molina 1930, appendix I). Several scholars have deprecated Laplace’s originality in this area of “probability of causes,” or Bayesian inference. [see Bayesian inferencefor a discussion of the probability of causes.] But Laplace did introduce an essential innovation here. Bayes had considered only the case where the a priori probabilities are equal (as an evident hypothesis when one is ignorant of these probabilities); Laplace extended Bayes’ theorem to cover the general case where these a priori probabilities are not necessarily equal. What has generally not been stressed enough is that Bayes’ theorem’whether generalized or not’is really only an interpretation of a standard relationship for conditional probabilities. [seeProbability, article onFormal Probability.] It is the interpretation of the formula itself that is important here. A discussion by Molina (1930) clearly establishes Laplace’s position and demonstrates previous misunderstandings of it.
Normal distribution . Laplace should have the major credit for discovering and demonstrating the central role of the normal distribution in the mathematical theory of probability and for determining its principal mathematical properties [seeDistributions, Statistical, article onSpecial Continuous Distributions].
De Moivre in 1733 had shown how to employ integrals of the function exp(—t2) to approximate sums of successive terms of the binomial expansion of (a + b)n when n is large, obtaining what we today call the normal approximation to the binomial probability distribution [seeDistributions, Statistical, article onApproximations TO Distributions]. (De Moivre’s analysis is readily available in Smith [1929] 1959, pp. 566–575.) Laplace extended this approach in two important directions.
Approximations to the normal integral. He developed (1781) a general method for approximating an arbitrary definite integral by a series expansion in terms of integrals of the function exp (’t2) and its derivatives, anticipating by a century the so-called Gram-Charlier Type A series expansion. He then utilized this technique to approximate various discrete and continuous probability distributions arising in games of chance and other problems in the calculus of probabilities. Remarking (1786a, p. 305) that the integral of exp(—t2) arises so frequently that it would be useful to have a table of its values for a succession of limits of integration, Laplace provided (1785, sec. vi; 1805; [1812] 1820, book 1, art. 27) the now well-known power-series, asymptotic-series, and continued fraction expansions for integrals of the function exp(—t2). These many results collectively constitute Laplace’s first great contribution to the central role of the normal distribution today, and throughout their development the function exp(—t2) seems to have been regarded exclusively as an approximating function —neither De Moivre nor Laplace appears to have regarded it or any corresponding expression as a law of error or even as a probability distribution in its own right.
Central limit theorem. Laplace’s second great contribution to the establishment of the leading role of the normal distribution is his discovery and proof of what we today call the (classical) central limit theorem. De Moivre’s result of 1733 is a special case of this theorem, as are many of Laplace’s results in his papers of 1781, 1785, and 1786, but the theorem itself appears for the first time in the important Memoire (1810, sec. vi), for the case of n independent errors from a common arbitrary symmetric discrete distribution on the interval (’a, -fa). Then (ibid., sec. vi), by considering the discrete distribution corresponding to a fine subdivision of his double exponential law of error
(1774, sec. v),\m exp (-m\x\), he illustrated the extension of this important theorem to symmetric continuous distributions on the interval (—?,+?). Gauss (1809, arts. 175-177) had deduced his law of error, Cexp(-h2x2), and thence his development of the method of least squares, from the principle of the arithmetic mean. Laplace’s central limit theorem provided an alternative justification, or “proof,” of this law of error as the limit of the distribution of the sum of n independent random errors as n ? ? when the relative contribution of each to their sum tends to 0 as n increases, and thus he provided a valid basis for the method of least squares when the individual results involved “are each determined by a very large number of observations, whatever be the laws of facility of the errors of these observations,” and hence “a reason for employing it in all cases” (1810, supplement, p. 353). [seeDistributions, Statistical, articleOn Special Continuous Distributions; See alsoLinear hypotheses, article onRegression.]
Laplace’s proof had only the limited precision current in his time, but it was later made more rigorous and more general by Aleksandr Mikhailovich Liapunov, Paul Lévy, and others. Laplace’s demonstration centers on the hypothesis that one deals with the sum of independent random variables, but in applications neither independence nor summation may be appropriate. Other functions of random variables can lead to other distributions; for example, if instead of a sum one deals with the greatest random variable, entirely different limit distributions can result (Frechet 1927). One of these distributions has found application in flood protection, breaking strength of materials, and other areas (Gumbel 1958).
Estimation. Laplace’s writings contain the seeds of ideas that have been carefully studied only in recent years: optimum point estimation, hypothesis testing, and confidence interval estimation. Laplace proposed (see 1774, sec. v) that when estimating a parameter, 6, one use that function T = T(Yl), (Y2,...) of the observations Y1, Y2,... for which the mean (or expected) absolute error of estimation, E[?T— ¸?], is a minimum for the given probability distribution of errors. For the case of three independent identically distributed observations he gave an explicit algorithm for finding such a function to use for estimating the location parameter of their common distribution, f(y —¸), when this is completely specified except for the value of 0. He subsequently extended (1781, sec. xxx) this procedure (which is the same as Pitman’s method of “close estimation”—see Pitman 1939) to cover n independent observations in the one-parameter case. Gauss“s reformulation (1821) of the method of least squares in terms of minimum mean-square error of estimation (i.e., min E[(T — ¸)2]) stems directly from this earlier work of Laplace’s.
Many modern statisticians see the problem of confidence intervals as one of providing a random interval that contains, with at least a specified probability, some parameter of the distribution sampled. Laplace, however, gave another—somewhat vague—interpretation of the interval (Oeuvres completes, vol. 7, pp. 286–287). Major differences of opinion persist about the interpretation of such intervals. [seeBayesian Inference; Estimation, article onConfidence Intervals AND Regions; Fiducial Inference.]
Applications to demography . Laplace was not satisfied merely to describe useful statistical methods; he never stopped applying them, most particularly to demography. In this area he was preceded by the great naturalist Buffon, whose early career was in mathematics, although this is not generally known, and who solved the famous problem of the needle thrown onto a table covered with parallel lines. His “Essai d’arithmetique morale” of 1777 deals with demographic statistics; he notes, for example, the propensity that most people have to use round numbers in stating their age. Laplace, using more general and more precise mathematical methods and ideas, was able to treat demographic problems much more intensively. Buffon had already noted the general preponderance of male over female births; Laplace noted that during the years 1745–1785, 393,386 boys and 377,555 girls were born in Paris, and he proved that given certain natural hypotheses, the probability that the chance of a male birth would be more than J is 1 — 1/N; in this example N is a very large number, greater than 1072. In an anomalous case cited by Buffon—the birth of 203 boys and 213 girls in the village of Vitteaux over a five-year period—the same probability is only about J. Laplace pointed out that there is no logical contradiction between the two cases, since it is only for large populations that the value of the sought probability is near unity—i.e., that the event is almost certain.
An exact census of the French population would have been a very difficult undertaking in Laplace’s time, and he therefore tried to estimate the population indirectly, by a rather curious method. He used in the approximation the rough constancy of the ratio between total population and annual number of births, so that if the numerical value of the ratio is known, one need only multiply it by the number of births to obtain an estimate of the
population. The government accepted his proposal that studies be made to determine annual births and total population for thirty ddpartements, selecting in each departement only those parishes whose mayors seemed sufficiently conscientious. On September 22, 1802 (the Republic’s New Year’s Day), the inhabitants of these parishes were counted, and the number of births in each of the three preceding years was also recorded. From the resulting ratio and an estimate of the total number of yearly births in France, Laplace estimated the population to be 25 million.
Laplace was also interested in statistics having to do with the duration of marriages and with life insurance. In addition, he studied the application of statistics to problems of social order, such as the validity of trial evidence and of court judgments, and the concept of “moral expectation.” Th.s concept, introduced by Daniel Bernoulli, is based on the observation that the richer a man is, the less concerned he is about any fixed moderate sum of money. Laplace added to this generalization the qualification “all other things being equal,” meaning that the benefit to be calculated depends in general on meaning that the benefit to be calculated depends in general on an infinity of circumstances that are impossible to evaluate and that are relative to the individual who is calculating it [see Utility]. The work that Laplace did on trial evidence and court judgments has been especially controversial; J. L. F. Bertrand, in his caustic way, treated it with complete disdain, but Borel ([1914] 1948, pp. 251–262) felt that Bertrand was too harsh.
Maurice Frechet
[Directly related is the entry Probability.Other relevant material may be found in Sociology,articleOn The early history of social research;and in the biographies of Gaussand Moivre.]
WORKS BY LAPLACE
1771 Recherches sur le calcul integral aux differences infiniment petites, aux differences finies. Accademia delle Scienze di Torino, Memorie 4:273–375. ? Volume 4 is dated “1766–1769.” (1774) 1891 Memoire sur la probabilite des causes par les evenements. Volume 8, pages 27–65 in Pierre Simon de Laplace, Oeuvres completes. Paris: GauthierVillars. ? Includes “Probleme in: Determiner le milieu que Ton doit prendre entre trois observations donnees d’un meme phenomene” on pages 41–48.
(1781) 1893 Mémoire sur les probabilités. Volume 9, pages 383–485 in Pierre Simon de Laplace, Oeuvres completes. Paris: Gauthier-Villars.
(1782) 1894 Mémoire sur les suites. Volume 10, pages 1–89 in Pierre Simon de Laplace, Oeuvres completes. Paris: Gauthier-Villars.
(1785) 1894 Mémoire sur les approximations des form-ules qui sont fonctions de trés grands nombres. Volume 10, pages 209–291 in Pierre Simon de Laplace, Oeuvres completes. Paris: Gauthier-Villars.
(1786a) 1894 Suite du mémoire sur les approximations des formules qui sont fonctions de trés grands nombres. Volume 10, pages 296–338 in Pierre Simon de Laplace, Oeuvres completes. Paris: Gauthier-Villars.
(1786b) 1895 Sur les naissances, les manages et les morts á Paris, depuis 1771 jusqu’en 1784, et dans toute 1’étendue de la France, pendant les annees 1781 et 1782. Volume 11, pages 35–46 in Pierre Simon de Laplace, Oeuvres completes. Paris: Gauthier-Villars.
(1796) 1836 Exposition du systéme du monde. 2 vols. 6th ed. Paris: Bachelier.
(1805) 1880 Traité de mecanique céleste. Part 2: Théories particuliéres des mouvements célestes. Volume 4 in Pierre Simon de Laplace, Oeuvres complétes. Paris: Gauthier-Villars.
(1810) 1898 Memoire sur les approximations des formules qui sont fonctions de trés grands nombres et sur leur application aux probabilités. Volume 12, pages 301–345 in Pierre Simon de Laplace, Oeuvres complétes. Paris: Gauthier-Villars. ? A supplement to the “Memoire” appears on pages 349–353.
1812) 1820 Théorie analytique des probabilités. 3d ed., rev. Paris: Courcier. ? Also published as Volume 1 of Oeuvres completes de Laplace.
(1814) 1951 A Philosophical Study on Probabilities. New York: Dover. ? First published as Essai philosophique sur les probabilités.
Oeuvres complétes de Laplace. 14 vols. Paris: GauthierVillars, 1878-1912.
SUPPLEMENTARY BIBLIOGRAPHY
Borel, Émile (1914) 1948 Le hasard. New ed., rev. & enl. Paris: Presses Universitaires de France.
Colbert-laplace, A. 1929 Letter to Karl Pearson, Dated 16 February 1929. Biometrika 21:203-204.
Dantzig, d. van 1955 Laplace, probabiliste et statisticien, et ses précurseurs. Archives Internationales d’histoire des sciences 8:27-37.
David, f. n. 1965 Some Notes on Laplace. Pages 30–44 in Jerzy Neyman and Lucien M. Le Cam (editors), Bernoulli, 1713; Bayes, 1763; Laplace, 1813. New York: Springer.
David, f. n.; and Barton, d. e. 1962 Combinatorial Chance. London: Griffin; New York: Hafner.
Duveen, denis i.; and Hahn, roger 1957 Laplace’s Succession to Bézout’s Post of Examinateur des fileves de 1’Artillerie. Isis 48:416-427.
Eisenhart, churchill 1964 The Meaning of “Least” in Least Squares. Journal of the Washington Academy of Sciences 54:24-33.
Feller, william (1950) 1957 An Introduction to Prob-ability Theory and Its Applications. Vol. 1. 2d ed. New York: Wiley.
FrÉchet, maurice 1927 Sur la loi de probabilité de 1’écart maximum. Polskie Towarzystwo Matematyczne, Annales: Rocznik 6:93-122.
FrÉchet, maurice 1940-1943 Les probabilites associees a un systeme d’evenements compatibles et dependants. Parts 1-2. Paris: Hermann.
Gauss, carl friedrich (1809) 1963 Theory of Motion of the Heavenly Bodies Moving About the Sun in Conic Sections. New York: Dover. ? First published in Latin.
Gauss, carl friedrich (1821) 1880 Theoria combinationis observationum erroribus minimis obnoxiae. Pars prior. Volume 4, pages 1–26 in Carl Friedrich Gauss Werke. Gottingen (Germany): Dieterichsche Universitats-Druckerei. ? A French translation was published
in Paris in 1855 under the title Methode des moindres carrés: Mémoires sur la combinaison des observations. Gumbel, e. j. 1958 Statistics of Extremes. New York:
Gumbel. e. j. 1958 Statistics od Extremes. New York: Columbia Univ. Press.
Jordan, charles (1939) 1947 Calculus of Finite Differences. 2d ed. New York: Chelsea.
Lagrange, joseph louis (1770-1773) 1868 Mémoire sur 1’utilité de la methode de prendre le milieu entre les résultats de plusieurs observations; dans lequel on examine les avantages de cette méthode par le calcul des probabilités; et ou Ton resout differents problemes relatifs a cette matiere. Volume 2, pages 173–234 in Joseph Louis Lagrange, Oeuvres de Lagrange. Paris: Gauthier-Villars.
Levy, paul 1925 Calcul des probabilites. Paris: GauthierVillars.
Macmahon, percy a. 1915 Combinatory Analysis. Vol. 1. Cambridge Univ. Press.
Merriman, mansfield 1877 A List of Writings Relating to the Method of Least Squares, With Hist ncal and Critical Notes. Connecticut Academy of Arts and Sciences, Transactions 4:151-232.
Moivre, abraham de (1718) 1756 The Doctrine of Chances: Or, a Method of Calculating the Probabilities of Events in Play. 3d ed. London: Millar.
Moivre, abraham de 1730 Miscellanea analytica de seriebus et quadraturis.... London: Tonson & Watts.
Moivre, abraham de (1733) 1959 A Method of Approximating the Sum of the Terms of the Binomial (a + b)n Expanded Into a Series, From Whence Are Deduced Some Practical Rules to Estimate the Degree of Assent Which Is to Be Given to Experiments. Volume 2, pages 566–575 in David Eugene Smith, A Source Book in Mathematics. New York: Dover. ? First published as “Approximatio ad summam terminorum binomii (a + b)n in seriem expansi.”
Molina, e. c. 1930 Theory of Probability: Some Comments on Laplace’s Théorie analytique. American Mathematical Society, Bulletin 36:369-392.
Newman, james r. 1956 Commentary on Pierre Simon de Laplace. Volume 2, pages 1316–1324 in James R. Newman (editor), The World of Mathematics. New York: Simon ? Schuster.
Pearson, karl 1929 Laplace: Being Extracts From the Lectures Delivered by Karl Pearson. Biometrika 21:202-216.
Pitman, e. j. g. 1939 The Estimation of the Location and Scale of Parameters of a Continuous Population of Any Given Form. Biometrika 30:391-421.
Riordan, john 1958 An Introduction to Combinatorial Analysis. New York: Wiley.
Simon, g. a. 1929 Les origines de Laplace: Sa géné-alogie, ses études. Biometrika 21:217-230.
Simpson, thomas 1756 A Letter to the Right Honourable George, Earl of Macclesfield, President of the Royal Society, on the Advantage of Taking the Mean of a Number of Observations, in Practical Astronomy. Royal Society of London, Philosophical Transactions 49:82-93.
Simpson, thomas 1757 An Attempt to Show the Advantage Arising by Taking the Mean of a Number of Observations in Practical Astronomy. Pages 64–75 in Thomas Simpson, Miscellaneous Tracts on Some Curious, and Very Interesting Subjects in Mechanics, Physical-astronomy, and Speculative Mathematics. London: Nourse.
Smith, david eugene (1929) 1959 A Source Book in Mathematics. 2 vols. New York: Dover.
Todhunter, isaac (1865) 1949 A History of the Mathematical Theory of Probability From the Time of Pascal to That of Laplace. New York: Chelsea.
Uspensky, james v. 1937 Introduction to Mathematical Probability. New York: McGraw-Hill.
Walker, helen m. 1929 Studies in the History of Statistical Method, With Special Reference to Certain Educational Problems. Baltimore: Williams ? Wilkins.
Whittaker, edmund 1949a Laplace. Mathematical Gazette 33:1-12.
Whittaker, edmund 1949b Laplace. American Mathematical Monthly 56:369-372.
Wilson, edwin b. 1923 First and Second Laws of Error. Journal of the American Statistical Association 18:841-851.