The Theory That Would Not Die

by Sharon Bertsch McGrayne

  To test his rule on a larger sample Laplace decided in 1781 to determine the size of the French population, the thermometer of its health and prosperity. A conscientious administrator in eastern France had carefully counted heads in several parishes; to estimate the population of the entire nation, he recommended multiplying the annual number of births in France by 26. His proposal produced what was thought to be France’s population, approximately 25.3 million. But no one knew how accurate his estimate was. Today’s demographers believe that France’s population had actually grown rapidly, to almost 28 million, because of fewer famines and because a governmenttrained midwife was touring the countryside promoting the use of soap and boiling water during childbirth.

  Using his probability of causes, Laplace combined his prior information from parish records about births and deaths throughout France with his new information about headcounts in eastern France. He was adjusting estimates of the nation’s population with more precise information from particular regions. In 1786 he reached a figure closer to modern estimates and calculated odds of 1,000 to 1 that his estimate was off by less than half a million. In 1802 he was able to advise Napoleon Bonaparte that a new census should be augmented with detailed samples of about a million residents in 30 representative departments scattered equally around France.

  As he worked on his birth and census studies during the monarchy’s last years, Laplace became involved in an inflammatory debate about France’s judicial system. Condorcet believed the social sciences should be as quantifiable as the physical sciences. To help transform absolutist France into an English-style constitutional monarchy, he wanted Laplace to use mathematics to explore a variety of issues. How confident can we be in a sentence handed down by judge or jury? How probable is it that voting by an assembly or judicial tribunal will establish the truth? Laplace agreed to apply his new theory of probability to questions about electoral procedures, the credibility of witnesses, decision making by judicial panels and juries, and procedures of representative bodies and judicial panels.

  Laplace took a dim view of most court judgments in France. Forensic science did not exist, so judicial systems everywhere relied on witness testimony. Taking a witness’s statement for an event, Laplace asked the probability that the witness or the judge might be truthful, misled, or simply mistaken. He estimated the prior odds of an accused person’s guilt at 50–50 and the probability that a juror was being truthful somewhat higher. Even at that, if a jury of eight voted by simple majority, the chance that they judged the accused’s guilt wrong would be 65/256, or more than one in four. Thus for both mathematical and religious reasons Laplace sided with the Enlightenment’s most radical demand, the abolition of capital punishment: “The possibility of atoning for these errors is the strongest argument of philosophers who have wanted to abolish the death penalty.”19 Laplace also used his rule for more complicated cases where a court must decide among contradictory witnesses or where the reliability of testimony decreases with each telling. For Laplace, these questions demonstrated that ancient biblical accounts by the Apostles lacked credibility.

  While still counting babies, Laplace returned to study the seeming instability of Saturn and Jupiter’s orbits, the problem that had helped sensitize him early in his career to uncertain data. He did not, however, use his new knowledge of probability to solve this important problem. He used other methods between 1785 and 1788 to determine that Jupiter and Saturn oscillate gently in an 877-year cycle around the sun and that the moon orbits Earth in a cycle millions of years long. The orbits of Jupiter, Saturn, and the moon were not exceptions to Newton’s gravitation but thrilling examples of it. The solar system was in equilibrium, and the world would not end. This discovery was the biggest advance in physical astronomy since Newton’s law of gravity.

  Despite Laplace’s astounding productivity, his life as a professional scientist was financially precarious. Fortunately, Paris in the 1700s had more educational institutions and scientific opportunities than anywhere else on Earth, and academy members could patch jobs together to make a respectable living. Laplace tripled his income by examining artillery and naval engineering students three or four months a year and serving as a scientist in the Duke of Orleans’ entourage. His increasingly secure position also gave him access to the government statistics he needed to develop and test his probability of causes.

  At the age of 39, with a bright future ahead of him, Laplace married 18-year-old Marie Anne Charlotte Courty de Romange. The average age of marriage for French women was 27, but Marie Anne came from a prosperous and recently ennobled family with multiple ties to his financial and social circle. A small street off the Boulevard Saint-Germain is named Courty for her family. The Laplaces would have two children; contraception, whether coitus interruptus or pessaries, was common, and the church itself campaigned against multiple childbirths because they endangered the lives of mothers. Some 16 months after the wedding a Parisian mob stormed the Bastille, and the French Revolution began.

  After the revolutionary government was attacked by foreign monarchies, France spent a decade at war. Few scientists or engineers emigrated, even during the Reign of Terror. Mobilized for the national defense, they organized the conscription of soldiers, collected raw materials for gunpowder, supervised munitions factories, drew military maps, and invented a secret weapon, reconnaissance balloons. Laplace worked throughout the upheaval and served as the central figure in one of the Revolution’s most important scientific projects, the metric reform to standardize weights and measures. It was Laplace who named the meter, centimeter, and millimeter.

  Nevertheless, during the 18 months of the Terror, as almost 17,000 French were executed and half a million imprisoned, his position became increasingly precarious. Radicals attacked the elite Royal Academy of Sciences, and publications denounced him as a modern charlatan and a “Newtonian idolator.” A month after the Royal Academy was abolished Laplace was arrested on suspicion of disloyalty to the Revolution but neighbors interceded and he was released the next day at 4 a.m. A few months later he was purged from the metric system commission as not “worthy of confidence as to [his] republican virtues and [his] hatred of kings.”20 His assistant, Jean-Baptiste Delambre, was arrested while measuring the meridian for the meter and then released. At one point Laplace was relieved of his part-time job examining artillery students, only to be given the same job at the École Polytechnique. Seven scientists, including several of Laplace’s closest friends and supporters, died during the Terror. Unlike Laplace, who took no part in radical politics, they had identified themselves with particular political factions. The most famous was Antoine Lavoisier, guillotined because he had been a royal tax collector. Condorcet, trying to escape from Paris, died in jail.

  The Revolution, however, transformed science from a popular hobby into a full-fledged profession. Laplace emerged from the chaos as a dean of French science, charged with building new secular educational institutions and training the next generation of scientists. For almost 50 years—from the 1780s until his death in 1827—France led world science as no other country has before or since. And for 30 of those years Laplace was among the most influential scientists of all time.

  As the best-selling author of books about the celestial system and the law of gravity, Laplace dedicated two volumes to a rising young general, Napoleon Bonaparte. Laplace had launched Napoleon on his military career by giving him a passing exam grade in military school. The two never became personal friends, but Napoleon appointed Laplace minister of the interior for a short time and then appointed him to the largely honorary Senate with a handsome salary and generous expense account that made him quite a rich man. Mme Laplace became a lady-in-waiting to Napoleon’s sister and received her own salary. With additional financing from Napoleon, Laplace and his friend the chemist Claude Berthollet turned their country homes in Arceuil, outside Paris, into the world’s only center for young postdoctoral scientists.

  At a reception in Josephine Bonaparte�
�s rose garden at Malmaison in 1802, the emperor, who was trying to engineer a rapprochement with the papacy, started a celebrated argument with Laplace about God, astronomy, and the heavens.

  “And who is the author of all this?” Napoleon demanded.

  Laplace replied calmly that a chain of natural causes would account for the construction and preservation of the celestial system.

  Napoleon complained that “Newton spoke of God in his book. I have perused yours but failed to find His name even once. Why?”

  “Sire,” Laplace replied magisterially, “I have no need of that hypothesis.”21

  Laplace’s answer, so different from Price’s idea that Bayes’ rule could prove the existence of God, became a symbol of a centuries-long process that would eventually exclude religion from the scientific study of physical phenomena. Laplace had long since separated his probability of causes from religious considerations: “The true object of the physical sciences is not the search for primary causes [that is, God] but the search for laws according to which phenomena are produced.”22 Scientific explanations of natural phenomena were triumphs of civilization whereas theological debates were fruitless because they could never be resolved.

  Laplace continued his research throughout France’s political upheavals. In 1810 he announced the central limit theorem, one of the great scientific and statistical discoveries of all time. It asserts that, with some exceptions, any average of a large number of similar terms will have a normal, bell-shaped distribution. Suddenly, the easy-to-use bell curve was a real mathematical construct. Laplace’s probability of causes had limited him to binomial problems, but his final proof of the central limit theorem let him deal with almost any kind of data.

  In providing the mathematical justification for taking the mean of many data points, the central limit theorem had a profound effect on the future of Bayes’ rule. At the age of 62, Laplace, its chief creator and proponent, made a remarkable about-face. He switched allegiances to an alternate, frequency-based approach he had also developed. From 1811 until his death 16 years later Laplace relied primarily on this approach, which twentieth-century theoreticians would use to almost obliterate Bayes’ rule.

  Laplace made the change because he realized that where large amounts of data were concerned, both approaches generally produce much the same results. The probability of causes was still useful in particularly uncertain cases because it was more powerful than frequentism. But science matured during Laplace’s lifetime. By the 1800s mathematicians had much more reliable data than they had had in his youth and dealing with trustworthy data was easier with frequentism. Mathematicians did not learn until the midtwentieth century that, even with great amounts of data, the two methods can sometimes seriously disagree.

  Looking back in 1813 on his 40-year quest to develop the probability of causes, Laplace described it as the primary method for researching unknown or complicated causes of natural phenomena. He referred to it fondly as his source of large numbers and the inspiration behind his development and use of generating functions.

  And finally, in the climax of one small part of his career, he proved the elegant, general version of his theorem that we now call Bayes’ rule. He had intuited its principle as a young man in 1774. In 1781 he found a way to use Bayes’ two-step process to derive the formula by making certain restrictive assumptions. Between 1810 and 1814 he finally realized what the general theorem had to be. It was the formula he had been dreaming about, one broad enough to allow him to distinguish highly probable hypotheses from less valid ones. With it, the entire process of learning from evidence was displayed:

  In modern terms, the equation says that P(C|E), the probability of a hypothesis (given information), equals Pprior(C), our initial estimate of its probability, times P(E|C), the probability of each new piece of information (under the hypothesis), divided by the sum of the probabilities of the data in all possible hypotheses.

  Undergraduates today study Laplace’s first version of the equation, which deals with discrete events such as coin tosses and births. Advanced and graduate students and researchers use calculus with his later equation to work with observations on a continuous range between two values, for example, all the temperatures between 32 and 33 degrees. With it, Laplace could estimate a value as being within such and such a range with a particular degree of probability.

  Laplace had owned Bayes’ rule in all but name since 1781. The formula, the method, and its masterful utilization all belong to Pierre Simon Laplace. He made probability-based statistics commonplace. By transforming a theory of gambling into practical mathematics, Laplace’s work dominated probability and statistics for a century. “In my mind,” Glenn Shafer of Rutgers University observed, “Laplace did everything, and we just read stuff back into Thomas Bayes. Laplace put it into modern terms. In a sense, everything is Laplacean.”23

  If advancing the world’s knowledge is important, Bayes’ rule should be called Laplace’s rule or, in modern parlance, BPL for Bayes-Price-Laplace. Sadly, a half century of usage forces us to give Bayes’ name to what was really Laplace’s achievement.

  Since discovering his first version of Bayes’ rule in 1774, Laplace had used it primarily to develop new mathematical techniques and had applied it most extensively to the social sciences, that is, demography and judicial reform. Not until 1815, at the age of 66, did he apply it to his first love, astronomy. He had received some astonishingly accurate tables compiled by his assistant Alexis Bouvard, the director of the Paris Observatory. Using Laplace’s probability of causes, Bouvard had calculated a large number of observations about the masses of Jupiter and Saturn, estimated the possible error for each entry, and then predicted the probable masses of the planets. Laplace was so delighted with the tables that, despite his aversion to gambling, he used Bayes’ rule to place a famous bet with his readers: odds were 11,000 to 1 that Bouvard’s results for Saturn were off by less than 1%. For Jupiter, the odds were a million to one. Space-age technology confirms that Laplace and Bouvard should have won both bets.

  Late in his career, Laplace also applied his probability of causes to a variety of calculations in Earth science, notably to the tides and to changes in barometric pressure. He used a nonnumerical common-sense version of his probability of causes to advance his famous nebular hypothesis: that the planets and their satellites in our solar system originated in a swirl of dust. And he compared three hypotheses about the orbits of 100 comets to confirm what he already knew: that the comets most probably originate within the sun’s sphere of influence.

  After the fall of Napoleon, France’s new king, Louis XVIII, bestowed the hereditary title of marquis on Laplace, the son of a village innkeeper. And on March 5, 1827, at the age of 78, Laplace died, almost exactly 100 years after his idol, Isaac Newton.

  Eulogies hailed Laplace as the Newton of France. He had brought modern science to students, governments, and the reading public and had developed probability into a formidable method for handling unknown and complex causes of natural phenomena. And in one small, relatively insignificant portion of his lifework he became the first to express and use what is now called Bayes’ rule. With it, he updated old knowledge with new, explained phenomena that previous centuries had ascribed to chance or to God’s will, and opened the way for future scientific exploration.

  Yet Laplace had built his probability theory on intuition. As far as he was concerned, “essentially, the theory of probability is nothing but good common sense reduced to mathematics. It provides an exact appreciation of what sound minds feel with a kind of instinct, frequently without being able to account for it.”24 Soon, however, scientists would begin confronting situations that intuition could not easily explain. Nature would prove to be far more complicated than even Laplace had envisioned. No sooner was the old man buried than critics began complaining about Laplace’s rule.

  3.

  many doubts, few defenders

  With Laplace gone, Bayes’ rule entered a tumultuous period when it w
as disdained, reformed, grudgingly tolerated, and finally nearly obliterated by battling theorists. Yet through it all the rule chugged sturdily along, helping to resolve practical problems involving the military, communications, social welfare, and medicine in the United States and Europe.

  The backdrop to the drama was a set of unsubstantiated but widely circulated charges against Laplace’s reputation. The English mathematician Augustus de Morgan wrote in The Penny Cyclopaedia of 1839 that Laplace failed to credit the work of others; the accusation was repeated without substantiation for 150 years until a detailed study by Stigler concluded it was groundless. During the 1880s an anti-Napoleonic and antimonarchical Frenchman named Maximilien Marie painted Laplace as a reactionary ultraroyalist; several English and American authors adopted Marie’s version unquestioningly. The Encyclopaedia Britannica of 1912 asserted that Laplace “aspired to the role of a politician, and . . . degraded to servility for the sake of a riband and a title.”1 In his long-lived but rather fanciful bestseller Men of Mathematics the American E. T. Bell titled the chapter on Laplace “From Peasant to Snob . . . Humble as Lincoln, proud as Lucifer.” Bell described Laplace as “grandiose,” “snobbish,” “pompous,” “coarse,” “smug,” “intimate with Napoleon,” and “perhaps the most conspicuous refutation of the pedagogical superstition that noble pursuits necessarily ennoble a man’s character.”2 Bell’s book, published in 1937, influenced an entire generation of mathematicians and scientists. In the 1960s an Anglo-American statistician Florence Nightingale David wrote without verification that Laplace met with “almost universal condemnation.”3 A scholarly American biography by historian Charles Coulston Gillispie and two collaborators, Robert Fox and Ivor Grattan-Guinness, hemmed and hawed. It began by stating categorically that “not a single testimonial bespeaking congeniality survives” but ended by listing Laplace’s “close personal attachments with other French scientists,” “warm and tranquil family life,” and the help he gave even to critics of his research.4