The theory that would not die pdf download

The theory that would not die pdf download

The Ship That Would Not Die,The Theory That Would Not Die by Sharon Bertsch McGrayne Book PDF Summary

17/05/ · The Theory That Would Not Die Book Summary "This account of how a once reviled theory, Baye’s rule, came to underpin modern life is both approachable and engrossing" The Theory That Would Not Die How Bayes' Rule - Free download as PDF File .pdf), Text File .txt) or read online for free. Scribd is the world's largest social reading and publishing site. The case for Solomonoff Induction is argued, a formal inductive framework which combines algorithmic information theory with the Bayesian framework and how this approach addresses The theory that would not die: how Bayes’ rule cracked the enigma code, hunted down Rus-sian submarines, and emerged triumphant from two centuries of controversy / Sharon Bertsch The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy ... read more

Hell was about to be unleashed on them in the largest single-ship kamikaze attack of World War II. On April 16, , the crewmen of the USS Laffey were battle hardened and prepared. They had engaged in combat off the Normandy coast in June They had been involved in three prior assaults of enemy positions in the Pacific-at Leyte and Lingayen in the Philippines and at Iwo Jima. They had seen kamikazes purposely crash into other destroyers and cruisers in their unit and had seen firsthand the bloody results of those crazed tactics. But nothing could have prepared the crew for this moment-an eighty-minute ordeal in which the single small ship was targeted by no fewer than twenty-two Japanese suicide aircraft.

By the time the unprecedented attack on the Laffey was finished, thirty-two sailors lay dead, more than seventy were wounded, and the ship was grievously damaged. Although she lay shrouded in smoke and fire for hours, the Laffey somehow survived, and the gutted American warship limped from Okinawa's shore for home, where the ship and crew would be feted as heroes. Using scores of personal interviews with survivors, the memoirs of crew members, and the sailors' wartime correspondence, historian and author John Wukovits breathes life into the story of this nearly forgotten historic event. The US Navy described the kamikaze attack on the Laffey "as one of the great sea epics of the war. To its adherents, it is an elegant statement about learning from experience. To its opponents, it is subjectivity run amok. In the first-ever account of Bayes' rule for general readers, Sharon Bertsch McGrayne explores this controversial theorem and the generations-long human drama surrounding it.

She reveals why respected statisticians rendered it professionally taboo for years—while practitioners relied on it to solve crises involving great uncertainty and scanty information, such as Alan Turing's work breaking Germany's Enigma code during World War II. McGrayne also explains how the advent of computer technology in the s proved to be a game-changer. Today, Bayes' rule is used everywhere from DNA de-coding to Homeland Security. Drawing on primary source material and interviews with statisticians and other scientists, The Theory That Would Not Die is the riveting account of how a seemingly simple theorem ignited one of the greatest controversies of all time.

The unimaginable has happened. The world has been plunged into all-out nuclear war. Sailing near the Arctic Circle, the U. Nathan James is relatively unscathed, but the future is grim and Captain Thomas is facing mutiny from the tattered remnants of his crew. With civilization in ruins, he urges those that remain—one-hundred-and-fifty-two men and twenty-six women—to pull together in search of land. When none of the women seems able to conceive, fear sets in. Will this be the end of humankind? Skip to content. The Ship That Would Not Die Download The Ship That Would Not Die full books in PDF, epub, and Kindle. The Ship That Would Not Die. Author : F. Julian Becton,Joseph Morschauser, III Publsiher : Unknown Total Pages : Release : Genre : History ISBN : GET BOOK. Download The Ship That Would Not Die Book in PDF, Epub and Kindle. Author : Stephen J. The Ship That Wouldn t Die. Author : Don Keith Publsiher : Penguin Total Pages : Release : Genre : History ISBN : GET BOOK.

Download The Ship That Wouldn t Die Book in PDF, Epub and Kindle. New Carissa. Author : Steven Michael Smith Publsiher : AuthorHouse Total Pages : 54 Release : Genre : Education ISBN : GET BOOK. Download New Carissa Book in PDF, Epub and Kindle. The Ship that Would Not Die. Wade, Jon Wakefield, Homer Warner, Frode Weierud, Robert B. Wilson, Wing H. Wong, Judith E. Zeh, and Arnold Zellner. I would like to thank two outside reviewers, Jim Berger and Andrew Dale; both read the manuscript carefully and made useful comments to improve it. Several friends and family members—Ruth Ann Bertsch, Cindy Vahey Bertsch, Fred Bertsch, Jean Colley, Genevra Gerhart, James Goodman, Carolyn Keating, Timothy W. Keller, Sharon C. Rutberg, Beverly Schaefer, and Audrey Jensen Weitkamp—made crucial comments.

I owe thanks to the mathematics library staff of the University of Washington. And my agent, Susan Rabiner, and editor, William Frucht, were steadfast in their support. Despite all this help, I am, of course, responsible for the errors in this book. xiii This page intentionally left blank part I enlightenment and the antibayesian reaction This page intentionally left blank 1. causes in the air Sometime during the s, the Reverend Thomas Bayes made the ingenious discovery that bears his name but then mysteriously abandoned it. It was rediscovered independently by a different and far more renowned man, Pierre Simon Laplace, who gave it its modern mathematical form and scientific application—and then moved on to other methods.

Yet at the same time, it solved practical questions that were unanswerable by any other means: the defenders of Captain Dreyfus used it to demonstrate his innocence; insurance actuaries used it to set rates; Alan Turing used it to decode the German Enigma cipher and arguably save the Allies from losing the Second World War; the U. Navy used it to search for a missing H-bomb and to locate Soviet subs; RAND Corporation used it to assess the likelihood of a nuclear accident; and Harvard and Chicago researchers used it to verify the authorship of the Federalist Papers. It was not until the twenty-first century that the method lost its stigma and was widely and enthusiastically embraced.

The story began with a simple thought experiment. As a member of the Presbyterian Church, 3 4 Enlightenment and the Anti-Bayesian Reaction a religious denomination persecuted for refusing to support the Church of England, he was considered a Dissenter or Non-Conformist. In he left for London, where his clergyman father ordained him and apparently employed him as an assistant minister. This was his only mathematical publication during his lifetime. About this time Bayes joined a second group of up-to-date amateur mathematicians. He had moved to a small congregation in a fashionable resort, the cold-water spa Tunbridge Wells. As an independently wealthy bachelor—his family had made a fortune manufacturing Sheffield steel Causes in the Air cutlery—he rented rooms, apparently from a Dissenting family. His religious duties—one Sunday sermon a week—were light.

And spa etiquette permitted Dissenters, Jews, Roman Catholics, and even foreigners to mix with English society, even with wealthy earls, as they could not elsewhere. A frequent visitor to Tunbridge Wells, Philip, the Second Earl of Stanhope, had been passionately interested in mathematics since childhood, but his guardian had banned its study as insufficiently genteel. When Stanhope was 20 and free to do as he liked, he seldom raised his eyes from Euclid. At some point Bayes joined the network. One day, for example, Stanhope sent Bayes a copy of a draft paper by a mathematician named Patrick Murdoch. Bayes disagreed with some of it and sent his comments back to Stanhope, who forwarded them to Murdoch, who in turn replied through Stanhope, and so on around and around.

In short, we can rely only on what we learn from experience. Hume argued that certain objects are constantly associated with each other. But the fact that umbrellas and rain appear together does not mean that umbrellas cause rain. The fact that the sun has risen thousands of times does not guarantee that it will do so the next day. Because we can seldom be certain that a particular cause will have a 5 6 Enlightenment and the Anti-Bayesian Reaction particular effect, we must be content with finding only probable causes and probable effects. Many mathematicians and scientists believed fervently that natural laws did indeed prove the existence of God, their First Cause. Today, probability, the mathematics of uncertainty, would be the obvious tool, but during the early s probability barely existed. Its only extensive application was to gambling, where it dealt with such basic issues as the odds of getting four aces in one poker hand.

De Moivre, who had spent several years in French prisons because he was a Protestant, had already solved that problem by working from cause to effect. But no one had figured out how to turn his work around backward to ask the so-called inverse question from effect to cause: what if a poker player deals himself four aces in each of three consecutive hands? What is the underlying chance or cause that his deck is loaded? Newton, who had died 20 years before, had stressed the importance of relying on observations, developed his theory of gravitation to explain them, and then used his theory to predict new observations. But Newton had not explained the cause of gravity or wrestled with the problem of how true his theory might be. In any event, problems involving cause and effect and uncertainty filled the air, and Bayes set out to deal with them quantitatively. Crystallizing the essence of the inverse probability problem in his mind, Bayes decided that his goal was to learn the approximate probability of a future event he knew nothing about except its past, that is, the number of times it had occurred or failed to occur.

To quantify the problem, he needed a number, and sometime between and he hit on an ingenious solution. As a starting point he would simply invent a number—he called it a guess—and refine it later as he gathered more information. Causes in the Air Next, he devised a thought experiment, a s version of a computer simulation. Stripping the problem to its basics, Bayes imagined a square table so level that a ball thrown on it would have the same chance of landing on one spot as on any other. Subsequent generations would call his construction a billiard table, but as a Dissenting minister Bayes would have disapproved of such games, and his experiment did not involve balls bouncing off table edges or colliding with one another.

As he envisioned it, a ball rolled randomly on the table could stop with equal probability anywhere. We can imagine him sitting with his back to the table so he cannot see anything on it. On a piece of paper he draws a square to represent the surface of the table. He begins by having an associate toss an imaginary cue ball onto the pretend tabletop. Because his back is turned, Bayes does not know where the cue ball has landed. Next, we picture him asking his colleague to throw a second ball onto the table and report whether it landed to the right or left of the cue ball.

If to the left, Bayes realizes that the cue ball is more likely to sit toward the right side of the table. He asks his colleague to make throw after throw after throw; gamblers and mathematicians already knew that the more times they tossed a coin, the more trustworthy their conclusions would be. What Bayes discovered is that, as more and more balls were thrown, each new piece of information made his imaginary cue ball wobble back and forth within a more limited area. As an extreme case, if all the subsequent tosses fell to the right of the first ball, Bayes would have to conclude that it probably sat on the far left-hand margin of his table. By contrast, if all the tosses landed to the left of the first ball, it probably sat on the far right. Eventually, given enough tosses of the ball, Bayes could narrow the range of places where the cue ball was apt to be. This approach could not produce a right answer. Bayes could never know precisely where the cue ball landed, but he could tell with increasing confidence that it was most probably within a particular range.

Using his knowledge of the present the left and right positions of the tossed balls , Bayes had figured out how to say something about the past 7 8 Enlightenment and the Anti-Bayesian Reaction the position of the first ball. He could even judge how confident he could be about his conclusion. Eventually, names were assigned to each part of his method: Prior for the probability of the initial belief; Likelihood for the probability of other hypotheses with objective new data; and Posterior for the probability of the newly revised belief. Each time the system is recalculated, the posterior becomes the prior of the new iteration. It was an evolving system, which each new bit of information pushed closer and closer to certitude. In short: Prior times likelihood is proportional to the posterior. In the more technical language of the statistician, the likelihood is the probability of competing hypotheses for the fixed data that have been observed.

Even worse, Bayes added that if he did not know enough to distinguish the position of the balls on his table, he would assume they were equally likely to fall anywhere on it. Assuming equal probabilities was a pragmatic approach for dealing with uncertain circumstances. In uncertain situ- Causes in the Air ations such as annuities or marine insurance policies, all parties were assigned equal shares and divided profits equally. Even prominent mathematicians assigned equal probabilities to gambling odds by assuming, with a remarkable lack of realism, that all tennis players or fighting cocks were equally skillful.

Despite their venerable history, equal probabilities would become a lightning rod for complaints that Bayes was quantifying ignorance. Today, some historians try to absolve him by saying he may have applied equal probabilities to his data the subsequent throws rather than to the initial, so-called prior toss. But this is also guesswork. And for many working statisticians, the question is irrelevant because in the tightly circumscribed case of balls that can roll anywhere on a carefully leveled surface both produce the same mathematical results. Whatever Bayes meant, the damage was done. For years to come, the message seemed clear: priors be damned. At this point, Bayes ended his discussion. He may have mentioned his discovery to others. Hartley was a Royal Society member who believed in cause-andeffect relationships. which shews that we may hope to determine the Proportions, and by degrees, the whole Nature, of unknown Causes, by a sufficient Observation of their Effects.

Modern-day sleuths have suggested Bayes or Stanhope, and in Stephen M. Stigler of the University of Chicago suggested that Nicholas Saunderson, a blind Cambridge mathematician, made the discovery instead of Bayes. No matter who talked about it, it seems highly unlikely that anyone other than Bayes made the breakthrough. Thirty years later Price was still referring to the work as that of Thomas Bayes. Instead of sending it off to the Royal Society for publication, he buried it among his papers, where it sat for roughly a 9 10 Enlightenment and the Anti-Bayesian Reaction decade. Perhaps he thought his discovery was useless; but if a pious clergyman like Bayes thought his work could prove the existence of God, surely he would have published it.

Some thought Bayes was too modest. Others wondered whether he was unsure about his mathematics. Whatever the reason, Bayes made an important contribution to a significant problem—and suppressed it. Price, another Presbyterian minister and amateur mathematician, achieved fame later as an advocate of civil liberties and of the American and French revolutions. Price ever had a superior. An English magazine thought Price would go down in American history beside Franklin, Washington, Lafayette, and Paine. Yet today Price is known primarily for the help he gave his friend Bayes. At first Price saw no reason to devote much time to the essay. Mathematical infelicities and imperfections marred the manuscript, and it looked impractical. Its continual iterations—throwing the ball over and over again and recalculating the formula each time—produced large numbers that would be difficult to calculate. In a cover letter to the Royal Society, Price supplied a religious reason for publishing the essay.

from final causes. the existence of the Deity. Critiquing Hume a few years later, Price used Bayes for the first and only time. By modern standards, we should refer to the Bayes-Price rule. A card player could start by believing his opponent played with a straight deck and then modify his opinion each time a new hand was dealt. Bayes combined judgments based on prior hunches with probabilities based on repeatable experiments. He introduced the signature features of Bayesian methods: an initial belief modified by objective new information.

He could move from observations of the world to abstractions about their probable cause. Given the revered status of his work today, it is also important to recognize what Bayes did not do. Nor did he develop his theorem into a powerful mathematical method. Above all, unlike Price, he did not mention Hume, religion, or God. Instead, he cautiously confined himself to the probability of events and did not mention hypothesizing, predicting, deciding, or taking action. He did not suggest possible uses for his work, whether in theology, science, or social science. Bayes did not even name his breakthrough. It would be called the probability of causes or inverse probability for the next years.

It would not be named Bayesian until the s. In short, Bayes took the first steps. He composed the prelude for what was to come. For the next two centuries few read the Bayes-Price article. In the end, this is the story of two friends, Dissenting clergymen and amateur mathematicians, whose labor had almost no impact. Almost, that is, except on the one person capable of doing something about it, the great French mathematician Pierre Simon Laplace. the man who did everything Just across the English Channel from Tunbridge Wells, about the time that Thomas Bayes was imagining his perfectly smooth table, the mayor of a tiny village in Normandy was celebrating the birth of a son, Pierre Simon Laplace, the future Einstein of his age. Pierre Simon, born on March 23, , and baptized two days later, came from several generations of literate and respected dignitaries.

By the time Pierre Simon was a teenager his father seems to have been his only close relative. The question was, what kind of schooling? Decades of religious warfare between Protestants and Catholics and several horrendous famines caused by cold weather had made France a determinedly secular country intent on developing its resources. Instead, the elder Laplace enrolled his son in a local primary and secondary school where Benedictine monks produced clergy 13 14 Enlightenment and the Anti-Bayesian Reaction for the church and soldiers, lawyers, and bureaucrats for the crown.

Thanks to the patronage of the Duke of Orleans, local day students like Pierre Simon attended free. The curriculum was conservative and Latin-based, heavy on copying, memorization, and philosophy. But it left Laplace with a fabulous memory and almost unbelievable perseverance. Contemporaries called it the Century of Lights and the Age of Science and Reason, and the popularization of science was its most important intellectual phenomenon. Given the almost dizzying curiosity of the times, it is not surprising that, shortly after his tenth birthday, Pierre Simon was profoundly affected by a spectacular scientific prediction. The French astronomers accurately pinpointed the date—mid-April plus or minus a month—when Europeans would be able to see the comet returning from its orbit around the sun. Years later Laplace said it was the event that made his generation realize that extraordinary events like comets, eclipses, and severe droughts were caused not by divine anger but by natural laws that mathematics could reveal.

Instead he went to the University of Caen, which was closer to home and had a solid theological program suitable for a future cleric. Yet even Caen had mathematical firebrands offering advanced lectures on differential and integral calculus. With it, they were forming equations and discovering a fabulous wealth of enticing new information about planets, their masses and details of their orbits. Laplace emerged from Caen a swashbuckling mathematical virtuoso eager to take on the scientific world. The Man Who Did Everything At graduation Laplace faced an anguishing dilemma.

young men, who are known for their debauchery. If Laplace had been willing to become an abbé, his father might have helped him financially, and Laplace could have combined church and science. A number of abbés supported themselves in science, the most famous being Jean Antoine Nollet, who demonstrated spectacular physics experiments to the paying public. For the edification of the king and queen of France, Nollet sent a charge of static electricity through a line of soldiers to make them leap comically into the air. Two abbés were even elected to the prestigious Royal Academy of Sciences. Still, the lot of most abbé-scientists was neither lucrative nor intellectually challenging. The majority found low-level jobs tutoring the sons of rich nobles or teaching elementary mathematics and science in secondary schools. University-level opportunities were limited because during the s professors transmitted knowledge from the past instead of doing original research.

But Caen had convinced Laplace that he wanted to do something quite new. He wanted to be a full-time, professional, secular, mathematical researcher. And he wanted to explore the new algebra-generated, data-rich world of science. To his father, an ambitious man in bucolic France, a career in mathematics must have seemed preposterous. Young Laplace made his move in the summer of , shortly after completing his studies at Caen. Years later Laplace could still recite passages from it. The school, located behind Les Invalides in Paris, provided Laplace with a salary, housing, meals, and money for wood to heat his room in winter. It was precisely the kind of job he had hoped to avoid. Many mathematically talented young men from modest families were employed in such institutions. But Laplace and his mentor were aiming far higher. Laplace wanted the challenge of doing basic research full time.

In striking contrast to the amateurism of the Royal Society of London, the French Royal Academy of Sciences was the most professional scientific institution in Europe. To augment their low salaries, academicians could use their prestige to cobble together various part-time jobs. Without financial support from the church or his father, however, Laplace had to work fast. Since most academy members were chosen on the basis of a long record of solid accomplishment, he would have to be elected over the heads of more senior men. And for that to happen, he needed to make a spectacular impact. Over the previous two centuries mathematical astronomy had made great strides.

Nicolaus Copernicus had moved Earth from the center of the solar system to a modest but accurate position among the planets; Johannes Kepler had connected the celestial bodies by simple laws; and Newton had introduced the concept of gravity. But Newton had described the motions The Man Who Did Everything of heavenly bodies roughly and without explanation. The burning scientific question of the day was whether the universe was stable. Was the end of the world at hand? Astronomers had long been aware of alarming evidence suggesting that the solar system was inherently unstable. Comparing the actual positions of the most remote known planets with centuries-old astronomical observations, they could see that Jupiter was slowly accelerating in its orbit around the sun while Saturn was slowing down.

Eventually, they thought, Jupiter would smash into the sun, and Saturn would spin off into space. Responding to the challenge, Laplace decided to make the stability of the universe his lifework. He said his tool would be mathematics and it would be like a telescope in the hands of an astronomer. He also wondered fleetingly whether comets might be disturbing the orbits of Jupiter and Saturn. But he changed his mind almost immediately. The problem was the data astronomers used. Working on the problem of Jupiter and Saturn, for example, Laplace would use observations made by Chinese astronomers in BC, Chaldeans in BC, Greeks in BC, Romans in AD , and Arabs in AD Obviously, not all data were 17 18 Enlightenment and the Anti-Bayesian Reaction equally valuable. The French academy was tackling the problem by encouraging the development of more precise telescopes and graduated arcs. And as algebra improved instrumentation, experimentalists were producing more quantitative results.

In a veritable information explosion, the sheer collection and systemization of data accelerated through the Western world. Just as the number of known plant and animal species expanded enormously during the s, so did knowledge about the physical universe. Even as Laplace arrived in Paris, the French and British academies were sending trained observers with state-of-the-art instrumentation to carefully selected locations around the globe to time Venus crossing the face of the sun; this was a critical part of Capt. By comparing all the measurements, French mathematicians would determine the approximate distance between the sun and Earth, a fundamental natural constant that would tell them the size of the solar system.

But sometimes even up-to-date expeditions provided contradictory data about whether, for instance, Earth was shaped like an American football or a pumpkin. Dealing with large amounts of complex data was emerging as a major scientific problem. Given a wealth of observations, how could scientists evaluate the facts at their disposal and choose the most valid? Observational astronomers typically averaged their three best observations of a particular phenomenon, but the practice was as straightforward as it was ad hoc; no one had ever tried to prove its validity empirically or theoretically. The mathematical theory of errors was in its infancy. Problems were ripe for the picking and, with his eye on membership in the Royal Academy, Laplace bombarded the society with 13 papers in five years.

He submitted hundreds of pages of powerful and original mathematics needed in astronomy, celestial mechanics, and important related issues. Laplace considered emigrating to Prussia or Russia to work in their academies. Analyzing large amounts of data was a formidable problem, and Laplace was already beginning to think it would require a fundamentally new way of thinking. He was beginning to see probability as a way to deal with the uncertainties pervading many events and their causes. The book had appeared in three editions between and , and Laplace may have read the version. Thomas Bayes had studied an earlier edition. Reading de Moivre, Laplace became more and more convinced that probability might help him deal with uncertainties in the solar system. Probability barely existed as a mathematical term, much less as a theory. Outside of gambling, it was applied in rudimentary form to philosophical questions like the existence of God and to commercial risk, including contracts, marine and life insurance, annuities, and money lending.

Young as he was, Laplace was confident enough in his mathematical judgment to disagree with his powerful patron. To Laplace, the movements of celestial bodies seemed so complex that he could not hope for precise solutions. Probability would not give him absolute answers, but it might show him which data were more likely to be correct. He began thinking about a method for deducing the probable causes of divergent, error-filled observations in astronomy. He was feeling his way toward a broad general theory for moving mathematically from known events back to their most probable causes. In a paper submitted and read to the academy in March, the former abbé compared ignorant mankind, not with God but with an imaginary intelligence capable of knowing All. His search for a probability of causes and his view of the deity were deeply congenial. Laplace was all of one piece and for that reason all the more formidable. He often said he did not believe in God, and not even his biographer could decide whether he was an atheist or a deist.

But his probability of causes was a mathematical expression of the universe, and for the rest of his days he updated his theories about God and the probability of causes as new evidence became available. They were not, but the threat of competition galvanized him. Dusting off one of his discarded manuscripts, Laplace transformed it into a broad method for determining the most likely causes of events and phenomena. In a giant and intellectually nimble leap, he realized he could inject these uncertainties into his thinking by considering all possible causes and then choosing among them. Laplace did not state his idea as an equation. He intuited it as a principle and described it only in words: the probability of a cause given an event is proportional to the probability of the event given its cause.

The Man Who Did Everything Armed with his principle, Laplace could do everything Thomas Bayes could have done—as long as he accepted the restrictive assumption that all his possible causes or hypotheses were equally likely. As a scientist, he needed to study the various possible causes of a phenomenon and then determine the best one. He did not yet know how to do that mathematically. He would need to make two more major breakthroughs and spend decades in thought. But he was the first mathematician to work with large data sets, and the proportionality of cause and effect would make it feasible to make complex numerical calculations using only goose quills and ink pots.

In a mémoire read aloud to the academy, Laplace first applied his new probability of causes to two gambling problems. In each case he understood intuitively what should happen but got bogged down trying to prove it mathematically. First, he imagined an urn filled with an unknown ratio of black and white tickets his cause. He drew a number of tickets from the urn and, based on that experience, asked for the probability that his next ticket would be white. Then in a frustrating battle to prove the answer he wrote no fewer than 45 equations covering four quarto-sized pages. His second gambling problem involved piquet, a game requiring both luck and skill. Two people start playing but stop midway through the game and have to figure out how to divide the kitty by estimating their relative skill levels the cause. Again, Laplace understood instinctively how to solve the problem but could not yet do so mathematically.

After dealing with gambling, which he loathed, Laplace moved happily on to the critical scientific problem faced by working astronomers. How should they deal with different observations of the same phenomenon? Even if observers repeated their measurements at the same time and place with the same instrument, their results could be slightly different each time. Trying to calculate a midvalue for such discrepant observations, Laplace limited himself to three observations but still needed seven pages of equations to formulate the problem. Scientifically, he understood the right answer—average the three data points—but he would have no mathematical justification for doing so until , when, without using the probability of causes, he invented the central limit theorem. Laplace was 15 when the Bayes-Price essay was published; it appeared in an English-language journal for the English gentry and was apparently never mentioned again.

Even French scientists who kept up with foreign journals thought Laplace was first and congratulated him wholeheartedly on his originality. Mathematics confirms that Laplace discovered the principle independently. Bayes solved a special problem about a flat table using a two-step process that involved a prior guess and new data. Laplace did not yet know about the initial guess but dealt with the problem generally, making it useful for a variety of problems. Bayes laboriously explained and illustrated why uniform probabilities were permissible; Laplace assumed them instinctively. The Englishman wanted to know the range of probabilities that something will happen in light of previous experience. Laplace wanted more: as a working scientist, he wanted to know the probability that certain measurements and numerical values associated with a phenomenon were realistic.

Most strikingly of all, Laplace at 25 was already steadfastly determined to develop his new method and make it useful. For the next 40 years he would work to clarify, simplify, expand, generalize, prove, and apply his new rule. He also made important advances in celestial mechanics, mathematics, physics, biology, Earth science, and statistics. He juggled projects, moving from one to another and then back to the first. Happily blazing trails through every field of science known to his age, he transformed and mathematized everything he touched. Although he was fast becoming the leading scientist of his era, the academy waited five years before electing him a member on March 31, His mémoire on the probability of causes was published a year later, in At the age of 24, Laplace was a professional researcher.

Strictly speaking, he did not produce a new formula but rather a statement about the first formula assuming equal probabilities for the causes. The statement gave him confidence that he was on the right track and told him that as long as all his prior hypotheses were equally probable, his earlier principle of was correct. Every time he got new information he could use the answer from his last solution as the starting point for another calculation. And by assuming that all his initial hypotheses were equally probable he could even derive his theorem. he accomplished in an acute and very ingenious, though slightly awkward, manner. It limited him to assigning equal probabilities to each of his initial hypotheses. As a scientist, he disapproved. If his method was ever going to reflect the actual state of affairs, he needed to be able to differentiate dubious data from more valid observations.

Calling all events or observations equally probable could be true only theoretically. Many dice, for example, that appeared perfectly cubed were actually skewed. In one case he started by assigning players equal probabilities of winning, but with each round of play their respective skills emerged and their probabilities changed. Probability problems require multiplying numbers over and over, whether tossing coin after coin or measuring and remeasuring 23 24 Enlightenment and the Anti-Bayesian Reaction an observation. The process generated huge numbers—nothing as large as those common today but definitely cumbersome for a man working alone without mechanical or electronic aids. He did not even get an assistant to help with calculations until about But to illustrate how tedious calculations with big numbers could be, he described multiplying 20, × 19, × 19, × 19,, etc.

and then dividing by 1 × 2 × 3 × 4 up to 10, Such big-number problems were new. Newton had calculated with geometry, not numbers. Many mathematicians, like Bayes, used thought experiments to separate real problems from abstract and methodological issues. But Laplace wanted to use mathematics to illuminate natural phenomena, and he insisted that theories had to be based on actual fact. Probability was propelling him into an unmanageable world. Armed with the Bayes—Price starting point, Laplace broke partway through the logjam that had stymied him for seven years. So far he had concentrated primarily on probability as a way to resolve error-prone astronomical observations. Now he switched gears to concentrate on finding the most probable causes of known events. To do so, he needed to practice with a big database of real and reliable values. But astronomy seldom provided extensive or controlled data, and the social sciences often involved so many possible causes that algebraic equations were useless.

Only one large amalgamation of truly trustworthy numbers existed in the s: parish records of births, christenings, marriages, and deaths. In the French government ordered all provincial officials to report birth and death figures regularly to Paris; and three years later, the Royal Academy published 60 years of data for the Paris region. The figures confirmed what the Englishman John Graunt had discovered in slightly more boys than girls were born, in a ratio that remained constant over many years. Soon he was assessing not gambling or astronomical statistics but infants. For anyone interested in large numbers, babies were ideal. First, they The Man Who Did Everything came in binomials, either boys or girls, and eighteenth-century mathematicians already knew how to treat binomials. the births of boys and girls. Absolutely not, Laplace replied firmly. A study based on a few facts cannot overrule a much larger one.

The calculations would be formidable. For example, if he had started with a ratio of newborn boys to girls and a sample of 58, boys, Laplace would have had to multiply. This was definitely not something anyone, not even the indomitable Laplace, wanted to do by hand. He started out, however, as Bayes had suggested, by pragmatically assigning equal probabilities to all his initial hunches, whether 50—50, 33—33—33, or 25—25—25— Because their sums equal one, multiplication would be easier. He employed equal probabilities only provisionally, as a starting point, and his final hypotheses would depend on all the observational data he could add. He was building the foundation of the modern theory of testing statistical hypotheses. Poring over records of christenings in Paris and births in London, he was soon willing to bet that boys would outnumber girls for the next years in Paris and for the next 8, years in London.

Among them were new generating functions, transforms, and asymptotic expansions. Computers have made many of his shortcuts unnecessary, but generating functions remain deeply embedded in mathematical analyses used for practical applications. Laplace used generating functions as a form of mathematical wizardry to trick a function he could deal with into providing him with the function he really wanted. To Laplace, these mathematical pyrotechnics seemed as obvious as common sense. He was soon asking whether boys were more apt to be born in certain geographic regions. Petersburg in the north, and French provinces in between. He concluded that climate could not explain the disparity in births. But would more boys than girls always be born? In building a mathematical model of scientific thinking, where a reasonable person could develop a hypothesis and then evaluate it relentlessly in light of new knowledge, he became the first modern Bayesian. His system was enormously sensitive to new information.

Just as each throw of a coin increases the probability of its being fair or rigged, so each additional birth record narrowed the range of uncertainties. By he was determining the influence of past events on the probability of future events and wondering how big his sample of newborns had to be. By then Laplace saw probability as the primary way to overcome uncertainty. a state of indecision,. Like a modern researcher, he competed and collaborated with others and published reports on his interim progress as he went. Above all, he was tenacious. Twenty-five years later he was still eagerly testing his probability of causes with new information. 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 But no one knew how accurate his estimate was. 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.

In he reached a figure closer to modern estimates and calculated odds of 1, to 1 that his estimate was off by less than half a million. In 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. 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 testi- 27 28 Enlightenment and the Anti-Bayesian Reaction mony. For Laplace, these questions demonstrated that ancient biblical accounts by the Apostles lacked credibility. He did not, however, use his new knowledge of probability to solve this important problem. He used other methods between and to determine that Jupiter and Saturn oscillate gently in an year cycle around the sun and that the moon orbits Earth in a cycle millions of years long.

The solar system was in equilibrium, and the world would not end. Fortunately, Paris in the s had more educational institutions and scientific opportunities than anywhere else on Earth, and academy members could patch jobs together to make a respectable living. 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 year-old Marie Anne Charlotte Courty de Romange. The average age of The Man Who Did Everything 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. It was Laplace who named the meter, centimeter, and millimeter.

Nevertheless, during the 18 months of the Terror, as almost 17, French were executed and half a million imprisoned, his position became increasingly precarious. 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. 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 29 30 Enlightenment and the Anti-Bayesian Reaction s until his death in —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. Laplace replied calmly that a chain of natural causes would account for the construction and preservation of the celestial system. I have perused yours but failed to find His name even once. In 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 mathemati- The Man Who Did Everything cal construct. At the age of 62, Laplace, its chief creator and proponent, made a remarkable about-face. He switched allegiances to an alternate, frequencybased approach he had also developed. 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. By the s 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 on his 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. He had intuited its principle as a young man in Between and 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.

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. The formula, the method, and its masterful utilization all belong to Pierre Simon Laplace. He made probability-based statistics commonplace. Laplace put it into modern terms. In a sense, everything is Laplacean. Not until , 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. 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 The Man Who Did Everything planets and their satellites in our solar system originated in a swirl of dust. And on March 5, , at the age of 78, Laplace died, almost exactly 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. Yet Laplace had built his probability theory on intuition.

It provides an exact appreciation of what sound minds feel with a kind of instinct, frequently without being able to account for it. Nature would prove to be far more complicated than even Laplace had envisioned. 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 English mathematician Augustus de Morgan wrote in The Penny Cyclopaedia of that Laplace failed to credit the work of others; the accusation was repeated without substantiation for years until a detailed study by Stigler concluded it was groundless. degraded to servility for the sake of a riband and a title. Humble as Lincoln, proud as Lucifer.

Such statements published by a writer of one nation about one of the most distinguished men of a second nation, and wholly unsubstantiated by references, are in every way deplorable. He did so by publicizing in the then-extraordinary fact that the number of dead letters in the Parisian postal system remained roughly constant from year to year. After the French government published a landmark series of statistics about the Paris region, it appeared that many irrational and godless criminal activities, including thefts, murders, and suicides, were also constants. By stable statistical ratios were firmly dissociated from divine providence, and Europe was swept by a veritable mania for the objective numbers needed by good government. Unsettled by rapid urbanization, industrialization, and the rise of a market economy, early Victorians formed private statistical societies to study filth, criminality, and numbers. The chest sizes of Scottish soldiers, the number of Prussian officers killed by kicking horses, the incidence of cholera victims— statistics were easy to collect.

Even women could do it. No mathematical analysis was necessary or expected. That most of the government bureaucrats collecting statistics were ignorant of and even hostile to mathematics did not matter. Facts, pure facts, were the order of the day. Gone was the idea that we can use probability to quantify our lack of knowledge. Gone was the search for causes conducted by Bayes, Price, and Laplace. The statistician has nothing to do with causation. The French Revolution and its 35 36 Enlightenment and the Anti-Bayesian Reaction aftermath shattered the idea that all rational people share the same beliefs. The Western world split between Romantics, who rejected science outright, and those who sought certainty in natural science and were enthralled by the objectivity of numbers, whether the number of knifings or of marriages at a particular age. coined into science.

Awash in newly collected data, the revisionists preferred to judge the probability of an event according to how frequently it occurred among many observations. Eventually, adherents of this frequency-based probability became known as frequentists or sampling theorists. Arguing that probabilities should be measured by objective frequencies of events rather than by subjective degrees of belief, they treated the two approaches as opposites, although Laplace had considered them basically equivalent. Two of his most popular applications of probability were condemned wholesale. Laplace had asked: given that the sun has risen thousands of times in the past, will it rise tomorrow? and given that the planets revolve in similar ways around the sun, is there a single cause of the solar system?

Sometimes, though, he and his followers began answering these questions by assuming 50—50 odds. The simplification would have been defensible had Laplace known nothing about the heavens. He had also started his study of male and female birthrates with 50—50 odds, although scientists already knew that the likelihood of a male birth is approximately 0. Laplace agreed that reducing scientific questions to chance boosted the odds in favor of his deep conviction that physical phenomena have natural causes rather than religious ones. He warned his readers about it. His followers also sometimes weighted their initial odds heavily in favor of natural Many Doubts, Few Defenders laws and weakened counterexamples.

Critics pounded away at the fact that chance was irrelevant to the questions at hand. Few of the critics tried to even imagine other kinds of priors. But it has been very widely accepted—by [Augustus] de Morgan, by [William] Jevons, by [Rudolf] Lotze, by [Emanuel] Czuber, and by Professor [Karl] Pearson—to name some representative writers of successive schools and periods. He had developed two theories of probability and shown that when large numbers are involved they lead to more or less the same results. But if natural science was the route to certain knowledge, how could it be subjective? Soon scientists were treating the two approaches as diametric opposites. Lacking a definitive experiment to decide the controversy and with Laplace demonstrating that both methods often lead to roughly the same result, the tiny world of probability experts would be hard put to settle the argument.

Research into probability mathematics petered out. Within two generations of his death Laplace was remembered largely for astronomy. By not a single copy of his massive treatise on probability was available in Parisian bookstores. The physicist James Clerk Maxwell learned about probability from Adolphe Quetelet, a Belgian factoid hunter, not from Laplace, and adopted frequency-based methods for statistical mechanics and the kinetic theory of gases. An American scientist and philosopher Charles Sanders Peirce promoted frequency-based probability during the late s and early s. Inverse Probability being dead, they should be decently buried out of sight, and not embalmed in text-books and examination papers. The indiscretions of great men should be quietly allowed to be forgotten.

The first time, Bayes himself had shelved it. The second time, Price revived it briefly before it again died of neglect. This time theoreticians buried it. The funeral was a trifle premature. In scattered niches far from the eyes of disapproving theoreticians, Bayes bubbled along, helping real-life practitioners assess evidence, combine every possible form of information, and cope with the gaps and uncertainties in their knowledge. Into this breach between theoretical disapproval and practical utility marched the French army, under the baton of a politically powerful mathematician named Joseph Louis François Bertrand. To illustrate, he told about the foolish peasants of Britanny who, looking for the possible causes of shipwrecks along their rocky coast, assigned equal odds to the tides and to the far more dangerous northwest winds.

Bertrand argued that equal prior odds should be confined to those rare cases when hypotheses really and truly were equally likely or when absolutely nothing was known about their likelihoods. Alfred Dreyfus, a French Jew and army officer, was falsely convicted of spying for Germany and condemned to life imprisonment. Almost the only evidence against Dreyfus was a letter he was accused of having sold to a German military attaché. Alphonse Many Doubts, Few Defenders Bertillon, a police criminologist who had invented an identification system based on body measurements, testified repeatedly that, according to probability mathematics, Dreyfus had most assuredly written the incriminating letter. Poincaré believed in frequency-based statistics. Poincaré considered it the only sensible way for a court of law to update a prior hypothesis with new evidence, and he regarded the forgery as a typical problem in Bayesian hypothesis testing.

This colossal error renders suspect all that follows. I do not understand why you are worried. I do not know if the accused will be condemned, but if he is, it will be on the basis of other proofs. Such arguments cannot impress unbiased men who have received a solid scientific education. The judges issued a compromise verdict, again finding Dreyfus guilty but reducing his sentence to five years. The public was outraged, however, and the president of the Republic issued a pardon two weeks later. Dreyfus was promoted and awarded the Legion of Honor, and government reforms were instituted to strictly separate church and state. Many American lawyers, unaware that probability helped to free Dreyfus, have considered his trial an example of mathematics run amok and a reason to limit the use of probability in criminal cases.

As the First World War approached, a French general and proponent of military aviation and tanks, Jean Baptiste Eugène Estienne, developed elaborate Bayesian tables telling field officers how to aim and fire. Estienne 39 40 Enlightenment and the Anti-Bayesian Reaction also developed a Bayesian method for testing ammunition. Mobilized for the national defense, professors of abstract mathematics developed Bayesian testing tables that required destroying only 20 cartridges in each lot of 20, Instead of conducting a predetermined number of tests, the army could stop when they were sure about the lot as a whole.

During the Second World War, American and British mathematicians discovered similar methods and called them operations research. In each case self-taught statisticians resorted to Bayes as a tool for making informed decisions, first about telephone communications and second about injured workers. The first crisis occurred when the financial panic of threatened the survival of the Bell telephone system owned by American Telephone and Telegraph Company. Unfortunately, Bell telephone circuits were often overloaded in the late morning and early afternoon, when too many customers tried to place calls at the same time. No company could afford to build a system to handle every call that could conceivably be made at peak times.

Edward C. Molina, an engineer in New York City, considered the uncertainties involved. Molina, whose family had emigrated from Portugal via France, was born in New York in The Bell System of phone companies was adopting a new mathematical approach to problem solving. Methods for utilizing both statistical and nonstatistical types of evidence were needed. The result was a cost-effective way for Bell to deal with uncertainties in telephone usage. To automate the system Molina conceived of the relay translator, which converted decimally dialed phone numbers into routing instructions. Then he used Bayes to analyze technical information and the economics of various combinations of switches, selectors, and trunking lines at particular exchanges. After women won the right to vote in , Bell feared a backlash if it fired all its operators, so it chose an automating method that merely halved their numbers.

Between the world wars, employment of operators dropped from 15 to 7 per 1, telephones even as toll call service increased. Probability assumed an important role in the Bell System, and Bayesian methods were used to develop basic sampling theory. Molina won prestigious awards, but his use of Bayes remained controversial among some Bell mathematicians, and he complained he had trouble publishing his research. Some of his problems may have stemmed from his colorful character. He followed the Russo-Japanese war so 41 42 Enlightenment and the Anti-Bayesian Reaction avidly that his colleagues nicknamed him, not fondly, General Molina. Government safety regulations were nonexistent, however, and 1 out of every industrial workers was killed on the job between and , and many more were injured.

Yet, unlike most of Europe, the United States had no system for insuring sick and injured workers, and most blue-collar families lived one paycheck away from needing charity. Federal judges ruled that injured employees could sue only if their bosses were personally at fault. In a U. Department of Labor statistician could think of no other social or legal reform in which the United States lagged so far behind other nations. The tide turned as growing numbers of workers joined the American Federation of Labor and as local juries started awarding generous settlements to their disabled peers.

At that point employers decided it was cheaper to treat occupational health as a predictable business expense than to trust juries and encourage unionization. In an avalanche of no-fault laws passed between and all but eight states began requiring employers to insure their workers immediately against occupational injuries and illness. This was the first, and for decades the only, social insurance in the United States. The legislation triggered an emergency. Normally, the price of an insurance premium reflects years of accumulated data about such factors as accident rates, medical costs, wages, industrywide trends, and particulars about individual companies. No such data existed in the United States. Not even the most industrialized states had enough occupational health statistics to price policies for all their industries. The industrial powerhouse of New York State had only enough experience to price policies for machine printers and garment workers; South Carolina had only enough for cotton spinners and weavers; and St.

Louis and Milwaukee for beer brewers. In Nebraska had only Many Doubts, Few Defenders 25 small manufacturers of any kind. Germany had collected accident statistics for 30 years, but its industrial conditions were safer, and because its data were collected nationwide, premiums could be based on industrywide information. It was a nightmare to keep any mathematically trained statistician awake at night—not that the United States had many. Still, premiums had to be priced accurately: high enough to keep the insurance company solvent for the life of its insurees and individualized enough to reward businesses with good safety records. In an extraordinary feat, Isaac M. Rubinow, a physician and statistician for the American Medical Association, organized by hand the analysis, classification, and tabulation of literally millions of insurance claims, primarily from Europe, as a two- or three-year stopgap until each state could accumulate statistics on its occupational casualties.

Whit- 43 44 Enlightenment and the Anti-Bayesian Reaction ney was an alumnus of Beloit College and had no graduate degrees but had taught mathematics and physics at the universities of Chicago, Nebraska, and Michigan.

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His experiments had a serious purpose, though. Laplace did not state his idea as an equation. He also made important advances in celestial mechanics, mathematics, physics, biology, Earth science, and statistics. The Ship That Wouldn t Die. No such data existed in the United States. He had also started his study of male and female birthrates with 50—50 odds, although scientists already knew that the likelihood of a male birth is approximately 0.

and I was torn between conflicting emotions: a. ISBN hardback 1. Andrews, Frank Anscombe, George Apostolakis, Robert A. Page F Jeffreys was an Earth scientist who studied earthquakes, tsunamis, and tides. Eventually, given enough tosses of the ball, Bayes could narrow the range of places where the cue ball was apt to be.