Insight Editions · No. 30
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
Sharon Bertsch McGrayne
Sharon Bertsch McGrayne recounts the history of Bayes' rule, tracing its controversial development and its applications in wartime codebreaking, submarine detection, medicine, and modern computing.
A study guide to The Theory That Would Not Die by Sharon Bertsch McGrayne. Shared free for personal reading; not for sale and not endorsed by the authors or publishers. Please read and buy the original.
Preface and Note to Readers
McGrayne opens with a comparison. Geologists had the evidence for continental drift in 1912 and argued for 50 years that continents cannot move. The fight over Bayes' rule lasted 150 years and concerned something broader: how we weigh evidence, change our minds and decide under uncertainty. It was not settled until the start of the twenty-first century.
The rule itself is one line: an initial belief, updated with objective new information, gives a new and improved belief. To converts this was learning from experience made exact; to opponents it was "subjectivity run amok." Bayes found it in the 1740s, his friend Richard Price published it, and Pierre Simon Laplace rediscovered it in 1774 and built it into the method we use. Only convention gives it Bayes' name.
After Laplace, academics declared the method dead while practical people kept using it: workers' compensation insurance, the Bell system in the panic of 1907, the defence of Dreyfus, Allied artillery, Turing's attack on Enigma, the hunt for a lost H-bomb and submarines, nuclear plant safety, smoking and heart disease, election-night forecasts on NBC. The reason for the fight was philosophical: Bayes is a measure of belief, and it claims we can learn from missing, inadequate and approximate data, which offended the ideal of objective science.
Bayes' rule is a flesh-and-blood story of a small group of beleaguered believers. Its fate was tangled up with wartime secrecy, it was a theorem waiting for a computer and software, and it won almost overnight in the 1990s, refreshed by physicists, computer scientists and AI researchers, because suddenly it worked.
By 2011 Bayes ran spam filters, Coast Guard searches, gene regulation studies, web recommendations and machine translation, and served as a model of how brains learn. The man who once called it "the crack cocaine of statistics... seductive, addictive and ultimately destructive" was recruiting Bayesians for Google. McGrayne adds a note: she uses "probability," "likelihood" and "odds" in their technical senses, which is why "probability" appears so often.
BELIEF + DATA → BETTER BELIEF. One line; 150 years of war over whether belief belongs in science.
Write one decision you made this week with incomplete data. Name your prior, the evidence that moved you, and the posterior you acted on. That is the book's subject, not a formula alone.
Invert the thesis: suppose Bayes won by philosophy, not by computation and field success. Then the secrecy of Bletchley, the actuarial underground and MCMC would be footnotes. McGrayne's whole arc says the opposite: pragmatism beat ideology once the computer arrived.
What the preface is asking
McGrayne frames Bayes as a flesh-and-blood story of beleaguered believers, not a dry theorem. Continental drift took 50 years; Bayes took 150. The stakes are how we weigh evidence, change our minds and decide under uncertainty. Wartime secrecy hid the method's best proofs; ivory-tower frequentism buried the rest. By 2011 the same rule ran spam filters, Coast Guard searches, gene studies, web recommendations and machine translation, and a former enemy was hiring Bayesians for Google.
1. Causes in the Air
The story starts with a thought experiment. Thomas Bayes (died 1761, aged 59) was a Presbyterian Dissenter, barred from English universities, who studied at Edinburgh and served as a minister. In his grandfather's generation 2,000 Dissenters had died in English prisons, and many good mathematicians were amateurs for that reason.
A minister among mathematicians
- 1731: a pamphlet arguing God gives people "the greatest happiness of which they are capable."
- After Bishop Berkeley attacked Newton's calculus and "infidel mathematicians," Bayes published a defence of Newton's calculus, his only mathematical publication in his lifetime.
- 1742: elected to the Royal Society, then a club of dues-paying gentlemen amateurs.
- At the spa of Tunbridge Wells, a wealthy bachelor (the family fortune came from Sheffield cutlery), he joined a peer-review network run by Philip, Second Earl of Stanhope, and Royal Society secretary John Canton.
Hume and the inverse problem
In 1748 David Hume argued that we can never be certain of cause and effect: umbrellas and rain go together without umbrellas causing rain, and the sun rising thousands of times does not guarantee tomorrow. The "design of the world" does not prove a creator. De Moivre's Doctrine of Chances worked from cause to effect (the odds of four aces). Nobody had worked backwards, from effect to cause: if a player deals himself four aces three hands running, how likely is it that the deck is loaded? That is the inverse probability problem.
The table experiment
Sometime between 1746 and 1749 Bayes imagined a perfectly level square table. With his back to it, a friend throws a cue ball, then throws more balls and reports only whether each lands left or right of the first. Each report narrows the region where the cue ball probably lies. Bayes could never know its exact position, but he could say with growing confidence that it lay between two bounds, and how confident he could be.
Initial belief + objective new data = a new and improved belief. Later names: the prior (initial belief), the likelihood (probability of the data under each hypothesis) and the posterior (revised belief). Each posterior becomes the next prior. In one line: prior times likelihood is proportional to the posterior. Historian Andrew Dale: "the likelihood is what remains of Bayes's Theorem once the prior is removed."
Two targets for attack
- The guess. Bayes made "a guess whereabouts it's probability is" and checked it. Critics would be horrified by a subjective hunch inside mathematics.
- Equal probabilities. When he knew nothing, he assumed every position equally likely, later called the law of insufficient reason or uniform prior. It had roots in the Church's usury rules (equal shares in uncertain ventures) and became the lightning rod for the charge that Bayes quantified ignorance.
Bayes then put the essay away among papers dated 1746 to 1749. In 1749 physician David Hartley mentioned an "ingenious friend" who had solved the inverse problem; Stephen Stigler (1999) floated the blind Cambridge mathematician Nicholas Saunderson, but McGrayne judges Bayes the author. Why Bayes suppressed it is unknown.
Richard Price
Minister, amateur mathematician and later a hero of the American and French revolutions. Admired by Franklin, Jefferson, Adams and Mary Wollstonecraft; Yale gave honorary degrees in 1781 to George Washington and Price. He spent nearly two years editing Bayes' papers, added references, cut background and, sadly, threw out Bayes' introduction.
Price saw the essay as an answer to Hume: reasoning from the world back to its cause showed "the existence of the Deity." Bayes' own text never mentions God. "An Essay toward solving a Problem in the Doctrine of Chances" appeared in the Philosophical Transactions in 1764 (read to the Society in December 1763), with a gambling title to dodge controversy. Nobody cited it for 17 years.
| Did | Did not |
|---|---|
| Turned probability from gambler's frequency into informed belief | Write the modern formula (he used Newton's geometric areas) |
| Combined hunches with repeatable data | Develop it into a general method |
| Reasoned from effects to probable causes | Mention hypothesis testing, decisions or uses |
| Pioneered the update loop | Name it; "Bayesian" dates from the 1950s |
McGrayne says it should be the Bayes-Price rule. Either way, it reached almost nobody except the one person who could use it.
Price, Hume and the delayed reception
Richard Price was not a footnote. He framed Bayes's essay as a reply to David Hume (1748): if causes cannot be proved from effects, design arguments for God collapse. Price cut Bayes's introduction, added an application and read the paper to the Royal Society in December 1763 (printed 1764). For 17 years almost nobody cited it. The gambling title ("Doctrine of Chances") was cover for theological stakes. Bayes himself had published almost nothing mathematical in life except a defence of Newton's fluxions against Berkeley. He was elected FRS in 1742 as a dues-paying amateur in a club, not as a professional.
- Essay written roughly 1746 to 1749; Bayes dies 1761; Price publishes 1764.
- Cue-ball thought experiment: prior flat across the table; each "left/right" report shrinks the credible zone.
- 17 years of near silence after publication.
HUME → TABLE → PRICE. Inverse problem; cue-ball thought experiment; friend who published.
Stand with your back to a desk. Have a friend drop a coin, then report only "left" or "right" of a hidden mark for five drops. Sketch how your credible interval for the mark shrinks. That is Bayes' table without the Latin.
Treat the uniform prior as "no information." On the logit scale it is not flat. Name one modern problem (spam, credit risk, rare disease) where a uniform prior would be reckless, and say what prior you would use instead.
Dissenter mathematics and why the essay slept
Bayes lived as a Presbyterian Dissenter: barred from Oxford and Cambridge, educated at Edinburgh, minister at Tunbridge Wells. His only lifetime mathematical publication defended Newton's calculus against Berkeley. The Royal Society of 1742 was still a club of dues-paying amateurs; Stanhope and Canton ran an informal peer-review network that circulated unpublished work. Bayes solved the inverse problem between 1746 and 1749, then filed the essay among dated papers. Price spent nearly two years editing, cut the introduction, and framed the result as an answer to Hume on design. The gambling title ("Doctrine of Chances") was political cover. Nobody cited it for 17 years. McGrayne's judgment stands: it should be the Bayes-Price rule.
Bayes, a Dissenting amateur, answered the inverse probability question with a table thought experiment and a guessed, uniform prior, then shelved it. Price edited and published it in 1764 for theological reasons. The guess and the equal priors would draw fire for 150 years.
2. The Man Who Did Everything
Pierre Simon Laplace, born 23 March 1749 in Beaumont-en-Auge, Normandy, the son of an innkeeper-mayor, is McGrayne's real hero: "the Einstein of his age." The return of Halley's comet in 1759, on schedule as predicted by Clairaut, Lalande and Nicole-Reine Lepaute, convinced his generation that nature obeys mathematical laws, not divine anger.
From Caen to Paris
At the University of Caen Laplace learned Leibniz's algebraic calculus and lost his faith. Refusing to become an abbé, he went to Paris in 1769 with a letter to d'Alembert, who within days got him a teaching job at the Royal Military School. Laplace did not go home for 20 years and skipped his father's funeral. To do research full time he needed election to the Royal Academy of Sciences, Europe's most professional scientific body. He sent it 13 papers in five years; it rejected him six times before electing him in 1773, at 24.
The data problem
The great question was whether the solar system was stable: Jupiter seemed to be speeding up and Saturn slowing down. Newton had invoked God to keep things in order. Laplace decided the problem was not Newton's theory but the data: observations from Chinese astronomers in 1100 BC to Arabs in AD 1000, of very unequal quality. Astronomers simply averaged their three best readings with no justification. Reading de Moivre in the school library, and against d'Alembert's view that probability was too subjective, Laplace turned to probability: "We owe to the frailty of the human mind one of the most delicate and ingenious of mathematical theories."
The 1774 memoir
Spurred by fear of competition, Laplace wrote the "Mémoire on the Probability of the Causes Given Events" (published 1774), the first version of what we call Bayes' rule, discovered independently: he was 15 when Bayes' essay appeared in an obscure English journal.
The probability of a cause given an event is proportional to the probability of the event given that cause. In modern notation, P(C|E) = P(E|C) / ΣP(E|C), assuming all causes equally likely at the start. Bayes solved a special table problem in two steps; Laplace stated a general principle for any problem and any set of causes.
He first tried it on gambling (an urn with an unknown ratio of tickets: 45 equations over four pages; dividing the kitty in an unfinished game of piquet), then on averaging astronomical observations, which he could not justify until 1810.
Bayes' starting guess arrives
In 1781 Price visited Paris and told Condorcet about Bayes. Laplace adopted the starting guess and could now recycle each answer as the next starting point. Condorcet's introduction credited Bayes, and Laplace later wrote that Bayes did it "in an acute and very ingenious, though slightly awkward, manner." Two problems remained: the method forced equal priors, which Laplace as a scientist disliked ("the science of chances must be used with care"), and it produced enormous numbers: 20,000 x 19,999 x ... divided by 1 x 2 x ... x 10,000, with no machines and no assistant until about 1785.
Babies, populations and juries
- Birth ratios. Paris data covering 60 years confirmed John Graunt's 1662 finding that slightly more boys are born. A 52:48 ratio with 58,000 boys would have meant multiplying 0.52 by itself 57,999 times, so Laplace invented generating functions, transforms and asymptotic expansions. He bet boys would outnumber girls for 179 years in Paris and 8,605 years in London. When Buffon found a Burgundy village with more girls for five years, Laplace replied that a few facts cannot overrule many. In 1812 he called it "a general law for the human race."
- Population of France. A formula of births x 26 gave about 25.3 million; modern estimates are nearer 28 million. Combining parish records with local headcounts, Laplace in 1786 gave odds of 1,000 to 1 that his estimate was within half a million, and in 1802 advised Napoleon on a sample census of about a million people in 30 departments.
- Juries. With 50:50 prior odds of guilt, an eight-person jury deciding by simple majority would be wrong with probability 65/256, more than one in four. Laplace used this to argue for abolishing the death penalty.
Laplace solved the Jupiter-Saturn problem without probability: they oscillate in an 877-year cycle, so the solar system is stable, the biggest advance in physical astronomy since Newton.
Revolution and Napoleon
Laplace married Marie Anne Charlotte Courty de Romanges (18 to his 39); the Bastille fell 16 months later. He named the metre, centimetre and millimetre, was arrested and released at 4 a.m., and was purged from the metric commission. Lavoisier was guillotined; Condorcet died in jail. Laplace emerged as a dean of French science; from the 1780s until his death France led world science. He had passed the young Napoleon in his military exam, served briefly as interior minister and sat in the Senate. When Napoleon asked in 1802 why God was absent from his book, Laplace answered: "Sire, I have no need of that hypothesis."
The general rule and the turn to frequencies
In 1810 Laplace announced the central limit theorem: averages of many similar terms are approximately normal. This justified the mean and let him handle almost any data. At 62 he switched allegiance to the frequency-based approach he had also developed, because with lots of reliable data the two approaches gave nearly the same answers. Not until the mid-twentieth century did anyone learn that they can seriously disagree even with big data. Between 1810 and 1814 he nevertheless wrote down the general form of Bayes' rule, with unequal priors:
P(C|E) = P_prior(C) x P(E|C) / Σ[P_prior(C') x P(E|C')]
In 1815, using tables by Alexis Bouvard, he bet 11,000 to 1 that Bouvard's mass of Saturn was within 1%, and a million to one for Jupiter. Space-age measurements say he would have won both bets. Glenn Shafer: "Laplace did everything, and we just read stuff back into Thomas Bayes." McGrayne proposes "BPL" (Bayes-Price-Laplace). Laplace, made a marquis by Louis XVIII, died on 5 March 1827, almost exactly 100 years after Newton. His creed: "the theory of probability is nothing but good common sense reduced to mathematics."
McGrayne says the academy "waited five years" and elected Laplace on 31 March 1773, then calls him 24 that year. Both fit: he arrived in 1769 and was rejected repeatedly from about 1771. She also suggests Laplace "invented" the central limit theorem in 1810; de Moivre had found the normal approximation to the binomial in 1733, and Laplace generalised it.
Laplace's industrial applications
Laplace rebuilt inverse probability in his 1774 memoir, then spent decades applying it. Sex ratios at birth, judicial error rates, triangulation and celestial mechanics all became Bayesian case studies. The 1786 population estimate for France multiplied births by about 26 and stated odds openly; modern estimates sit near 28 million for the period, so he ran roughly 10% low with honest uncertainty. The Théorie analytique des probabilités (1812) is the manual Bayes never wrote. Late-life frequency talk gave frequentists a quote weapon; the chapter's operational reading is that Laplace used inverse probability when data were scarce and frequencies when they were plentiful.
LAPLACE = INDUSTRIAL BAYES. Births, juries, orbits, France's population; then a drift toward frequencies.
After s successes in n trials, Laplace's succession rule says the next trial has probability (s+1)/(n+2). Compute it for 0/0, 1/1 and 9/10. Notice how the uniform prior still pulls hard when n is small.
Quote Laplace's late frequency talk as a renunciation of inverse probability. The chapter shows he used inverse probability when data were scarce and frequencies when they were plentiful. If you erase that distinction, you misread both Laplace and the later frequentist attack.
Laplace's working method
Born 23 March 1749 in Beaumont-en-Auge, Laplace rode the post-Halley conviction that nature obeys laws. In Paris under d'Alembert's patronage he rebuilt inverse probability from first principles (1774 memoir), then spent forty years applying it: sex ratio at birth, judicial error, triangulation, celestial mechanics. The 1786 population estimate of France (multiplier about 26 on births) came with odds he stated openly. The Théorie analytique des probabilités (1812) is the industrial manual Bayes never wrote. Late in life, with more data, Laplace emphasised frequencies; frequentists later quoted that turn against Bayesian "theology." The chapter's lesson is operational: the formula worked in his hands long before philosophy settled what probability means.
3. Many Doubts, Few Defenders
After Laplace died, Bayes' rule was disdained, reformed, grudgingly tolerated and nearly wiped out by theorists, while practical people kept using it for the military, telephones, insurance and medicine.
Laplace's reputation and the cult of facts
Laplace's name was blackened by unsupported charges: De Morgan (1839) said he failed to credit others (Stigler later showed this was groundless), Maximilien Marie called him an ultra-royalist, the 1912 Britannica called him servile, and E. T. Bell's Men of Mathematics (1937) called him "From Peasant to Snob." Karl Pearson called the Britannica writer "one of the most superficial writers that ever obscured the history of science."
Laplace also started a craze for statistics. In 1827 he publicised that the number of dead letters in the Paris post was nearly constant each year; murders, thefts and suicides turned out to be stable too. Victorian statistical societies collected facts, pure facts, with no mathematics. A correspondent told Florence Nightingale in 1861: "The statistician has nothing to do with causation." "Subjective" became an insult.
The frequentist turn
John Stuart Mill called probability "an aberration of the intellect" and "ignorance... coined into science." The revisionists defined probability as the relative frequency of events over many observations; they became the frequentists. Critics fixed on Laplace's use of 50:50 starting odds for questions he understood well, such as whether the sun will rise tomorrow. His sunrise bet, based on 5,000 years of history, was 1,825,214 to 1. Keynes noted it was rejected by Boole (arbitrary), Venn (contrary to experience) and Bertrand (ridiculous), but accepted by De Morgan, Jevons, Lotze, Czuber and Karl Pearson. Critics equated Bayes with equal priors and condemned the whole rule.
By 1850 no copy of Laplace's probability treatise was for sale in Paris. Maxwell took frequency methods from Quetelet; Peirce promoted frequentism. In 1891 George Chrystal wrote the obituary: "The laws of... Inverse Probability being dead, they should be decently buried out of sight." It was the third death of Bayes' rule: shelved by Bayes, neglected after Price, buried by theorists.
Practical Bayes survives
Joseph Bertrand reformed Bayes for French artillery officers facing unknown enemy positions, air density, wind and hand-forged cannons. He insisted equal priors be used only when hypotheses really were equally likely, mocking Breton peasants who gave equal odds to tides and to deadly north-west winds as causes of shipwrecks. French and Russian gunners used his textbook from the 1880s to the Second World War. At Dreyfus's 1899 retrial, Henri Poincaré, a frequentist, used Bayesian reasoning to demolish Alphonse Bertillon's "mathematical gibberish" about the handwriting: "This colossal error renders suspect all that follows." The judges, all trained on Bertrand, still convicted, but reduced the sentence; a pardon followed two weeks later.
During the First World War General Jean Baptiste Estienne produced Bayesian firing tables and a Bayesian ammunition test: with France's industry captured, mathematicians designed tables that destroyed only 20 cartridges per lot of 20,000 and let testers stop as soon as they were sure, a forerunner of sequential analysis.
Molina and the Bell system
The 1907 panic nearly sank AT&T's Bell system, and regulators wanted proof of better service. Lines were overloaded in late morning and early afternoon but idle 80% of the time. Edward C. Molina (born 1877, no college degree) taught himself to become America's leading expert on Bayes-Laplace probability. He insisted on distinguishing Bayes' original theorem from Laplace's general version, which combines observations with "collateral" information. He used it to size telephone systems and to plan automation with his relay translator. Operators fell from 15 to 7 per 1,000 telephones between the wars. Bell kept many of his Bayesian papers as proprietary secrets.
Credibility: Bayes for workers' compensation
Between 1890 and 1910 one in 318 American industrial workers was killed on the job. Between 1911 and 1920 all but eight states passed no-fault workers' compensation laws, the first US social insurance. Premiums had to be set overnight with almost no data: Nebraska in 1909 had only 25 small manufacturers. "When will Nebraska be able to determine its pure premium on 'suspenders without buckles'?"
- Isaac Rubinow tabulated millions of mostly European claims ("every scrap of information" must be used) and founded the Casualty Actuarial Society (1914) with 11 actuaries.
- In spring 1918 Albert Whitney's committee stripped Bayes down: assume every business in a class faces equal risk and every actuary is equally skilled at adding judgment. Industry experience is the prior; the firm's own history is the new data.
Premium = Z x (firm's own experience) + (1 - Z) x (class experience), with Z = P / (P + K), where P measures the volume of the firm's own experience and K is a constant. Small firms are pulled toward the class mean; large firms rely more on themselves. This is the shrinkage effect Charles Stein would explain in the 1950s. "Not so great a discovery as E = mc²... but it has made life much easier for insurance men for generations." Asked where the theory came from, Whitney said: "In Michelbacher's dining room."
Pearson, Fisher and the anti-Bayesian trio
Karl Pearson, atheist, socialist and eugenicist, ruled British statistics and used Bayes "till better are forthcoming." Keynes (1921) said it had "a smack of astrology, of alchemy"; Julian Coolidge: "We use Bayes' formula with a sigh."
Nearly blind, rumpled, quick to take offence: Kruskal called his life "a sequence of scientific fights." Founder of the Cambridge University Eugenics Society (1911), funded through Leonard Darwin (he wrote 200+ reviews for the Eugenics Society's journal, 1914 to 1934). At Rothamsted from 1919 he created randomisation, significance tests, maximum likelihood, analysis of variance and experimental design; Statistical Methods for Research Workers (1925) made frequency the default method. On Bayes: "an impenetrable jungle," "founded upon an error, and must be wholly rejected," equal priors a "staggering falsity."
Egon Pearson in 1925 ran the largest Bayesian study between Laplace and the 1960s (taxis with LX plates, pipe smokers on Euston Road, chestnut colts) looking for "nature's prior." With Jerzy Neyman he then built Neyman-Pearson hypothesis testing (1933): reject a hypothesis only in favour of a more probable alternative. Neyman called the equal-prior shortcut "illegitimate."
| Frequentist (Fisher, Neyman) | Bayesian |
|---|---|
| Probability of the data given the hypothesis | Probability of the hypothesis given the data |
| Only repeatable events | Single events (rain tomorrow) allowed |
| Each data set a separate problem | Every datum updates the belief |
| No subjective priors | Priors encode prior knowledge |
Fisher and Neyman then fell out for 30 years (the 1934 Royal Society Latin square row). At UCL Neyman's group drank India tea at 3.30 and Fisher's drank China tea after 4.15. Collective noun for statisticians: "A quarrel." Neyman made Berkeley an anti-Bayesian stronghold.
Betting on belief
Three men independently proposed that belief can be measured by a bet: Émile Borel (1924), Frank Ramsey (1926 talk, adding utility and expected-utility maximisation; he died at 26) and Bruno de Finetti ("the art of guessing"), who gave subjective probability its mathematical foundation. Others: Erik Essen-Möller's 1938 paternity index, Bayesian in all but name, used in US courts for 50 years until DNA; Lowell Reed at Johns Hopkins (1936) set X-ray doses using only 27 cats.
Jeffreys keeps Bayes alive
Harold Jeffreys, Cambridge geophysicist (fellow for 75 years), friends with Fisher despite disagreeing. From earthquake wave arrival times he inferred in 1926 that the Earth's core is liquid, probably molten iron. He devised formal rules for choosing priors and applied probability to hypotheses themselves.
A p-value is the probability of data at least as extreme as observed, assuming the hypothesis. Jeffreys found it strange to reject a hypothesis because of outcomes that did not happen. "There has not been a single date in the history of the law of gravitation when a modern significance test would not have rejected all laws." Jim Berger: with p = 0.01 the odds against the hypothesis can still be only about 9 or 10 to 1, "not earth-shaking." Steven Goodman (1999): "The p-value is almost nothing sensible you can think of."
The Fisher-Jeffreys debate in the Proceedings of the Royal Society ran until editors stopped it. Jeffreys' Theory of Probability (1939) was the only systematic guide to Bayesian science; Fisher said it had "a logical mistake on the first page which invalidates all the 395 formulae." Jeffreys lost in practice because Fisher was persuasive, social scientists wanted objective methods, quantum physics used frequencies, Fisher's cookbook was easier (Jeffreys offered five evidence grades hidden in Appendix B) and Jeffreys ignored decision-making, which war would make central.
The preface says Jeffreys deduced "(erroneously)" that the core is molten iron. He was half right: the outer core is liquid iron-nickel, but Inge Lehmann found a solid inner core in 1936. Jeffreys also rejected continental drift as late as 1970, which McGrayne notes. Fisher's eugenics gets only brief mention here.
Jeffreys' complaint became mainstream. The ASA statement on p-values (2016), the 2019 Nature call (800+ signatories) to retire "statistical significance," and Benjamin and 71 co-authors' (2018) proposal to move the threshold to 0.005 all echo him. Bayes factors, the modern form of Jeffreys' test, are standard in psychology software such as JASP. The ASA President's Task Force (2021) defended p-values when properly used, so the debate is still open.
Taboo and the geophysicist's fort
Mill, Boole, then Fisher and Neyman-Pearson redefined probability as long-run frequency. Equal-prior inverse probability looked unscientific. Jeffreys's Theory of Probability (1939) kept scientific induction Bayesian almost alone in the academy. Actuaries kept Whitney's 1918 credibility formula in work clothes. Ramsey (1926) and de Finetti made belief a coherent bet. The exile is institutional: textbooks and journals enforced a taboo that money and war ignored.
MILL-BOOLE-FISHER vs JEFFREYS. Subjective priors become shameful; one geophysicist holds the fort.
Write Fisher's objection to inverse probability in one sentence, then Jeffreys's reply in one sentence. If you cannot, reread the chapter's contrast table before going on.
Assume frequentism is "objective" and Bayes is "subjective." Then explain how choosing a 5% significance level, a stopping rule and a sample size are not subjective. The shadow exposes the chapter's real fight: which subjectivity is allowed to wear a lab coat.
The exile's machinery
After Laplace, Mill, Boole and later Fisher and Neyman-Pearson made equal-prior inverse probability look unscientific. Probability was redefined as long-run frequency in a hypothetical sequence. Jeffreys's Theory of Probability (1939) kept inverse probability alive for geophysics and induction, almost alone in the academy. Actuaries quietly kept credibility formulae that are Bayes in work clothes (Whitney, 1918). The chapter's point is institutional: textbooks and journals enforced a taboo that practitioners ignored when money or lives were on the line.
4. Bayes Goes to War
By 1939 Bayes was taboo among statisticians. But war asked a question frequentism could not answer quickly: how do you make life-and-death decisions fast, without waiting for complete information? In deep secrecy, some of the century's best mathematicians answered with Bayes.
The stakes
- 1 of 4 reflector combinations, each settable 26 ways
- 3 of 8 rotors: up to 336 wheel orders
- More than 150 billion plugboard combinations
- 17,000 clip positions and 17,000 starting positions (half a million on four-rotor machines)
- Many settings changed every two days, sometimes every 8 or 24 hours
The Poles got there first: Marian Rejewski used group theory to work out the rotor wiring, and by early 1938 the Poles read 75% of German army and air force traffic. In July 1939, at a safe house in the Pyry Forest, they handed their methods and a replica Enigma to the French and British. Britain's codebreakers at the Government Code and Cypher School (GC&CS) were mostly classicists and crossword solvers; mathematicians were "strange fellows." Britain's top statisticians were sidelined: Jeffreys ignored, Fisher distrusted for writing to a German colleague, Neyman too slow and theoretical. Wartime data fell to actuaries, physicists and pure mathematicians who did not know Bayes was unscientific. "Their ignorance proved fortunate."
Turing, Banburismus and the ban
Alan Turing, 27, reported to Bletchley Park on 4 September 1939. He redesigned the Polish bombe, an electromechanical machine that eliminated impossible wheel settings (prototype in March 1940, with Gordon Welchman and Harold "Doc" Keen). Then, because "no one else was doing anything about it and I could have it to myself," he took on naval Enigma.
A bombe took 18 minutes per wheel order, so testing all 336 could take four days. Turing invented Banburismus, named after the Banbury printers who made its long punched card strips. Wrens punched intercepts into the sheets and slid pairs over each other to count coincident letters. Bayesian scoring of the results identified two of the three wheels and cut the wheel orders to test from 336 to as few as 18. Cribs came from predictable texts: weather reports, "Beacons lit as ordered," and the word "ein," found in 90% of messages (clerks catalogued 17,000 encipherments of it by hand).
Turing needed a unit of evidence. The ban is the weight of evidence that multiplies the odds for a hypothesis by 10; he usually worked in decibans, "about the smallest change in weight of evidence that is directly perceptible to human intuition." Because bans are logarithms, evidence is added rather than multiplied. When a hypothesis reached odds of about 50 to 1, the codebreakers treated it as settled. The ban is essentially the same as Shannon's bit, found at Bell Labs at about the same time.
Turing also targeted a required amount of evidence and stopped sampling once he had it, an early form of sequential analysis. Statistical assistant I. J. "Jack" Good, hired partly for his chess, asked "Aren't you essentially using Bayes' theorem?" Turing: "I suppose." Good told George Barnard (against the rules) about sequential Bayes factors, and Barnard said the Ministry of Supply used the same idea to accept or reject lots, roughly halving the tests needed. Abraham Wald usually gets the credit; Good argued Turing, Wald and Barnard all deserve it.
Pinches, blackouts and Churchill
Banburismus needed the U-boats' bigram tables, which had to be "pinched." Ian Fleming's Operation Ruthless (a fake ditched German plane) was cancelled; material came from weather ships off Iceland and a trawler off Norway. By June 1941 Bletchley read U-boat traffic within an hour; for 23 days no North Atlantic convoy was attacked. In October 1941 Turing, Welchman, Hugh Alexander and Stuart Milner-Barry wrote straight to Churchill for staff; he ordered "Action this day."
A fourth rotor in February 1942 caused the Great Blackout. On 30 October 1942 Anthony Fasson, Colin Grazier and Tommy Brown (16) boarded the sinking U-559 off Egypt; the two officers drowned, but the tables reached Bletchley, which was reading U-boat traffic again by mid-December. Turing sailed to the US in November 1942, found the US Navy ignoring Banburismus, and took tea at Bell Labs with Claude Shannon, who was using Bayes in information theory and cryptography. In a perfectly secret system, Shannon showed in 1949, prior equals posterior: nothing is learned.
Kolmogorov's guns and Koopman's search
In Moscow, Andrei Kolmogorov, who had axiomatised probability in 1933 and was personally a frequentist, told the artillery that 50:50 priors were "not only arbitrary but surely wrong," then recommended them anyway for repeated fire at a small area, with deliberate dispersion of shots. His 1941 paper passed the censors because they could not understand it.
US Navy O.R. (led by Philip Morse and George Kimball) assigned Bernard Koopman to narrow the 10,000-square-mile circles from radio direction-finding. He began with a prior for the U-boat's location ("no rational prospector would search a region for mineral deposits" without one), then folded in detection data such as dirty windscreens and hazy 7x50 binoculars (crews were a quarter as efficient as lab lookouts). His rule: spread search effort according to the logarithm of the probability, update with Bayes every two hours if nothing is found. In one worked case with 4 planes and 5 lanes, two planes go to the 40% lane, two to the 30% lane, none to the 10% lanes. Minesweepers adopted it.
Colossus and D-Day
German high command used Lorenz teleprinter ciphers ("Tunny"). Turing's Turingery (July 1942) began with a 50:50 guess and added clues until odds reached 50 to 1. Tommy Flowers built Colossus, the first large-scale electronic digital computer, which broke its first message on 5 February 1944; Colossus II was ready by 1 June. McGrayne tells how a decrypted Hitler order to Rommel reached Eisenhower, who said "We go tomorrow." Ten Colossi were working by the war's end. Turing's Bayesian scoring became "a major aid" in breaking Lorenz wheel patterns. Edward Simpson, Ian Cassels and Jimmy Whitworth independently built Bayesian methods for Japan's JN-25.
Secrecy and Turing's fate
In 1945 Churchill ordered the evidence destroyed; most Colossi were broken up, and the story stayed secret until after 1973 (Britain read Soviet Lorenz traffic in the Cold War). Effects: the world believed the ENIAC came first; Turing, given only an OBE, arrived at the National Physical Laboratory "as a nobody." He designed the ACE stored-program computer (built 1950, then the fastest in the world), moved to Manchester, wrote early software and proposed the Turing Test. Arrested in 1952 for homosexual activity, he lost his clearance, chose chemical castration over prison and died on 7 June 1954. In 1956 Britain knighted the spy Anthony Blunt. Good went on to write about 900 articles on Bayes and became a leader of its revival.
- The Eisenhower "We go tomorrow" story rests on later recollections (notably Tommy Flowers'). Historians do not document a Hitler order reaching Eisenhower on 5 June, and McGrayne gives it as fact.
- Turing's death was ruled suicide; some biographers (for example Jack Copeland) argue accidental cyanide poisoning cannot be ruled out.
- McGrayne dates the U-559 pinch to December 1942; the boarding was on 30 October 1942, and the tables let Bletchley break the four-rotor traffic on 13 December 1942.
- Wald developed sequential analysis independently in 1943; credit is shared.
After Gordon Brown's 2009 apology came a royal pardon (2013), the "Alan Turing law" pardoning thousands of others (2017) and Turing on the £50 note (2021). Many GC&CS files have since been released to the National Archives. Turing's deciban scoring is the log-likelihood-ratio used today in sequential testing, speech recognition and the log-probabilities that LLMs add token by token.
Atlantic stakes and the Polish gift
The Poles moved first. A decade before 1939, Polish intelligence trained German-speaking maths students; actuarial mathematician Marián Rejewski used group theory to recover wheel wiring. By early 1938 Poland read about 75% of German army and air force traffic. Just before invasion they handed methods and an Enigma to Britain and France at Pyry Forest. Britain still lacked statisticians who knew Bayes was "unscientific"; that ignorance proved lucky. Actuaries, biologists and pure mathematicians analysed wartime data without the taboo.
Banburismus as sequential Bayes
Alan Turing arrived 4 September 1939. With Welchman and engineer Keen he refined the bombe (prototype March 1940). Naval Enigma he claimed for himself because nobody else wanted it. Banburismus scored overlapping message strips (Banbury sheets punched by Wrens) with bans: additive log-evidence that let analysts stop when enough evidence arrived rather than after a fixed sample size. Good later called this sequential analysis. If it worked, two of three wheel settings fell and bombe search shrank from 336 orders toward as few as 18. Turing and Good, trained in pure maths not statistics, had not been "poisoned" by anti-Bayesian teaching; Jeffreys at Cambridge was the rare academic defender they might have overheard.
Blackout, pinch and silence
Wolf packs sank 2,177 merchant ships in the first 40 months. The fourth rotor (February 1942) caused the Great Blackout. The U-559 boarding on 30 October 1942 (McGrayne says December; A closer look corrects) recovered tables that restored reading by 13 December 1942. Churchill's "Action this day" note (October 1941) forced resources. After victory, Colossi were broken up; Turing received an OBE; Good wrote about 900 Bayesian papers but could not cite Enigma. The chapter's thesis: Bayes won a war and lost the peace because the proof stayed classified.
BAN → BANBURISMUS → BOMBE. Log-odds evidence, sequential stopping, Enigma broken in secret.
Score two hypotheses in decibans: start at 0, add +10 for evidence that multiplies odds by 10, add -10 for the reverse. After four mixed clues, state your posterior odds. That is Turing's additive evidence.
Treat Banburismus as "just cryptography." Without the ban (log-likelihood ratio) and sequential stopping, the bombe search stays combinatorially hopeless. Erase Bayes and the chapter becomes a hardware story that cannot explain why naval Enigma fell when it did.
Why naval Enigma needed Bayes
Army Enigma was hard; naval Enigma with rotors, rings, plugboard and daily keys was worse. Reflector settings (26), wheel orders (336 for 3 of 8), clip and starting positions (about 17,000 each) and plugboard combinations (over 150 billion) multiplied. The Poles' prewar break and the Pyry handoff gave methods; Turing's Banburismus used Bayesian scoring of message overlaps to cut the search before the bombe ran. Good refined the weights; Kolmogorov and Koopman developed related ideas for the Soviets and the US Navy. The U-559 pinch (30 Oct 1942) ended the four-rotor blackout. Secrecy then buried the method: Churchill ordered evidence destroyed; Turing left the war decorated only with an OBE.
Turing, Good, Kolmogorov, Koopman and Shannon used Bayes in secret to break Enigma and Lorenz, aim guns and hunt submarines. The ban made evidence additive and sequential stopping saved time. Secrecy then buried the successes, and Bayes left the war as vilified as ever.
5. Dead and Buried Again
This is the book's shortest chapter, a bleak interlude between Bletchley Park and the postwar revival. Its point is simple: winning the war did not rescue Bayes. Because its successes were classified, the rule came out of the Second World War more suspect than before.
Ridicule and suspicion
Statistics books and papers stressed "repeatedly and self-righteously" that they did not use the rule. When Jack Good, Turing's wartime assistant, presented Bayesian methods at the Royal Statistical Society, the next speaker opened with "After that nonsense..." Good could not answer with the strongest evidence he had: the Official Secrets Act barred him from saying that the same reasoning had helped break naval Enigma.
The word itself was still narrow. "Bayes" meant equal priors (Laplace's principle of insufficient reason), not the broader idea of updating beliefs as observations arrive. Critics attacked the crude version, and the version Turing actually used stayed secret.
- The National Bureau of Standards suppressed a 1950s report to the Army's Aberdeen Proving Ground, its weapons-testing centre, because the study used subjective Bayesian methods.
- During Senator Joseph McCarthy's anticommunist campaign, a bureau statistician half-jokingly called a colleague "un-American because [he] was Bayesian, and... undermining the United States Government."
- At Harvard Business School, professors called their Bayesian colleagues "socialists and so-called scientists." This was the school where Raiffa and Schlaifer would later build decision analysis (Chapter 11).
A prominent statistician wrote that there remained "a lingering idea that there is something 'not quite nice,' something unsound, about the whole concept of inverse probability." The default identity was clear: unless declared otherwise, a statistician was considered a frequentist.
An invisible community
The Bayesian community was small, isolated and nearly unpublished. The prewar foundations lay unread:
- Frank Ramsey (Cambridge) argued in "Truth and Probability" (1926) that probability is a coherent degree of belief measurable by betting. He died in 1930, aged 26.
- Bruno de Finetti (Italy) independently built subjective probability in the early 1930s ("probability does not exist" as an objective thing) and proved his exchangeability theorem.
- Harold Jeffreys (Cambridge) published Theory of Probability (1939), the main scientific defence of inverse probability (Chapter 3).
The institutions belonged to the other side:
- Nearly every paper in the Annals of Mathematical Statistics dealt with questions framed by Jerzy Neyman's frequentist work of the 1930s.
- Thanks to Fisher's genetics and the strong anti-Bayesian stance of Oscar Kempthorne at Iowa State, agricultural research at most US land-grant universities relied on frequentism. Agriculture was then the main client of applied statistics.
- Gertrude Cox, a leading experimental designer and the 1956 president of the American Statistical Association, surveyed the future of statistics and barely mentioned Bayes.
- The first practical article telling scientists how to do a Bayesian analysis appeared only in 1963.
"The probability is either 0 or 1"
Even civilian researchers working for the military knew little about Bayes in 1950. At RAND, the Air Force think tank in Santa Monica, an economist preparing a research budget asked visiting statistician David Blackwell how to assess the probability of a major war within five years.
"Oh, that question just doesn't make sense. Probability applies to a long sequence of repeatable events, and this is clearly a unique situation. The probability is either 0 or 1, but we won't know for five years."
The economist nodded: "I was afraid you were going to say that. I have spoken to several other statisticians, and they have all told me the same thing." Strict frequentism could say nothing about one-off risks, which were exactly the questions Cold War planners needed answered: war, nuclear accidents, untested weapons. Blackwell soon changed his mind. Working at RAND on duels and games of timing, he became one of the leading Bayesian decision theorists, co-writing Theory of Games and Statistical Decisions (1954) with M. A. Girshick. In 1965 he became the first Black member of the US National Academy of Sciences.
Frequentist probability needs a repeatable experiment. "Will this war happen?", "Is this reactor safe?" and "Where is this submarine?" happen once. Bayesian probability treats them as degrees of belief that evidence can update. Every later chapter that rescues Bayes (Madansky's bomb accidents, Rasmussen's reactors, Craven's searches) turns on this kind of question.
The low point
Dennis Lindley summed up the period: "The upstart Bayesian movement is being contained, largely by being ignored." Another statistician remembered: "A lot of us thought [Bayes] was dead and buried."
- McGrayne gives Cox's title as president of the "American Statistical Society"; the body is the American Statistical Association.
- She does not name the 1963 "first practical article." It is usually identified as Edwards, Lindman and Savage, "Bayesian Statistical Inference for Psychological Research" (Psychological Review, 1963).
- The chapter relies on remembered remarks (the McCarthy jibe, the Harvard insult) with no surviving documents. They show the mood, not formal policy.
- "Dead" overstates it: in the same years Savage was writing The Foundations of Statistics (1954), and actuaries were using credibility theory (Chapter 6).
CLASSIFIED SUCCESS = PUBLIC FAILURE. Winning the war made Bayes look worse, not better.
List three reasons a classified success cannot rehabilitate a public theory. Compare with one modern dual-use method (cryptography, epidemiology models, credit scoring).
Suppose Good had been free to cite Enigma in 1950. Would Fisherism have collapsed overnight? The chapter's darker claim is that ideology plus institutional power outlast even decisive field proof until tools make the method cheap.
The bleak interlude
"Bayes" still meant equal priors, not updating. Good could not answer "After that nonsense..." with Bletchley. The National Bureau of Standards suppressed Bayesian army work; McCarthy-era jokes called Bayesians un-American; Harvard Business School colleagues called them socialists. Default identity: a statistician was a frequentist unless declared otherwise. The chapter is short on purpose: it is the trough between secret victory and public revival.
6. Arthur Bailey
The first public challenge to anti-Bayesian orthodoxy after the war came not from the military or from university mathematicians but from a Bible-quoting insurance actuary, Arthur L. Bailey.
An outsider in bananas
Bailey's father had been fired and blackballed by every bank in Boston for telling his employers they should not lend large sums to local politicians; even Arthur's schoolmates stopped inviting him to parties. Turning his back on the New England establishment, he studied statistics in the actuarial programme at Michigan (BS 1928), where he met his wife, Helen, later an actuary at John Hancock Mutual Life.
His first job was, he liked to say, "in bananas": the statistics department of United Fruit in Boston. When the Depression eliminated it, he drove a fruit truck and chased escaped tarantulas down Boston streets. In 1937, after nine years in bananas, he moved to New York to set premiums for automobile, aircraft, manufacturing, burglary and theft risks at the American Mutual Alliance, a consortium of mutual insurers. He lived quietly in unpretentious suburbs, hiked with his four children, annotated Gray's Botany with the sites of wild orchids, and preferred church and community to the fair-weather friends of his youth.
The scandal of the sledgehammer
Bailey arrived a modern sophisticate, taught by professors under the influence of Fisher and Neyman that priors were "more horrid than 'spit.'" Statisticians were to hold no prior opinions, use only directly relevant observations, and reject peripheral, nonstatistical information. No standard method even existed for weighing the credibility of prior knowledge.
So he was horrified to see "hard-shelled underwriters" using the semi-empirical, "sledge hammer" Bayesian techniques built in 1918 for workers' compensation. Actuaries refused to discard what they knew: next year's premium was this year's rate, modified by new claims. They did not ask what the rate should be, but "How much should the present rates be changed?"
McGrayne's illustration: a Bayesian forecasting how much ice cream someone will eat next year combines the person's recent consumption (the new data) with national dessert trends (the prior). The actuary's version: start from the current rate, then move it toward the firm's own claims experience in proportion to how credible that experience is. A small firm with a few years of data barely moves its rate; a large one with a long record moves it a lot.
Bailey spent his first year in New York trying to prove that "all of the fancy actuarial [Bayesian] procedures of the casualty business were mathematically unsound." After a year of intense mental struggle he concluded that the sledgehammer worked, and that "the hard-shelled underwriters were recognizing certain facts of life neglected by the statistical theorists." Giving more weight to a large body of data than to a small sample felt "logical and reasonable." Only a "suicidal" actuary would use Fisher's maximum likelihood, which gives zero probability to events not yet seen: many firms file no claims at all, so premiums would be too low to cover future losses.
He spent the Second World War on the problem, alone, cut off from academic thinkers and from colleagues who "scratched their heads" at his brilliance.
The keeper of consciences
In 1947 Bailey became chief actuary of the New York State Insurance Department. An executive called him "the keeper of our consciences": while colleagues drank in hotel bars at conferences, Bailey sipped soft drinks and read the Bible in slack moments. Some actuaries said "all manner of nasty things" about him, but respected his integrity.
His paper opened with a biblical warrant for priors, quoting Mark: "all things are possible to him that believeth." Reviewing Albert Whitney's 1918 mathematics, he showed that the Credibility theory at the heart of actuarial thought had Bayesian roots, and he built methods to meld every scrap of information into the body of data, giving supplementary evidence partial weight according to its credibility, its subjective believability. That let actuaries integrate thousands of old and new rates across employers, activities and places consistently. His library held a 1940 reprint of Bayes's essays prefaced by Bell Telephone's Edward Molina, and like Molina he used Laplace's more complex and precise system rather than Bayes's own.
22 May 1950.
Bailey names credibility as Bayes in front of the actuarial profession. Permission structure changes; formulas stay.
Bailey\'s presentation day: the actuarial room hears that credibility is Bayes, said out loud.
By 1950 Bailey was a vice president of the Kemper Insurance Group in Chicago and a regular after-dinner speaker at Casualty Actuarial Society banquets. On 22 May 1950 he read "Credibility Procedures: Laplace's Generalization of Bayes' Rule and the Combination of Collateral Knowledge with Observed Data."
| Move in the speech | What he said |
|---|---|
| Praise | actuaries had staged the only organised revolt against frequentist sampling philosophy, marching "a step ahead" |
| Admission | their practice went "beyond anything that has been proven mathematically" |
| Defence | "it works. . . . It does work!" |
| Revelation | the beloved Credibility formula was Bayes' theorem; Richard Price would be called an actuary today |
| Attack | Bayes's imaginary table turned against the frequentists and Fisher |
| Call | reinstate prior knowledge in statistical theory |
Richard von Mises of Harvard wrote that he hoped the speech would make "the unjustified and unreasonable attacks on the Bayes theory, initiated by R. A. Fisher, fade out." Bailey did not live to campaign: a smoker since college, he died of a heart attack on 12 August 1954, aged 49.
A forecast with no data
At the Insurance Company of North America's Christmas party in 1954, the chief executive, dressed as Santa Claus, asked chief actuary L. H. Longley-Cook, a Bailey admirer, the probability of two airliners colliding in midair. It had never happened, so any orthodox statistician had to answer no. "I really don't like these things mixed with martinis," Longley-Cook drawled, but within a year more Americans would travel by air than by rail. On 6 January 1955, using data on airline accidents in general, he forecast "anything from 0 to 4 air carrier-to-air carrier collisions over the next ten years" and advised raising rates and buying reinsurance. In 1956 a DC-7 and a Constellation collided over the Grand Canyon (128 dead); in 1960 a DC-8 and a Constellation collided over New York (133 dead).
The son, the safe driver, and Zurich
In the 1960s car casualties soared so high that half of living Americans could expect to be injured in a crash: no uniform signage, drivers and vehicles tested once in a lifetime if at all, light drunk-driving penalties, cars designed without safety in mind. Merit discounts for good drivers were considered unsound because one car has too little credibility. Bailey's son Robert A. Bailey, with Leroy J. Simon, used Bayes to update US statistics with data from Canada's safe-driving discounts, justifying merit rates. Robert also rated insurers themselves using subjective information, down to managers' drinking habits. Eventually the industry's data grew so vast that, in McGrayne's words, Bayes' rule, like the slide rule, became obsolete for this job.
To the few actuaries who understood him, Bailey was a da Vinci who had led the profession out of its dark ages. The news reached academics haphazardly: Allen Mayerson at Michigan wrote on Bailey in the early 1960s; Savage, then in Ann Arbor, visited de Finetti, and at a Trieste conference de Finetti told most statisticians about Bailey for the first time. Hans Bühlmann, who had studied at Berkeley in the 1950s "when it was kind of dangerous to pronounce the Bayesian point of view," took up the challenge and built a general Bayesian theory of credibility. Renaming the prior a "structural function," he believed he helped Continental Europe escape the "religious" quarrels still ahead for Anglo-Americans. Bühlmann later became president of ETH Zurich.
Bühlmann's theory (1967) and the Bühlmann-Straub model (1970) remain standard material in actuarial textbooks. Modern pricing mostly uses generalised linear models and machine learning, but credibility weighting survives inside them as partial pooling in hierarchical (mixed-effects) models, the same shrinkage of thin data toward a group prior that Bailey defended.
Whitney to Bailey's speech
Industrial death rates near 1 in 318 workers (1890 to 1910) and post-1911 no-fault compensation laws forced rates for tiny employer pools. Nebraska (1909) had 25 small manufacturers of data. The Casualty Actuarial Society (1914, Rubinow) and Whitney's credibility formula (1918) blended individual and class experience: weight Z = n/(n+k). Bailey's 22 May 1950 paper said credibility is Bayes. Actuaries priced under uncertainty daily; they could not wait for infinite identical trials. Money forced honesty before philosophy did.
CREDIBILITY = BAYES FOR PREMIUMS. Actuaries never needed permission from Fisher.
A small employer has 2 claims in a year; the industry average is 5. Credibility weight Z = n/(n+k). Try k=10 and n=2: how much do you trust the employer versus the industry? That is Bailey's world.
Call credibility "just a weighted average" and deny it is Bayesian. Then derive the weights without a prior-likelihood story. The shadow shows why Bailey's 1950 speech ("credibility is Bayes") was both obvious to practitioners and radioactive to theorists.
Insurance kept the flame
Workers' compensation after 1911 needed rates for tiny risk pools. Whitney's credibility formula (1918) and the Casualty Actuarial Society's practice blended individual and class experience. Bailey's 22 May 1950 paper said the quiet part aloud: credibility is Bayes. Actuaries priced risk under uncertainty every day; they could not wait for infinite sequences of identical trials. Chapter 6 is the bridge from exile to revival: money forced honesty.
Deepen plus: 22 May 1950 and the theology fight
Bailey's 22 May 1950 presentation day is the actuarial hinge: the room hears that credibility is Bayes, said out loud. Some hear liberation; some hear heresy. The formulas do not change; the permission structure does. Forecasting with no class data forces the prior into the open, which is why new products and new territories terrify purely frequentist price books.
Savage's 1954 foundations and Good's hierarchical parables gave philosophers and statisticians a scripture and a folk tale. Subjective versus objective priors became a food fight because computers could not yet settle arguments by fitting models that ate both. Lindley's mordant advice to attend funerals was sociology, not maths: paradigms shift when chairs turn over.
- Insurance kept practical Bayes alive while universities banned it.
- Bailey set out to debunk actuarial updating and instead proved Credibility was Laplace's form of Bayes.
- Maximum likelihood fails for rare events: zero claims do not mean zero risk.
- A Christmas-party question showed Bayes could price events with no history; Bühlmann carried the idea into general theory.
7. From Tool to Theology
Postwar money (the Office of Naval Research, then from 1950 the NSF) multiplied theoretical statisticians a hundredfold and moved the field's capital from Britain to the US. Neyman's Berkeley ("Jesus and his disciples") hired frequentists for 40 years; Neyman and Fisher kept feuding (Fisher feared graduates with "a dense fog in the place where their brains ought to be"). To sympathisers, frequentism looked like a Rube Goldberg set of ad hoc tests; if you disliked the result, you ran a different test. Bayes offered one coherent rationale. Three men set out to make it respectable.
- I. J. Good (1916 to 2009). Turing's assistant. Probability and the Weighing of Evidence (drafted 1946, published 1950). Stayed in secret cryptography (the NSA cut a passage naming him from Kahn's The Codebreakers, 1967). At Virginia Tech from 1967, paid one dollar more than the football coach; licence plate 007 IJG; 800+ articles on his "Private List."
- Dennis Lindley (1923 to 2013). Taught in an air-raid shelter by M. P. Meshenberg; wanted to make statistics "a respectable branch of mathematics." Built Bayesian groups at Aberystwyth and UCL ("as though a Jehovah's Witness had been elected Pope") and about ten UK Bayesian departments. Never made FRS. "Bayesian statistics is not a branch of statistics. It is a way of looking at the whole of statistics."
- L. J. "Jimmie" Savage (1917 to 1971). Nearly blind, thought "feebleminded" at school, rescued by G. Y. Rainich's class taught in total darkness. Worked with Milton Friedman in wartime; co-founded Chicago's statistics department with Allen Wallis. The Foundations of Statistics (1954).
Good's bird watcher
To hint at a classified Turing technique, Good wrote about a bird watcher who sees 180 species, many only once. Frequency counting gives unseen species probability zero, claiming they can never be found. Turing gave them a small non-zero probability, estimated from how many species were seen exactly once, to predict rare letter groups in the next Enigma message. This Good-Turing estimator later became standard in DNA analysis and language modelling.
Savage's Bible
Lindley (1953) and Savage both set out only to put ordinary statistics on axioms and found the axioms led to Bayes. "We were both fools," said Lindley. Savage, building on Borel, Ramsey and de Finetti, combined probability (uncertainty) with utility (reward): rational people choose to minimise expected loss. Probability is personal, like a horse-race bet. His answer to "what about objectivity?": with enough data, people with different priors converge. Scientists disagree when data are scarce and agree when they pile up.
All the information the data carry about the hypothesis is in the likelihood. Practical effect: researchers may stop an experiment whenever they choose without spoiling the analysis, and they judge what did happen, not what might have happened under a sampling plan. Savage said he became a Bayesian "only through recognition of the likelihood principle."
Savage's book never used the word "Bayesian" and his examples were whimsical (does aspirin curl rabbits' ears? the speed of neon light through beer?). Fermi used Bayes (derived from Gauss) in 1953; Feynman suggested it for comparing physical theories.
Theology and the food fight
Robert Kass: Bayes' theorem gives "the cosmic religious feeling," but believers "become blinded to its fundamental vulnerability... its magical powers depend on the validity of its probabilistic inputs." Savage turned "messianic" (Kruskal): disagreement meant you were "inimical, or stupid." "It was a huge food fight... They hated each other." One wit wished Bayesians would follow Bayes and "publish posthumously." Leaving Chicago in 1960 Savage said, "I proved the Bayesian argument in 1954. None of you have found a flaw... yet you still deny it." He moved to Michigan, then Yale (1964), and died in 1971 aged 53; the local paper's obituary headline: "Authored Book on Gambling."
McGrayne admits the zealotry hurt. The subjective versus objective framing that Savage and Lindley pushed is less sharp today: most applied work uses weakly informative or reference priors and checks sensitivity, a middle way the 1960s rhetoric left out. Kass's warning, that outputs are only as good as inputs, is the lasting lesson.
Good-Turing smoothing underpinned n-gram language models for decades (with Katz back-off and Kneser-Ney). Neural LLMs replaced explicit smoothing, but the problem is the same: give non-zero probability to unseen sequences, which softmax outputs do automatically. Estimating "missing mass" is now used to judge how much of a domain a training corpus covers.
RAMSEY → de FINETTI → SAVAGE → LINDLEY. Belief as coherent betting; foundations before software.
State a probability for rain tomorrow, then a price at which you would buy or sell a $1 claim that pays if it rains. If your price and your stated probability disagree, you are Dutch-bookable. That is Ramsey-de Finetti coherence in a coffee queue.
Treat Savage's Foundations as theology that never touched data. Then explain Bailey, Cornfield and Raiffa. The chapter's real claim is that foundations without computation still left Bayes a minority sect; Chapter 16 supplies the missing engine.
Making belief coherent
Ramsey (1926) and de Finetti measured belief by bets; Jeffreys kept scientific induction Bayesian; Good brought wartime practice into print; Savage (1954) axiomatised subjective probability and decision. Lindley carried the flag in Britain. The conversion stories (Lindley's "road to Damascus") matter: this was identity, not a toolbox. Without computers the theology could not yet run large models, which is why business, medicine and the military appear next as the proving grounds.
8. Jerome Cornfield, Lung Cancer, and Heart Attacks
Bayes entered medicine through one man: Jerome Cornfield, whose only degree was a BA in history, and who used the rule to help identify the causes of lung cancer and heart attacks.
A sudden epidemic
The man and the bridge
Cornfield, son of Russian Jewish immigrants (NYU 1933), joined the Labor Department during the Depression at $26.31 a week ("if you were Jewish, you could work for the Department of Labor") and learned statistics at the USDA Graduate School run by W. Edwards Deming, who published Bayes' essay with Molina's introduction. He helped revise the Consumer Price Index, smoked a Turkish water pipe beside his Marchant calculator, and moved to the new NIH in 1947. He and his wife quit 2½-pack-a-day habits.
Cornfield took the lung cancer rate in the whole population as the prior and combined it with the share of smokers among patients and controls. Bayes' rule then gave the risk for smokers and non-smokers. It showed that, with well-matched controls, retrospective case-control data can measure the strength of a link, the basis of the odds ratio. Cornfield made much of modern epidemiology possible; case-control studies identified thalidomide as a cause of birth defects in 1961.
Prospective proof and Fisher's resistance
Doll and Hill then followed about 40,000 British doctors; Hammond and Horn followed 187,783 American men aged 50 to 69 for over 3½ years. Heavy smokers were 22 to 24 times more likely to get lung cancer, and 42% and 57% more likely to get heart and circulatory disease.
Fisher and Neyman, both heavy smokers, rejected the conclusion (Fisher also consulted for the tobacco industry). From 1955 Fisher argued that lung cancer might cause smoking, or that a hidden gene caused both. Joseph Berkson of the Mayo Clinic doubted cigarettes could cause both cancer and heart disease.
Cornfield and five co-authors answered every alternative in a paper that reads like a legal brief: - If smokers have 9 times the risk, any hidden factor must be even more strongly linked to smoking, and nothing like it had been seen (now called the Cornfield inequality, the first sensitivity analysis). - Cancer causing smoking? Cancers diagnosed after 50 would have had to start before 18, the median age smokers start. - The gene would have to explain dose response (60 times more common in two-pack smokers), decline after quitting, mouse skin tumours, sex differences and pipe versus cigarette differences. "A point is reached... when a continuously modified hypothesis becomes difficult to entertain seriously."
The 1964 Surgeon General's report ("cigarette smoking is causally related to lung cancer in men") cited Cornfield.
Framingham and logistic risk
Cornfield helped plan the Framingham Heart Study (begun 1948). Cross-tabulation broke down: 10 risk factors at three levels make 59,049 cells, needing 600,000 people for 10 per cell. Using the cardiovascular death rate as prior and data on 1,329 men (92 heart events, 1948 to 1958), Bayes' rule produced a posterior in the form of a logistic regression. The key factors besides age: cholesterol, smoking, heart abnormalities and blood pressure, with no safe threshold. Cardiovascular death rates fell 60% between 1960 and 1996. The multiple logistic risk function became one of epidemiology's main tools.
Conversion
Cornfield brought Jeffreys' betting odds, now the Bayes factor, into an NIH multicentre trial. Frequentist rules made mouse cancer screens run six more observations after the result was clear and forbade looking at interim results; Bayes let him stop after two solidly adverse ones. "If maintenance of [Fisher's] significance level interferes with the release of interim results, all I can say is so much the worse for the significance level." "Bayes' theorem has come back from the cemetery." Few NIH colleagues followed; NIH trials did not use Bayes for another 30 years. He chaired statistics at George Washington University, advised on Kinsey, voting rights and Three Mile Island, and became ASA president in 1974 ("Statistics... bedfellow of the sciences"). He died of pancreatic cancer in 1979: "You spend your whole life practicing your humor for the times when you really need it."
- McGrayne attributes the logistic risk function to Cornfield's 1962 analysis; the classic multivariate logistic paper is Truett, Cornfield and Kannel (1967). Cornfield's 1962 work used discriminant analysis that leads to the logistic form.
- His role in designing Framingham in 1948 is overstated in some accounts; the study was run by the US Public Health Service, and Cornfield's main contributions were analytical.
- "Paid consultant" is how McGrayne describes Fisher; the documented tie was consulting for the Tobacco Manufacturers' Standing Committee.
Cornfield's inequality grew into sensitivity analysis for unmeasured confounding, for example the E-value (VanderWeele and Ding, 2017), now routine in observational studies. Pearl's causal graphs and Hernán and Robins' target trial emulation formalise his "legal brief" reasoning. Bayesian versions of these methods are common in pharmacoepidemiology and real-world evidence submissions to regulators.
From case-control to logistic risk
England and Wales lung-cancer deaths reached 321 per million by 1952. Doll-Hill and Wynder-Graham showed almost no non-smoking male cases, but patients ask about personal risk. Cornfield (1951) used the population rate as prior and smoking shares as likelihood. Prospective cohorts (British doctors; Hammond-Horn) showed 22 to 24× risk for heavy smokers. The Cornfield inequality (1959) forced any confounder to outmuscle the smoking link; the 1964 Surgeon General report leaned on it. Framingham's many risk factors exploded cross-tabs; Bayes plus discriminant ideas yielded the logistic risk function. "Bayes' theorem has come back from the cemetery."
CASE-CONTROL → PERSONAL RISK → LOGISTIC. Cornfield's bridge; inequality; Framingham.
A disease has prevalence 1%. A case-control study finds smokers are 9× more common among cases than controls. In one paragraph, explain why that does not by itself equal "a smoker has 9% risk," and what population rate Cornfield used as prior to finish the job.
Accept Fisher's "hidden gene" as equally live after the Cornfield inequality. Then state how strong the gene-smoking association would need to be to explain a 9× risk. If you cannot name a factor that strong, you have rediscovered why the 1964 Surgeon General report cited Cornfield.
Epidemiology learns to update
Lung cancer deaths hit 321 per million in England and Wales by 1952. Doll-Hill and Wynder-Graham case-control work showed almost no non-smokers among male cases, but patients ask about personal risk. Cornfield's 1951 bridge used the population rate as prior and smoking shares as likelihood. Prospective cohorts (British doctors; Hammond-Horn) then showed 22 to 24× risk for heavy smokers. The Cornfield inequality (1959) forced any confounder to be stronger than the smoking link. Framingham's many risk factors made cross-tabs explode; Bayes plus discriminant ideas yielded the logistic risk function. "Bayes' theorem has come back from the cemetery."
Deepen plus: sensitivity and logistic risk
Cornfield's sensitivity argument is Bayesian in spirit even when dressed as bounds: to erase the smoking association, a confounder would need relative risks and prevalences nobody could point to. That is prior knowledge about the size of lurking variables. Framingham then made multivariate risk operational for clinicians who needed a number for this patient, not a p-value for a population. The chapter is the book's clearest answer to "does Bayes save lives?"
9. There's Always a First Time
In summer 1957 Savage, visiting RAND in Santa Monica, encouraged two young men to calculate a life-and-death number: the probability that a thermonuclear weapon explodes by mistake.
RAND, founded with Curtis LeMay's backing about ten years earlier, was the Air Force's "university without students," and it often challenged Strategic Air Command (SAC). There Savage met Fred Iklé, a Swiss-born demographer studying accidental or deranged nuclear use ("Our method of preventing nuclear war rests on... the mass killing of hostages"). Savage handed the problem to Albert Madansky, 23, a new Chicago PhD who had paid for graduate school working for Arthur Bailey.
The probability of the impossible
No atomic or hydrogen bomb had ever exploded by accident, so SAC concluded none would. Frequentist tradition agreed: Jakob Bernoulli (1713) said highly improbable events do not happen, Cournot and Kolmogorov said the same in other words, and Fisher's relative frequency gave zero accidents in the past, so zero probability in the future. Madansky: "As long as you are set that the probability is going to be zero, then nothing's going to change your mind."
Lindley: if you give the hypothesis that the moon is made of green cheese a prior of zero, "whole armies of astronauts coming back bearing green cheese cannot convince him." Hence never assign a prior of exactly 0 or 1, after Cromwell's 1650 plea to the Church of Scotland: "I beseech you, in the bowels of Christ, think it possible you may be mistaken." "If you're willing to admit a shred of disbelief," said Madansky, "you can let Bayes' theorem work."
Madansky had 16 classified "dramatic incidents" from 1950 to 1958: drops, jettisons, crashes. In 1950 alone, high explosives killed 13 near Kirtland AFB, 16 near Lebanon, Ohio, and 19 (including General Robert Travis) in California. In 1958 a B-47 dropped an unarmed bomb on Walter Gregg's garden at Mars Bluff, South Carolina, leaving a crater 30 feet deep; the Air Force paid $54,000. Ballistic missile reliability engineers were already using Bayes quietly; "It was just a natural thing to do."
- A minimally informative prior that put some probability on accidents, not all concentrated at zero.
- Counting "accident opportunities": weapons x lifetime x times handled or flown. "A probability that is very small for a single operation, say one in a million, can become significant if this operation will occur 10,000 times."
- Indirect data: B-52s averaged 5 major accidents per 100,000 flying hours; about 3 bombs per 1,000 weapon-carrying flights were dropped or jettisoned; 80% of crashes happened within 3 miles of a base.
- Planned scale: 1,800 bombers, about 15% airborne at all times.
- Pencil-and-paper maths using what Raiffa and Schlaifer would later call conjugate priors ("I was just doing it ad hoc").
- Result: an expanded airborne alert would likely bring 19 "conspicuous" weapon accidents a year.
The report, RM-2251 "On the Risk of an Accidental or Unauthorized Nuclear Detonation" (15 October 1958; Iklé with Madansky and psychiatrist Gerald Aronson), was classified until 9 May 2000. It recommended the two-man rule, coded combination locks inside warheads, arming only over enemy territory and other safeguards; an appendix even proposed screening crews with isolation and LSD. To RAND's surprise LeMay asked for a copy and issued a "blizzard" of orders, including the two-man rule and coded locks. Few locks were fitted until Kennedy: four days after his inauguration a B-52 broke up over Goldsboro, North Carolina, and a hydrogen bomb fell into a swamp. JFK was told of more than 60 nuclear weapon accidents and pushed safety and locks (permissive action links).
- McGrayne calls the Goldsboro bombs 24-megaton weapons. They were Mark 39s of about 3.8 megatons. Her "only one of six safety devices functioned" follows popular accounts; a memo declassified in 2013 (by Eric Schlosser) says a single low-voltage switch stopped one bomb from detonating.
- Madansky's specific forecast of 19 accidents a year is hard to check against the record, but the Palomares (1966) and Thule (1968) accidents bore out the direction of his warning.
Collisions and conspicuous accidents
INA asked whether two commercial airliners could collide when none ever had. Actuary Longley-Cook answered on 6 January 1955: expect 0 to 4 collisions in ten years; raise rates; buy reinsurance. Grand Canyon (1956, 128 dead) and New York (1960, 133 dead) followed. At RAND, Albert Madansky turned B-52 major-accident rates (5 per 100,000 hours), fleet size (about 1,800 bombers, 15% airborne) and handling counts into an estimate of about 19 conspicuous nuclear-weapon accidents a year. Teledyne's later shuttle analysis (Chapter 16) reused the same rare-event logic against NASA's 1 in 100,000 booster claim.
- Refuse a prior of zero.
- Decompose opportunity (platforms × hours × handlings).
- Borrow rates from analogous systems.
- Propagate uncertainty, not a single heroic number.
- Buy insurance or redesign when the posterior mass sits in the tail that kills.
NEVER HAPPENED ≠ PROBABILITY ZERO. Airliners, bombs on bombers; expert prior plus rare-event math.
Chance of accident per operation = 10^-6. Approximate P(at least one in N operations) ≈ 1 - e^(-N/10^6). Compute for N = 10,000 and N = 1,000,000. Small risks accumulate.
Set the prior on catastrophe to zero because "it has never happened." Then no amount of opportunity data can raise it. That is the frequentist trap the chapter exists to break.
Pricing the impossible
INA asked whether two commercial airliners could collide; Longley-Cook (6 Jan 1955) gave 0 to 4 collisions in ten years and urged reinsurance. Grand Canyon (1956, 128 dead) and New York (1960, 133 dead) followed. At RAND, Madansky estimated conspicuous nuclear-weapon accidents from B-52 rates and fleet exposure: about 19 a year under the book's figures. The chapter's rule: a first time has a probability; experts and analogy supply the prior when history is empty.
Bayes gave a non-zero probability to an event that had never happened. Madansky's count of accident opportunities and his minimal prior helped push SAC toward the two-man rule and coded locks, safeguards that mattered at Goldsboro and later.
10. 46,656 Varieties
In the 1960s the prior was still the sticking point, and Bayesian theories multiplied. Good counted "at least 46656 different interpretations": subjective, objective, empirical, hierarchical, fuzzy, quasi and more. "Ye shall know them by their posteriors." "Inverse probability" gave way to "Bayesian inference," and English-language writing pushed Laplace into the background. In 2008 Lindley, aged 85, conceded that Laplace mattered more than Bayes ("But I am biased: the French let us down during WW2 and then there was the ghastly de Gaulle").
Box and Stein's shrinkage
George Box, Fisher's son-in-law, founded Wisconsin's statistics department in 1960 and found his Foundations course turning Bayesian week by week. "Comparing averages looked right to me until Stein."
When estimating several related means (egg yields of chicken breeds, batting averages, roofers' compensation exposure), you get better total accuracy by shrinking each estimate toward the overall mean than by using each average alone. A batter hitting .400 early in a season is probably better than average but not a .400 hitter. James and Stein (1961) simplified it into the same formula as Whitney's 1918 Credibility (x = P + z(p - P)). Bayesians said it was a prior in disguise, a "hyperdistribution" of chicken-ness. Stein called subjective priors "completely inappropriate."
Box found Bayesian methods improved time-series forecasting where Stein's trick alone failed: "You can't take numbers out of context." He wrote a Christmas song: "There's no Theorem like Bayes theorem / Like no theorem we know."
Cracks in the frequentist wall
- Fiducial probability. Fisher's alternative to Bayes (1930). Lindley showed in 1958 that with uniform priors it gives Bayesian answers, and that it does not obey the rules of probability. Savage: "a bold attempt to make the Bayesian omelet without breaking the Bayesian eggs." Fisher was furious.
- Birnbaum (1962) derived the likelihood principle from principles frequentists accept. Savage called it "a historic occasion."
- Lindley's paradox (1957). With huge samples, a test can reject a precise null at tiny p while Bayes strongly favours it. Example: Robert Jahn's Princeton psychokinesis study (104,490,000 trials, 18,471 excess hits, p about 0.00015) "proves" psychokinesis to a frequentist and refutes it to a Bayesian.
- Edwards, Lindman and Savage (1963): even with n = 20, Bayesian answers could be ten times larger than p-values.
Fisher told Box, "Thinking that you don't know and thinking that the probabilities of all the possibilities are equal is not the same thing." But Fisher used Bayes when he knew the prior, as with the pedigrees of his mice. Compromise came as empirical Bayes, which estimates the prior from data (Egon Pearson 1925, Turing in the war, Herbert Robbins 1955), though it had little impact until the late 1970s.
Institutions and applications
Morris DeGroot wrote the first well-known Bayesian decision theory text; Arnold Zellner brought Bayes into economics and set up conferences. The ISBA and the ASA Bayesian section came only in the early 1990s. Applications stayed rare:
| Field | What happened |
|---|---|
| Medicine | Ledley and Lusted (1959, in Science); Homer Warner's 1961 computer diagnosis of congenital heart disease from 1,000 children beat old cardiologists; Gorry and Barnett cut 54 tests to 7 or 8. Doctors did not adopt it. |
| NSA | F. T. Leahy (1960): Bayes "was used in almost all the successful cryptanalysis at the National Security Agency." Six Top Secret Umbra articles, 1960 to 1972. |
| CIA | Bayesian forecasting experiments were found more insightful but too slow without computers, and dropped. |
| Law | Laurence Tribe (1971) attacked mathematical evidence for "enshrouding the [legal] process in mathematical obscurity"; courtrooms closed to Bayes. |
McGrayne's verdict: the revival of the 1950s and 1960s was mostly theory. Until Bayesians proved in public that their methods worked better, they were stuck.
Stein's idea is now ordinary practice: hierarchical (multilevel) models, ridge regression and weight decay in neural networks (equivalent to Gaussian priors), empirical Bayes in genomics (limma, DESeq2), sports analytics and small-area estimation in official statistics. Lindley's paradox is the standard argument for lower significance thresholds in big-data and particle-physics work (the 5-sigma rule).
Shrinkage before the name was cool
HIERARCHY BEATS FLAT. Shrinkage across groups; Good's 46,656 Dirichlet varieties; Robbins's empirical Bayes.
Three hospitals report complication rates 1/100, 5/100 and 20/20. Without pooling, the third is "100%." With a hierarchical prior, all three shrink toward a common mean. Sketch the direction of shrinkage for each.
Refuse all pooling as "ecological bias." Then explain how insurers, schools and baseball batting averages become unusable with rare events. Hierarchy is the chapter's answer to the bias-variance fight.
Many parameters, one structure
Good enumerated 46,656 varieties of prior for multinomial problems to show how rich the Bayesian zoo already was. Empirical Bayes (Robbins) estimated the prior from the ensemble of units. Shrinkage toward the grand mean is the practical payoff: noisy rates in small cells borrow strength. Modern descendants include random-effects models, multilevel regression and James-Stein estimators. The chapter prepares the ground for MCMC hierarchies in Chapter 16.
11. Business Decisions
By the 1960s the shortage of real applications was embarrassing. John Pratt complained of "too many minor advances extracted from real or mythical problems, sanitized, rigorized, polished." The logic was simple ("You can't have an opinion afterwards without having had an opinion before"), but making belief precise required heavy calculation, with no software and few computers. The first serious attempt came from Harvard Business School.
- Robert Schlaifer (1914 to 1994). Classicist (Harvard PhD on ancient Greek slavery, 1940) whose only maths course was calculus taken for a cash prize. Wartime sonar work, then a 600-page history of aircraft engines. Assigned to teach statistical quality control, he felt like "a jackass" teaching frequentism, threw away the books and reinvented Bayesian decision theory from scratch. Four packs a day, 75 to 80 hours a week, "imperious and hierarchical." Savage: "hot as a pistol, sharp as a knife."
- Howard Raiffa (1924 to 2016). City College; anti-Semitism steered him into actuarial maths at Michigan, where he never heard the word "Bayes." Drifted into subjectivism through game theory and Wald's decision rules (Wald privately said every good decision procedure is Bayesian). A closet Bayesian at Columbia; colleagues accused him of bringing in "squishy judgmental, psychological stuff." Harvard 1957 (Cochran: "Well, you'll grow up"). Later "Mr. Decision Tree," a co-founder of the Kennedy School and of the Harvard negotiation course, and adviser to McGeorge Bundy.
Why business needed Bayes
Frequentism works for repeatable phenomena. A manager launching a product or setting a price has no repeated trials. "Under uncertainty, the businessman is forced, in effect, to gamble." Schlaifer reasoned: whatever prior information executives have beats none; sample data should update it; and the real question is whether more information is worth its cost. Raiffa wanted to tell a manager "I now believe there is a .92 probability that p is greater than .25," which significance tests forbid. He used the same logic on a Columbia oral exam: the faculty's strong prior opinion of sociology student James Coleman should outweigh one bad hour. Coleman passed and became famous.
The toolkit
Schlaifer's Probability and Statistics for Business Decisions (1959, $11.50; he made McGraw-Hill pulp the first printing to fix errors) was the first wholly Bayesian textbook. With Raiffa he aimed to make Bayes operational:
- Decision trees. Map the sequence of choices and chance events: act now, or pay to gather information first? The best-known practical form of Bayes.
- Tree-flipping. Reverse the order of a chance node, putting test results before the test, which is Bayes' rule without saying so. "The worst thing you can do is to use Bayes' theorem. It's too complicated... We did it by tree-flipping."
- Conjugate priors. Choose prior families whose posterior has the same shape (normal in, normal out), so repeated updating stays easy. Made unnecessary later by MCMC.
Expert opinion and the DUU seminar
Nobody knew if experts' judgments could be quantified. Pratt's wife Joy, who promoted films, learned from nightly feedback to forecast cinema attendance accurately. Executives turned out to love being asked: DuPont (1962) assessed prior demand for artificial shoe leather; Ford engineers' opinions let it use smaller samples. A typical engineering problem might have 20 uncertainties: 12 handled by single guesses, 5 needing tests, 2 important enough for formal expert interviews. A weekly seminar on decision making under uncertainty (1961 to 1965) covered utility, portfolios, group decisions and behavioural anomalies. Applied Statistical Decision Theory (1961) set Bayesian statistics' direction for two decades. Raiffa's 11-month Ford Foundation programme (1960 to 1961) trained the next generation of business school deans.
Why it faded
Bayesian decision analysis never replaced "the old stuff" in business schools and shrank to a few weeks of study. Schlaifer was a theorist: he set aside problems he could not solve, did little consulting and stripped cases (cottage cheese packaging, IBM quality control) of their messy detail. Students burned his MERC text page by page on the steps of Baker Library ("I'd rather be among those whose books are burned than those who burn books"). Pratt and Raiffa finished their joint 875-page book 30 years later. Lindley thought Schlaifer "one of the most original minds that I have ever met."
McGrayne credits the Harvard pair with much of decision analysis. Ronald Howard at Stanford coined the term "decision analysis" (1964) and, with SRI, did more of the consulting that put it into industry; the field had several founders.
Decision trees became influence diagrams and Bayesian networks, used in oil and gas, pharma portfolio planning and health technology assessment (NICE and others use value-of-information analysis). In tech, Bayesian A/B testing and multi-armed bandits (Thompson sampling) apply Schlaifer's "is more information worth its cost?" millions of times a day.
Schlaifer reinvented Bayes; Raiffa made it teachable
Robert Schlaifer (Amherst classics, Harvard PhD on Greek slavery 1940, wartime sonar at the Underwater Sound Lab, then a 600-page aircraft-engine history) was assigned statistical quality control at Harvard Business School with almost no maths training. Frequentism felt useless for one-off product launches. He threw the books out and rebuilt decision theory from subjective priors plus sample data. Savage called him "hot as a pistol." He smoked four packs a day, worked 75 to 80 hours a week and nicknamed colleagues ("Uncle Howard," "Great Man Pratt," "Arturchick").
Howard Raiffa came via City College, wartime anti-Semitism that pushed him into actuarial exams, and Michigan degrees. He slipped into subjective probabilities through von Neumann-Morgenstern game theory, then taught Wald's decision functions at Columbia (Wald privately told Hilda von Mises he was Bayesian but dared not say so). Columbia colleagues accused him of "squishy judgmental" science. At Harvard (1957) Cochran said "you'll grow up"; Schlaifer hugged him. Each morning Raiffa tutored Schlaifer in linear algebra; the next day Schlaifer conjectured theorems and applied them to inventory and pricing. They never cited journals. Betting dollars on elections and sports made belief tangible.
What operationalised and what faded
Schlaifer's 1959 Probability and Statistics for Business Decisions ($11.50) was the first wholly Bayesian textbook; he forced McGraw-Hill to pulp the first printing. Tools: decision trees, tree-flipping (Bayes without the name) and conjugate priors. DuPont (1962) and Ford used expert priors to shrink samples. The DUU seminar (1961 to 1965) and Raiffa's Ford Foundation programme trained deans. Students later burned the MERC text on Baker Library steps. Decision analysis shrank to a few weeks because Schlaifer stayed a theorist, cases were sanitised and computers were still scarce. The toolkit waited for Chapter 16.
TREE → FLIP → DECIDE. Schlaifer-Raiffa operational Bayes for one-off managerial gambles.
Draw a two-branch decision tree: run a $10k market test or launch now. Invent P(good sales), P(positive test | good) and P(positive test | bad). Flip the tree to get P(good | positive). Decide whether the test is worth $10k.
Keep the beautiful tree and skip eliciting the prior from the actual executive. The chapter's failure mode is exactly that: theory without messy consulting. Decision analysis then shrinks to a short elective.
Making Bayes operational at Harvard Business School
Schlaifer (classicist, aircraft-engine historian, four packs a day) reinvented Bayesian decision theory because frequentism felt like teaching jackasses to gamble. Raiffa arrived from Columbia as a closet Bayesian who had slipped into subjective probabilities through game theory and Wald. Together they built decision trees, tree-flipping (Bayes without the name) and conjugate priors. Schlaifer's 1959 textbook was the first wholly Bayesian business text; he made McGraw-Hill pulp the first printing to fix errors. Experts at DuPont and Ford learned they liked being asked for priors. The DUU seminar (1961 to 1965) trained a generation. It faded in curricula because Schlaifer stayed a theorist, cases were sanitised and computation was still brutal. The toolkit waited for cheaper machines.
Deepen plus: value of information in one screen
Schlaifer's question is still the one product managers dodge: is another A/B test week worth the delay? Decision trees make the dodge visible. Tree-flipping is Bayes for people who refuse the formula. Conjugate priors were a 1960s prosthetic for missing MCMC; today the prosthetic is a sampler, but the decision question is unchanged. Howard's Stanford line and SRI consulting carried more industrial deployments than HBS classroom burns suggest; credit is shared.
12. Who Wrote The Federalist?
Both were classification problems, so Mosteller looked for a large one of his own.
The puzzle
Between 1787 and 1788 Hamilton, Jay and Madison wrote 85 anonymous essays urging New York to ratify the Constitution. 12 were claimed by both Hamilton and Madison. Their styles were almost identical (in 1941 Mosteller learned, counting sentence lengths, that "people cannot count, at least not very high"), so he decided to "leave general style as a poor bet and pay attention to words." In 1955 he recruited David Wallace of Chicago, who had absorbed Savage's teaching and urged Bayes: "when you have large numbers of parameters... the difference between Bayes and non-Bayes grows immense." The aim grew into a serious test of Bayesian against frequentist discrimination: "apply a 200-year-old mathematical theorem to a 175-year-old historical problem."
- Data: 94,000 words known to be Hamilton's, 114,000 Madison's. Topic words ("war," "executive") were dropped; function words ("in," "an," "of," "upon") kept.
- Method: 50:50 prior odds for each paper, updated word by word using 30 words; first calibrated on 57 papers of known authorship, then applied to the 12 disputed ones. Worked in odds, which were easier than probabilities.
- Breakthrough: historian Douglass Adair pointed to a 1916 finding: Hamilton wrote "while," Madison "whilst." Hand concordances on adding-machine tape revealed "upon" (Hamilton about twice per paper, Madison almost never), the best single discriminator by a factor of four; "enough" was another Hamilton marker.
- Labour: an army of 100 helpers (80 Harvard students), new Fortran code, MIT computer time and slide-rule checks; 25 technical problems solved and four parallel analyses compared.
- Result (1964): odds "satisfyingly fat" that Madison wrote all 12; weakest was No. 55 at 240 to 1.
"We went in thinking that everything depended on what sort of prior information you used." Instead, the choice of model for the data mattered far more than the choice of prior. Priors still matter when data are scarce or contested.
Mosteller (founder of Harvard's statistics department, a pioneer of meta-analysis and randomised trials, TV teacher to more than a million viewers in 1961, author of 57 books) worked in small teams; he recruited Persi Diaconis, a former magician, on his first day. The Federalist study was hailed as "Herculean" and still counted as the greatest single Bayesian case study in 1990, yet no one followed it up. Mosteller: "I think Bayesian methods are about to take off. But then I've been saying that for twenty-five years."
The Federalist study founded computational stylometry. Function-word methods (Burrows' Delta and successors) helped unmask Robert Galbraith as J. K. Rowling (2013) and are used to study disputed authorship of works by Shakespeare and others. In 2026 the hot problem is detecting AI-written text. Detectors based on token probabilities are unreliable and biased against non-native writers, and watermarking schemes are now being proposed to replace them.
Function words as likelihoods
Hamilton and Madison both claimed disputed essays. Mosteller and Wallace assembled about 94,000 words of known Hamilton and 114,000 of Madison, chose roughly 30 function words ("upon," "whilst," "while"), calibrated on 57 known papers, set each disputed essay's prior to 50:50 and updated word by word in odds. Verdict (1964): Madison wrote all 12; weakest odds still 240 to 1. It was the largest single Bayesian study of its day and a computational stress test before MCMC. The method is stylometry: content-free markers carry author signal. Forensic misuse without population calibration is the Shadow the chapter implies.
WORDS AS LIKELIHOODS. Mosteller-Wallace; function words; Madison by huge odds.
Pick two emails you wrote and two a colleague wrote. Count "however," "therefore," "I" and "we." Without reading content, guess authorship of a held-out paragraph. That is stylometry with a tiny feature set.
Treat function-word frequencies as proof of guilt in court without a validated population model. The Federalist study had known-author calibration; forensic linguistics often does not. The shadow is misuse, not the method.
The largest Bayesian study of its day
Hamilton and Madison both claimed disputed Federalist essays. Mosteller and Wallace used 94,000 words of known Hamilton and 114,000 of Madison, about 30 function words ("upon," "whilst," "while"), calibrated on 57 known papers, started each disputed essay at 50:50, and updated word by word in odds. Result (1964): Madison wrote all 12; weakest odds still 240 to 1. It was scholarship and a computational stress test before MCMC.
Function words act like stable author IDs. Multiply likelihoods; let the prior lose when the counts speak.
Deepen plus: stylometry and election night
Mosteller and Wallace did not invent word counting; they made it Bayesian. Function words (upon, whilst, enough, thereupon) behave like stable author fingerprints because writers rarely revise them for style. Likelihood ratios for Madison versus Hamilton multiply across many words; a prior that either man is equally likely a priori is then overwhelmed by the data for most disputed essays. Remaining uncertainty clusters where samples are short or editing may have mixed hands. The public lesson was larger than authorship: historical hypotheses can carry probabilities without becoming frequency nonsense.
Tukey's election work is the streaming cousin. Partial precincts arrive as a biased sample of the electorate; the prior from past elections and demographics is the only honest way to interpret early leads. Call too early and you mislead the country; call too late and you waste the franchise of projection. Cold War secrecy around related polling methods meant the academic literature lagged the practice, which is why McGrayne's chapter reads like espionage as much as statistics.
| Item | Figure |
|---|---|
| Disputed Federalist essays under study | about a dozen |
| Method | word-rate likelihoods + priors |
| Election use-case | sequential update on partial returns |
Mosteller and Wallace ran the largest public civilian Bayesian analysis since Laplace and found Madison wrote the disputed papers. Their lesson, models matter more than priors when data are plentiful, answered the old charge of subjectivity.
13. The Cold Warrior
The Federalist study impressed a small professional world. John Tukey, a star of Cold War science, gave Bayes the chance to perform before 20 million television viewers. The question McGrayne poses: would statisticians learn from it that Bayes had come of age?
A man inside the secrets
Today Tukey is remembered for coining "bit" and "software." In his time he worked two jobs 30 miles apart, professor of statistics at Princeton and scientist at Bell Labs, and advised five presidents, the NSA and the CIA.
| Cold War role | What the chapter records |
|---|---|
| Wartime operations research | how a B-29 over Europe should aim its machine-gun fire |
| Nike | broad-brushed aerodynamics, trajectory and warhead of the first surface-to-air missile system, with pencil and paper |
| U-2 | helped persuade Eisenhower to build the spy plane |
| H-bomb | evaluated the Teller-Ulam design in early 1951; "supervisor, Military Systems Analyst" at Forrestal Research Center (Project Matterhorn), 1951 to 1956 |
| Test ban | in 1959 showed Soviet delegates that seismograms could tell underground tests from earthquakes, which helped make the 1963 Partial Test Ban Treaty possible |
| Cryptography | CIA and NSA advisory boards from 1952; decades on the board of the Institute for Defense Analyses, housed behind an eight-foot brick wall in Von Neumann Hall until protests forced it off campus in 1970 |
He also supervised more than 50 PhD students and put his name to about 800 publications with over 105 coauthors, most often Fred Mosteller. The stories are legion: dozing or reading mail in the back row and then rising to dissect the talk; opening a 1977 Imperial College lecture cross-legged on the podium with "Comments, queries, suggestions?" and eating twelve prunes until someone asked a question; 14,000 paperbacks; the same black polo shirt for 40 years.
Why NBC, and why Tukey said yes
NBC's Huntley-Brinkley Report reached 20 million viewers a night, and its owner RCA wanted to show off its computers. With memories of the polling disasters of 1936 and 1948 and a razor-close Kennedy-Nixon race, executives went looking for help.
- Polling reform: commercial pollsters were slow to adopt random sampling. On the Kinsey committee Tukey said he would prefer a random sample of three to a Kinsey sample of 300.
- Computers: RCA built mainframes for the military and big business. Tukey had sat on von Neumann's computer design committee in 1945, though his own hardware was pencil and paper.
- Real data: the topologist turned "data analyst" wanted "the rocky road of real problems in preference to the smooth road of unreal assumptions."
- The chase: in his words, "the best education in real-time statistics that anybody could have."
Election night
On 8 November 1960 Kennedy won by about 120,000 votes out of 70 million, the tightest race since 1916. Tukey's team was ready to call it at 2.30 a.m., but nervous executives locked them in a room without phones until 8 a.m. He had at least kept NBC from wrongly declaring Nixon, and stayed 18 years, through 1980, with a team that came to include ENIAC co-inventor John Mauchly, Jerome Cornfield, Richard Link (his first graduate student and Louis Harris's chief statistician), Robert Abelson, David Brillinger and, from 1964, David Wallace, fresh from the Federalist.
Wallace expected a holiday from Bayes, since Tukey was thought to look down on it, and instead found it "securely ensconced" in the program: "I immediately thought, this is all very Bayesian." Fienberg called it "a form of empirical Bayes, where the past results were used to construct the prior distribution."
Before the election: build a prior from pre-election polls, political scientists' opinions and voting histories of precincts, counties, cities and states. On the night: watch partial returns for swings from past behaviour and update the odds. Wallace's example: "I don't know what's happening in county A, but county B is very similar and it's showing a swing 5% toward Republicans," so county A might be going the same way, "but not give it great weight, because you do have to come up with a number." Tukey's instruction: take a group of similar counties, weight the data, give zero weight to counties with no data, "and update it all the time," working upward from rural and urban groups to regions and states. Wallace: "very Bayesian, with a hierarchical model and historically based prior variances."
When nights went wrong
- 1964: NBC put seven RCA computers on the floor of Studio 8H behind Huntley. They did not work, whether from an unfinished operating system or the heat of the studio lights, and sat "like so many limp rags" while the team used hand calculators. LBJ's 61% made it easy.
- In another year the team called California and New York early, then watched contrary late figures for two tense hours before patterns came back into line. Another tight race kept them working from Tuesday afternoon to Thursday afternoon.
- Wallace's lesson: turnout was harder to project than vote shares. Machine votes arrive faster, reporting is biased by area, "sometimes there's chicanery." When a student called election data the one place "all your assumptions are totally valid," Wallace was appalled: "It's a highly biased sample."
After 1980 NBC switched to exit polls: cheaper, more photogenic and chatty, the opposite of Tukey's secret mathematics.
The private man and the public sceptic
Colleagues called Tukey "a very private man," an "elliptical and enigmatic Delphic Oracle in a black polo shirt." When he kept interrupting George Box's Princeton seminar, Box called a show of hands, won, and found Tukey surprised: "In some ways, he was a very clever eight-year-old." Pratt: "Tukey could argue on both sides of anything."
| Target | Tukey's words |
|---|---|
| Bayes | "There are many classes of problems where Bayesian analyses are reasonable, mainly classes with which I have little acquaintance." Yet: "Discarding Bayesian techniques would be a real mistake; trying to use them everywhere . . . a considerably greater mistake." |
| Unified frameworks | "the greatest danger I see from Bayesian analysis stems from the belief that everything that is important can be stuffed into a single quantitative framework" |
| Fisher | "Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise." |
| Objectivity | "an heirloom" and "a fallacy": engineers are not expected to design identical bridges, so why should statisticians reach identical results? |
After 64 pages hunting for the foundations of Fisher's fiducial probability, he decided "the belief in a unified structure for inference is a dangerous form of hubris." Fisher, in one version of the story, stalked out and left the Tukeys to find their own way out of his house. Tukey's own creed became exploratory data analysis. And he renamed things: a colleague counted about 50 coinages. Bayes, on his lips, was "borrowing strength."
Why the silence?
Tukey forbade colleagues to write or talk about the NBC methods and never wrote about them himself, calling them proprietary to RCA, even after RCA sold its computer division to Sperry Rand. Late in life he conceded that "probably the best excuse for Bayesian analysis is the need . . . to combine information from other bodies of data, the general views of experts, etc.," urged rejecting "the fetish of objectivity," and argued before Congress that the Census should adjust undercounts of minorities using similar regions, a Bayesian argument the Bureau had not adopted as of 2010.
McGrayne assembles the circumstantial case. Bayes suits decoders, who start with guesses and must minimise time to a solution. Tukey sat on NSA's Science Advisory Board, which met twice a year at Fort Meade on code breaking; his friend William O. Baker of Bell Labs said Tukey helped decrypt Enigma and Soviet codes. He invited I. J. Good, Turing's wartime assistant, to lecture at Bell Labs in October 1955 (and was found next day lying on the floor, having understood it all), and introduced Good to Cornfield. He coined "bit" for Claude Shannon, who had used Bayes in wartime cryptography. Wallace: "If you go to the secret coding agencies, you'd find that Bayes had a larger history." Brillinger: "I have no problem thinking that he might have." Whether Tukey did hands-on cryptography is not known.
The cost was historical. Bayes played a starring role on national television for almost two decades without most statisticians knowing. The only large computerised Bayesian application in the public domain in the 1960s remained the Federalist study (1964); it would be 11 years before the next, and 28 years after 1980 before a presidential forecast used Bayes again, when Nate Silver at FiveThirtyEight in 2008 used hierarchical Bayes, weighting pollsters by track record, sample size and recency, and called 49 of 50 states.
- McGrayne lists "linear programming" among Tukey's coinages; that term comes from George Dantzig and Tjalling Koopmans (late 1940s). "Data analysis" and "bit" are Tukey's.
- The claim that Tukey helped decrypt Enigma rests on a remark by Bell Labs head William Baker and has not been documented.
- The link between Tukey's silence and cryptography is McGrayne's inference, as she admits.
After 2008, hierarchical Bayesian models (FiveThirtyEight, The Economist's model built with Andrew Gelman, Stan-based forecasts) became the standard. 2016 taught the turnout and correlated-error lesson Wallace described: models that treated state errors as independent gave Trump as little as 1% to 15%, while FiveThirtyEight's correlated model gave about 29%. MRP (multilevel regression and post-stratification), Tukey's borrowing strength in modern form, is now widely used by pollsters such as YouGov.
Election night as hierarchical Bayes
NBC hired John Tukey to call races from partial returns. Priors came from polls, past elections and demographic similarity; likelihoods from counted precincts; posteriors decided when to go on air. Hierarchical borrowing kept rural and urban cells from contaminating each other. One over-performing urban precinct without a prior invents a fake swing. The chapter pairs with Cornfield and Raiffa: Bayes as a machine for decisions under the clock.
PRIOR POLL + NEW VOTES → CALL. Tukey at NBC; hierarchical models under election-night pressure.
An early precinct is 62% for candidate A; last month's poll said 51%. Write a one-line update: posterior mean ≈ weighted blend of poll and precinct. State what weight you would give 200 votes versus a 1,000-person poll.
Broadcast a call from one over-performing urban precinct and ignore the prior. That is how election desks invent fake swings. Tukey's discipline was hierarchical borrowing across similar precincts, not vibes.
Calling elections without waiting for California
NBC hired John Tukey and a team to call races from partial returns. The prior came from polls, past elections and demographic similarity; the likelihood from counted precincts; the posterior decided when to go on air. Hierarchical structure kept rural and urban cells from contaminating each other. The chapter sits with Cornfield and Raiffa: Bayes as a machine for decisions under the clock, not for journal articles.
14. Three Mile Island
In 1967 Mosteller and Tukey summed up: "the battle of Bayes has raged for more than two centuries, sometimes violently, sometimes almost placidly." By the 1970s Bayes was in the doldrums.
Treading water
Leaders scattered or died: Savage died in 1971; Jeffreys and physicist Edwin T. Jaynes campaigned in vain in physics; Lindley quit administration in 1977; Good went to Virginia Tech; Madansky went into business; Box turned to quality control; Raiffa to public policy. When Jim Berger converted in the 1970s he could follow the whole field. The first international Bayesian conference (Valencia, 1979) drew perhaps 100 people. At Berger's first conference (1976) "half the room [was] yelling at the other half": subjective (Savage) against objective (Jeffreys) priors.
- The messianic dream faded. Mosteller and Tukey favoured frequency-based or "gentle" priors that data could easily overwhelm.
- Box, Hunter and Hunter's Statistics for Experimenters (1978) left Bayes out to sell, and became a bestseller.
- Philosopher Richard Swinburne (1979) calculated God's existence as more than 50% likely (and later the Resurrection at about 97%), which did Bayes' reputation no good.
- Berkeley and Stanford stayed frequentist; anti-Bayesian signs hung on office doors.
- Herman Chernoff to Susan Holmes: "Start out as a Bayesian thinking about it, and you'll get the right answer. Then you can justify it whichever way you like."
Recriminations flew: Lindley scolded Diaconis for using frequency methods ("And you're our leading Bayesian"). De Finetti predicted a Bayesian paradigm shift after 2020; Bradley Efron put the probability of a Bayesian twenty-first century at 0.15. Asked how to spread Bayesian theory, Lindley said: "Attend funerals."
The Rasmussen Report
Eisenhower's Atoms for Peace (1953) launched nuclear power; by the early 1970s 50 privately run plants operated with no comprehensive safety study. Congress, debating liability, got the Atomic Energy Commission to order one, led by MIT physicist-engineer Norman Rasmussen (born 1927).
- No core-melt data existed, so, like Madansky, the team worked from failure rates of pumps, valves and other components.
- Where those were too sparse, they used expert opinion, politically explosive in a post-Vietnam climate of distrust and "No Nukes" protests.
- To dodge the fight over method they used Raiffa's decision trees (event and fault trees) and called the approach "subjectivistic." The only mention of Bayes was hidden in Appendix III: "the Bayesian interpretation can also be used."
- Findings: core damage was more likely than experts had thought, but its consequences not always catastrophic; human error and radioactive release outside the building were key risks.
In January 1979 the Nuclear Regulatory Commission withdrew support for the report. Two months later the core of Three Mile Island Unit 2 was badly damaged, just as The China Syndrome opened. The report looked prescient: it had flagged human error and even the kind of event sequence that occurred. In 1981 two industry studies of Zion and Indian Point finally used Bayes openly to combine generic failure rates with plant-specific data. Probabilistic risk assessment spread to chemicals, waste repositories, planetary protection on Mars, bridges and mineral exploration. "Avoiding the word 'Bayes'... is no longer necessary, or an option."
WASH-1400 was criticised in its time (the Lewis Committee, 1978) for overconfident uncertainty bounds and misleading summaries, not only for subjective inputs. McGrayne presents it mainly as vindicated. Its event-tree method survived; its headline numbers did not.
McGrayne wrote that no US plant had been ordered since 1978. Vogtle 3 and 4 in Georgia, ordered in 2008, entered service in 2023 and 2024, years late and billions over budget. Bayesian PRA is now built into NRC rules (risk-informed regulation), and the Fukushima accident (2011) prompted Bayesian re-analysis of tsunami and external-hazard priors. AI data-centre demand has revived interest in small modular reactors, whose licensing relies heavily on probabilistic safety cases.
WASH-1400, withdrawal and vindication
About 50 US plants in the early 1970s lacked a full probabilistic safety study. WASH-1400 (Rasmussen, 1974) built event trees and attached probabilities. The Lewis Committee (1978) attacked overconfident uncertainty bounds; the NRC withdrew support in January 1979. Three Mile Island followed in two months. By 1981, Zion and Indian Point studies used Bayes openly. The chapter's lesson is institutional as much as technical: rare-event risk needs priors and hierarchies; withdrawing a report under political heat does not erase the posterior that later looks right.
PRA + BAYES → RARE NUCLEAR RISK. Rasmussen WASH-1400; withdrawn, then vindicated by TMI.
A plant has three independent safety layers, each failing with probability 0.01. Approximate the joint failure probability. Then spoil independence with a common-cause prior. That tension is PRA.
Read WASH-1400's withdrawal as proof Bayes failed. Two months later Three Mile Island made the report look prescient. The shadow is confusing institutional panic with model invalidity.
Probabilistic risk before and after TMI
WASH-1400 (Rasmussen, 1974) used event trees and probabilities for reactor accidents when about 50 US plants lacked a full safety study. The Lewis Committee (1978) attacked overconfident bounds; the NRC withdrew support in January 1979. Three Mile Island followed in two months. By 1981, Zion and Indian Point studies used Bayes openly. The chapter's moral: rare-event risk needs priors, hierarchies and honesty about uncertainty, not a promise of precise frequencies from thin history.
Deepen plus: from Atoms for Peace to risk-informed rules
The political economy matters. Liability debates in Congress forced a number when the industry preferred a slogan. Rasmussen's team answered with trees because trees scale: thousands of component failures beat a handful of full-plant accidents. Expert opinion entered where data ended; that was honest and radioactive in a culture that equated "subjective" with "rigged." Hiding the word Bayes bought temporary peace and long-term confusion.
Three Mile Island was not WASH-1400's twin, but it rhymed: human error, stuck valves, misunderstood instruments, release pathways. After Zion and Indian Point, Bayesian updating of generic rates with plant-specific evidence became speakable. Probabilistic risk assessment then hopped industries because the structure travels: hazard, tree, rates, uncertainty, decision.
| Item | Figure |
|---|---|
| WASH-1400 | 1974 |
| NRC withdrawal | Jan 1979 |
| TMI Unit 2 | Mar 1979 |
| Open Bayes at Zion / Indian Point | 1981 |
15. The Navy Searches
Despite Koopman's wartime success, the US Navy took up Bayes slowly and grudgingly during the Cold War, at first wanting only the look of statistics. A series of emergencies forced the issue.
Palomares: the lost H-bomb (1966)
On 17 January 1966, during Operation Chrome Dome airborne alert, a B-52 carrying four hydrogen bombs (about 1 megaton each) collided with its KC-135 tanker six miles above Palomares, Spain. Seven of 11 crew died. Three bombs were found on land within 24 hours; two had cracked open, spreading plutonium over 558 acres. The fourth was missing. It was at least the 29th serious US nuclear weapons accident.
The Pentagon called John Craven, the Navy's civilian chief scientist ("the Navy's Oracle at Delphi," who got a C in statistics) and veteran of the Thresher search. Asked for a "search doctrine" overnight, he remembered Raiffa's idea of pooling experts' bets. With no time to convene anyone, he did "a lot of imagineering," guessing what each expert would bet, across seven scenarios:
- The bomb stayed in the bomber wreckage.
- It lay in debris along the collision path.
- It fell free, away from the wreckage.
- One parachute opened and carried it out to sea.
- Both parachutes opened and carried it farther.
- None of the above.
- A Spanish fisherman saw it enter the water (added after talking to Francisco Simó Orts).
He hired Daniel H. Wagner, Associates to "put the cloak of authenticity" on it; Tony Richardson, 26, a PhD of seven months, went to Spain.
Richardson found ships sweeping areas their sensors could not actually see. For each grid cell he computed the SEP: the probability that the bomb would have been found if it were there. Many were zero. The underlying Bayesian logic: an unsuccessful search lowers the probability of the cells searched and raises it everywhere else, like a missing sock that, after a thorough search of the bedroom, is more likely in the laundry.
Rear Admiral William "Bull Dog" Guest ("I didn't think we were getting a teenager") used SEPs to document coverage but stuck to his "plan of squares" and never updated Craven's priors. Orts had seen a grey parachute with "half a man, with the insides trailing" splash down. Lieutenant Commander Brad Mooney, raised among New Hampshire lobstermen, took him to sea; Orts twice pointed to the same spot. The likeliest site was a deep canyon only the submersible Alvin could reach. President Johnson, told "We can only tell you the probability," demanded a date. On 15 March 1966, as professors endorsed the Bayesian plan, Alvin found the bomb on its 19th dive, "like a big body in a shroud," about 2,850 feet down and within a mile of Orts's spot. Orts was awarded $10,000 in 1971. Richardson's verdict: "Bayes was a sidelight to the H-bomb search." Without updating, there was no Bayes.
K-129 and Scorpion (1968)
When Soviet sub K-129 vanished, US sensors heard a bang exactly on the International Date Line at 40°N, far from the Soviet search. Craven had Wagner run a probability analysis without telling them the target. In May 1968 the USS Scorpion disappeared with 99 men somewhere on a 3,000-mile route home. Secret hydrophone "blips" 400 miles south-west of the Azores narrowed the area to a few square miles, and this time Craven ran a full Bayesian search from the start.
- On 18 July 1968 Craven and Captain Frank Andrews gave Richardson and Larry Stone a "brain dump": nine scenarios for the sinking, each weighted by credibility.
- Monte Carlo prior. Richardson simulated "little hypothetical submarines": pick a scenario by its weight, then random speed, heading and time to hit bottom; repeat 10,000 times; count the points in each of 172 cells covering 140 square miles. Cells E5 (1,250 hits) and B7 (1,096) stood out. Academic Bayesians would not adopt Monte Carlo for another 20 years.
- Updating with daily search effectiveness from the research ship Mizar's camera sled. Searched cells fall, unsearched ones rise; keep searching the highest-probability cell. Bayes became "the group memory of the search."
- Problems: untested sensors, poor navigation, and a map that took a month to reach the ship (Stone hand-carried it to the Azores on 12 August).
- On 28 October 1968, on the sled's 74th run, Scorpion appeared in cell F6, 260 yards from the high-probability cell, near a shiny metal fragment found early on. "Close enough for government work." Andrews reckoned Bayes was a day and a half behind.
Stone's Theory of Optimal Search (1975) became the classic text. At Suez (after 1973) Wagner used three conjugate priors (Poisson, binomial, negative binomial) to estimate unexploded ordnance left in the canal; nobody mentioned Bayes to the disposal teams.
Moving targets: Coast Guard and Soviet subs
Joseph Discenza discovered the Coast Guard had been using Koopman's Bayesian search theory by hand since the war. With Wagner it built a computer system (1972 to 1974) with Monte Carlo particles (10,000 possible positions) and an early Kalman-type filter for drifting targets. Rules: experts weight scenarios privately, then pool; never discard a scenario ("Discarding one of the pieces of information is in effect making the subjective judgment that its weight is zero"). In 1974 it found three more survivors of a tuna boat after a chance rescue; in 1976 a sailor adrift 28 days.
Kalman (1960) and Kalman-Bucy (1961) filters update estimates as each observation arrives. Kalman denied any link to Bayes; Masanao Aoki showed in 1967 that the filter follows from Bayes' rule.
In 1976, in Naples, Richardson fused sonobuoy, radar and intelligence data on Soviet officers' habits to track a Soviet Echo II submarine. Pressed to move his ships east, Vice Admiral John Nicholson waited because Richardson put the chance the sub had not yet passed the Sicily Straits at about 55%. It had not. Later the Soviet sub rammed the destroyer USS Voge, apparently trying to cut its trailing sonar wire. "Most everyone became a believer." At a 1979 NATO symposium, fisheries biologist Ray Hilborn was converted: "Everyone who is actually involved in the real world does things in a Bayesian way."
- McGrayne calls K-129 "the source of" The Hunt for Red October; Clancy's novel was inspired by the 1975 Storozhevoy mutiny. K-129 was the target of the CIA's Project Azorian (1974), which raised part of the submarine with the Glomar Explorer; the book says the US "was unable to raise it."
- Craven's "rogue submarine" theory of K-129 is his own speculation and is not accepted by Russian sources.
- The cause of Scorpion's loss is still disputed (hot-running torpedo, battery explosion or other).
Stone and Metron's Bayesian reanalysis located Air France 447 in 2011 within a week of searching the high-probability area after two years of failure. Australia's DSTG published a full Bayesian particle-filter analysis for MH370 (2015 to 2017); renewed private seabed search activity continued into 2025. SAROPS, the US Coast Guard's Monte Carlo system, remains standard. Particle filters are core to robot localisation (SLAM), self-driving stacks and tracking.
Palomares: half Bayes under pressure
16 to 17 January 1966: a B-52 with four H-bombs collides with a KC-135 tanker over Palomares, Spain. Each bomb about ten feet long, yield on the order of one million tons TNT. Three bombs found in 24 hours; the fourth vanishes at sea. This was at least the 29th serious US nuclear-weapons accident; several bombs remain in oceans and swamps. Public denial lasted days; Radio Moscow scooped US briefings.
John Craven (Caltech physics, Special Projects / Polaris, Thresher recovery mandate after 1963) had a C in statistics and a Bayesian habit from Rufus Reber's Koopman tables and from hearing of Raiffa. In one night he invents seven scenarios, weights them by "imagineering" and hires Daniel Wagner's firm. Henry Richardson invents Search Effectiveness Probability after watching ships sweep blind. Admiral Guest ("Bull Dog") wants certification that the bomb is not there more than a finding; he never updates Craven's priors. Cost of Aircraft Salvops Med: about $12 million (1966). Alvin finds the bomb on dive 19 (15 March 1966), near fisherman Francisco Simó Orts's mark. Palomares is Bayes as window dressing that still almost worked.
Scorpion: full Bayes
SCENARIO WEIGHTS → GRID → UPDATE ON FAILURE. Palomares (half Bayes); Scorpion (full Bayes); Coast Guard particles.
A lost phone is in one of five rooms with prior probabilities 0.4, 0.3, 0.15, 0.1, 0.05. You search the top room with detection probability 0.8 and fail. Renormalise the five probabilities. Where do you search next?
Hire mathematicians to certify a search after the fact without updating the prior when cells fail. That was Palomares's temptation. Richardson's SEP maths without posterior updates is documentation, not Bayes.
From Palomares to Scorpion to SAROPS
16. Eureka!
By 1980 computers were flooding science with data and many unknowns at once: the curse of high dimensionality. Bayes had already survived five near-fatal blows (shelved by Bayes, ignored after Price, abandoned by Laplace, banned by frequentists, hidden by the military). Could an eighteenth-century rule designed for scarce data cope?
Still impractical
Bayesians numbered about a hundred. A 1982 meeting called "Practical Bayesian Statistics" sounded like an oxymoron; organiser Philip Dawid admitted Bayesian computation "of any complexity was still essentially impossible." Lindley and Adrian Smith's hierarchical models (1972) "fell flat." Bayes himself was not even in the Dictionary of National Biography. Yet:
An Air Force review by Teledyne Energy Systems used a Bayesian analysis of prior experience, 32 confirmed failures in 1,902 rocket motor launches, and put the probability of a booster failure at 1 in 35. NASA's estimate was 1 in 100,000. On 28 January 1986, on the shuttle programme's 25th launch, Challenger exploded, killing all seven crew.
Early breakthroughs
- DuMouchel and Harris (1983) used hierarchical Bayes meta-analysis to borrow evidence from mice, hamster cells and chemistry about diesel exhaust: a typical urban resident's lung cancer risk from diesel was negligible; a pack-a-day smoker's was 420,000 times greater.
- Adrian Raftery (Dublin) compared models of nineteenth-century British coal-mine explosions and found an abrupt drop around 1889 to 1890, matching the founding of the militant Miners' Federation in 1889: "a Eureka moment." His short 1986 paper on Bayesian model comparison (the BIC) swept sociology: "we should be comparing the models, not just looking for possibly minor discrepancies between one of them and the data."
- Image restoration. Bobby Hunt used Bayes at Los Alamos (1973 to 1974); Julian Besag modelled neighbouring pixels from diseased tomato plants; Ulf Grenander's Brown group tackled millions of unknowns. Stuart and Donald Geman (1984) invented the Gibbs sampler, named on a whim after Whitman's Sampler chocolates and the physicist Josiah Willard Gibbs. Valentin Turchin had found it in Russia in 1971.
Lindley later called it "a major mistake of my professional life, not to have appreciated the need for computing rather than mathematical analysis." Smith saw it: "Efficient numerical integration procedures are the key to more widespread use of Bayesian methods" (1984). He translated de Finetti, co-founded the Valencia meetings with Lindley, Bernardo and DeGroot (from 1979), and set students building software.
MCMC: the forgotten tool
| Year | Step |
|---|---|
| 1906 | Markov invents chains (tested on vowels and consonants in Pushkin) |
| 1930s | Fermi computes neutron paths by hand at night |
| 1946 to 1949 | Ulam, von Neumann, Metropolis: Monte Carlo on ENIAC for the H-bomb |
| 1953 | Metropolis, the Rosenbluths and the Tellers publish the algorithm in a chemistry journal |
| 1970 | Hastings generalises it in Biometrika; ignored; he left research |
| 1977 | EM algorithm (Dempster, Laird, Rubin) |
| 1984 | Geman and Geman: Gibbs sampler for images |
| 1987 | Tanner and Wong: data augmentation |
| 1990 | Gelfand and Smith: sampling-based Bayesian inference |
| 1991 | BUGS software released |
On sabbatical at Nottingham, Alan Gelfand read Tanner and Wong; David Clayton pointed him to the Gemans. The pieces fell together: replace impossible integrals with sampling. "Break the problem into tiny pieces that are easy to solve and then do millions of iterations. So you replace one high-dimensional draw with lots of low-dimensional draws that are easy. That's how you break the curse of high-dimensionality."
Markov chain Monte Carlo draws a long chain of samples, each depending only on the one before, which settles into the posterior distribution. Averages over the samples answer any question about the posterior. After 250 years Bayesians could finally use realistic priors and likelihoods. "Exact" came to mean "simulated." Within years Bayesians could do more than frequentists.
The 1990 paper used the B-word only five times in 12 pages; a follow-up with Hills and Racine-Poon spoke of "universality" and problems "hitherto regarded as intractable." Luke Tierney tied it to Metropolis, and the name MCMC stuck. Robert and Casella called it an "epiphany." Some 80 scientists came to a 1991 workshop in Ohio. Jun Liu and Charles Lawrence used it to find motifs in DNA and proteins; particle filters and real-time uses in finance and AI followed. Valencia attendance quadrupled. In 1993 Bayes entered the Dictionary of National Biography.
BUGS and medicine
David Spiegelhalter at the MRC Biostatistics Unit in Cambridge linked Gibbs sampling to graphical models and released free BUGS (Bayesian inference Using Gibbs Sampling) in 1991. It "caused the biggest single jump in Bayesian popularity," spreading to ecology, health economics, genetics, fisheries and more. Peter Green's reversible-jump MCMC (1995) let analysts move between models (stroke risk: blood pressure, exercise, diabetes, aspirin).
- Universal HIV screening (1991) would flood a low-prevalence population with false positives (Johnson and Gastwirth).
- A woman screened yearly for 10 years is almost certain to get at least one false-positive mammogram; biopsies cost $1,000 to $2,000.
- Mammogram sensitivity about 85 to 90%: a negative result does not rule out a lump found months later.
- PSA testing: most positives are false, and negatives give little reassurance.
- Bayesian reasoning entered physicians' licensing exams in 1992; clinical risk scores for heart attack, DVT and embolism are rough Bayes.
Judea Pearl's 1988 book popularised Bayesian networks and revived AI; David Heckerman's lymph-node diagnosis program won an ACM award (1990) and he went to Microsoft. The FDA accepted Bayes for medical devices but resisted it for drugs; England's NHS reimbursement work was "very Bayesian without using the B-word."
Wildlife: Bayesian analysis showed bowhead whales increasing (1993, Zeh and Raftery, combining visual and acoustic counts); Paul Wade showed in five years rather than seven that Cook Inlet belugas were declining, leading to a hunting moratorium in May 1999. The National Research Council (1998) urged Bayesian fish-stock assessment. David Blackwell, once the frequentist at RAND, had predicted Bayes would grow through "actuaries, engineers, business people... who actually like the Bayesian approach and use it." Diaconis, 1997: "Our time is now." Smith became the first Bayesian president of the Royal Statistical Society (1995), then left statistics, saying the paradigm was settled: plug it in and "turn the Bayesian crank."
- McGrayne credits the EM algorithm to "Arthur Dempster and his student Nan Laird"; the 1977 paper was by Dempster, Laird and Rubin. Her claim that the NSA secretly developed EM earlier is unsourced.
- "Federal Drug Administration" should be the Food and Drug Administration.
- The "Wildlife Protection Act" amendment under Clinton is unclear; the relevant change was the 1994 Marine Mammal Protection Act amendments (potential biological removal), which use precautionary rather than explicitly Bayesian rules.
- MCMC's "victory" also brought problems the chapter skips: convergence diagnostics, slow mixing and hidden prior sensitivity.
- Software. Stan (HMC/NUTS), PyMC 6 (May 2026: Numba default, Nutpie sampling, ArviZ 1.x), NumPyro, BlackJAX. Variational options: Pathfinder, DADVI (pymc-extras). Ongoing NUTS work (Fisher-divergence mass matrices, low-rank-plus-diagonal preconditioning; Walnuts/Nutpie, 2025 to 2026).
- Medicine. FDA January 2026 draft guidance, Use of Bayesian Methodology in Clinical Trials of Drug and Biological Products (primary inference; prespecified priors; simulation of operating characteristics; comment window into March 2026). ICH E20 adaptive designs treat Bayesian options as mainstream.
- People. Adrian Smith served as President of the Royal Society (2020 to 2025).
- Risk screening. COVID-19 testing (2020 to 2022) retaught Appendix B's base-rate lesson in public.
Crisis of practicality, then Gibbs
By 1980 data flooded mainframes; Bayesians numbered maybe a hundred. A 1982 meeting titled "Practical Bayesian Statistics" was a joke; Dawid admitted complex Bayesian computation was "essentially impossible." Lindley and Adrian Smith's hierarchical models were too stylised; textbooks would take 20 years to catch up. Lindley later called his neglect of computing "a major mistake." Smith italicised the manifesto: efficient numerical integration is the key.
Meanwhile applications nibbled. Teledyne's shuttle review (1983) used 32 failures in 1,902 rocket launches for a Bayesian 1 in 35 booster risk against NASA's 1 in 100,000. DuMouchel and Harris borrowed animal diesel-emission data for human cancer risk. Raftery's short Bayesian paper on abrupt coal-mine accident changes beat his dense frequentist twin and swept sociology. Bobby Hunt used Bayes for classified image restoration at Los Alamos (1973 to 1974); Congress heard him on the Kennedy assassination images without the word Bayes. Besag studied tomato epidemics on lattices; Donald and Stuart Geman named the Gibbs sampler after a Whitman's Sampler chocolate map (1984 IEEE paper). Russian Turchin had the idea in 1971, unread in the West.
MCMC arrives in statistics
Gelfand and Smith (1990) reintroduced Gibbs sampling to statisticians; Metropolis (1953) and Hastings (1970) waited in physics. BUGS (1991) put the method in medicine and epidemiology. Valencia Bayesian meetings (from 1979) built camaraderie under attack. Diaconis noted the field had become oddly non-competitive. The computer finally matched the logic: hierarchies that were sermons in 1970 became weekend projects.
Stan (HMC/NUTS), PyMC 6 (May 2026: Numba default; nutpie Rust NUTS as default when installed, ~2× faster; low-rank mass-matrix adaptation; ArviZ 1.x / xarray DataTree), NumPyro, BlackJAX. Variational: Pathfinder, DADVI. FDA January 2026 draft guidance on Bayesian primary inference for drugs and biologics (PDUFA VII commitment; comments into March 2026).
GIBBS → MCMC → BUGS → STAN. The computer finally matches the logic.
Explain to a colleague in four sentences why a posterior integral that is hopeless in closed form becomes routine once you can simulate draws from the posterior. Use the words chain, burn-in and summary.
Run MCMC, ignore convergence diagnostics, and treat the pretty density plot as truth. The chapter's victory can become a new priesthood that hides prior sensitivity and slow mixing behind software defaults.
When computation caught up
Metropolis (1953), Hastings (1970), Geman and Geman (1984) and Gelfand and Smith (1990) turned high-dimensional posteriors into simulated chains. BUGS (1991) put Gibbs sampling in applied hands. Hierarchical models that were sermons in 1970 became weekend projects. FDA January 2026 draft guidance puts Bayesian primary inference for drugs and biologics on the official agenda.
17. Rosetta Stones
Two and a half centuries on, Bayes has "taken wing," soaring through science and the Internet. The war has cooled: computer scientists care about results, not philosophy, and theorists accept Tukey's maxim, "far better an approximate answer to the right question... than an exact answer to the wrong question." Bayes and Fisher's likelihood ended their cold war in a grand synthesis behind modern modelling. Bradley Efron, author of a classic 1986 defence of frequentism, now says Bayes is "one of the great branches of statistical inference." Robert Kass notes what neither Bayes nor Laplace saw: accumulating data makes open-minded observers converge on the truth.
Fashionable, but not everywhere
- Still resistant: the US census (partisan politics) and Anglo-American courts. A British appeal court in the 1990s (the Adams rape case) ruled that Bayesian reasoning plunged the jury into needless theory and complexity.
- Accepted: British and European forensic science, which uses the "likelihood ratio" approach (Turing's bans by another name), and a 1994 New Jersey case showing troopers targeted African American drivers.
- History: in 2000 Fraser Neiman combined Jefferson's visit calendar with DNA evidence; assuming 50:50 prior odds, he found it about 99% probable that Thomas Jefferson fathered Sally Hemings's children.
Money
- Bayesian ideas run through Nobel economics (Markowitz 1990, Harsanyi 1994, and others); Kahneman and Tversky (Nobel 2002) showed people are not intuitive Bayesians.
- Alan Greenspan (2004): "the risk-management approach to monetary policymaking is an application of Bayesian decisionmaking." After 2008 he blamed data that covered only "a period of euphoria," not the theory. Alan Blinder thought it was only a metaphor.
- Renaissance Technologies hired IBM's Bayesian speech team under Peter Brown and Robert Mercer in 1993; its Medallion fund updates priors with "a trillion bytes of data a day."
Needles in haystacks
Tom Loredo extracted information from the 18 neutrinos of Supernova 1987A, a one-off event that frequency methods could not handle; Bayes is now standard in astrostatistics and particle physics. Susan Holmes gives unseen amino acids a small probability, the Good-Turing trick from Bletchley. Daphne Koller infers gene regulation from noisy microarrays: "like inferring who knows whom in Paris from which windows are lit at 2, 3 and 4 a.m."
Sebastian Thrun's driverless car Stanley won the 2005 DARPA Grand Challenge ($2 million) using probabilistic filters to estimate obstacles: "Every bolt of that car was Bayesian" (Diaconis). It now sits in the Smithsonian. A Carnegie Mellon and GM team won the 2007 Urban Challenge. Military automatic target recognition, missile tracking and antiterror screening use Bayes too, though screening for rare terrorists hits the same false-positive wall as mass HIV testing.
The web
- Spam: Heckerman, Horvitz and colleagues at Microsoft built Bayesian antispam filters (1998); phrases like "our price" push a spam score toward 0.9999.
- Microsoft: Bayesian networks in Windows troubleshooting, handwriting and the much-mocked Clippy.
- Recommenders: the Netflix Prize (won 2009) leaned on empirical Bayes and model averaging (Volinsky and Bell).
- Google tries naive Bayes first on big problems and uses Bayesian networks for synonyms and spelling.
The noisy channel and the Tower of Babel
At IBM in the 1970s, Mercer and Brown's team treated speech as a signal through a noisy channel: find the word sequence W that maximises P(W | sound) ∝ P(sound | W) x P(W). The prior P(W) is a language model learned from huge text. They wore T-shirts printed with Bayes' theorem. In the late 1980s the Canadian Hansard (about 100 million words, three million aligned sentence pairs) became their Rosetta Stone; the landmark 1990 paper applied the same rule to translation. "Every time I fire a linguist, the performance goes up" was the era's joke (attributed to IBM's Fred Jelinek).
Google later mined UN documents (200 billion words) and won NIST's 2005 contest. The lesson: progress came from more data, not cleverer algorithms.
Learning and the Bayesian brain
- Cognitive Tutors (Carnegie Mellon, John Anderson) track each pupil's "skillometer," a Bayesian estimate of mastery; students learned as much in about a third of the time.
- Emery Brown and Matthew Wilson (1998) decoded a rat's location from about 30 hippocampal place cells with Kalman filters; Andrew Schwartz's monkeys fed themselves with neurally controlled robot arms.
- Daniel Wolpert's virtual tennis showed people combine priors about bounces with noisy sight like good Bayesians. Hypothesis: we have Bayesian brains.
Diaconis: "Bayes is still young... It hasn't settled down yet... We're just starting."
- The CMU and GM Urban Challenge win was in 2007, two years (not "the next year") after Stanley.
- Britain's Forensic Science Service closed in 2012; Bayesian evaluation continues through private providers and ENFSI guidance.
- The "Bayesian brain" remains contested: critics argue that almost any behaviour can be fitted by some prior, so the theory risks being unfalsifiable.
- Much of what the chapter calls Bayesian (naive Bayes, Kalman filters, collaborative filtering) uses Bayes' rule without full Bayesian treatment of uncertainty, as Jordan and Dawid's objection to the name "Bayesian networks" suggests.
Classical naive Bayes still powers spam and text baselines. Modern ML leans on stochastic gradient, dropout-as-noise and ensembles more than full posteriors. Active Bayesian lines as of October 2026: Bayesian LoRA / subspace VI for fine-tuned LLMs (e.g. ScalaBL), Bayesian prompt ensembles for black-box models, and textual MCMC over prompts (Textual Bayes / MHLP). LLMs emit token probabilities; those are not automatic posteriors over world hypotheses. Treat "Bayesian LLM" claims as research programmes, not settled inheritance from Chapter 17.
Dialects of the same update
Canadian Hansard parallel text became a translation Rosetta Stone; IBM speech teams moved into finance (Renaissance Technologies, 1993); Microsoft shipped Bayesian spam filters (1998); Stanley won DARPA's Grand Challenge (2005) with probabilistic sensing; Brown and Wilson decoded rat place cells (1998); forensic labs report likelihood ratios that are Turing's bans under another name. McGrayne's 2011 horizon is now substrate for search, ads, robotics and much of ML.
Naive Bayes still baselines spam and text. Production ML leans on SGD, dropout and ensembles more than full posteriors. Active lines: Bayesian LoRA / subspace VI (e.g. ScalaBL), Bayesian prompt ensembles, textual MCMC over prompts (Textual Bayes / MHLP). Token softmaxes are not posteriors over world hypotheses. Treat "Bayesian LLM" as a research programme, not a Chapter 17 inheritance.
TRANSLATION, SPAM, ROBOTS, BRAINS. The same update rule, many dialects.
Open a spam folder. List three words that raise P(spam | mail) and three that lower it. You just hand-built a naive Bayes spam filter.
Call an LLM "Bayesian" because it outputs probabilities. Token softmaxes are not posterior distributions over hypotheses about the world. Real Bayesian LLM work (prompt ensembles, LoRA subspace inference, textual MCMC) is narrower and harder.
After victory, ubiquity
Canadian Hansard as a parallel corpus; IBM speech teams hired into finance; Microsoft spam filters (1998); Stanley's DARPA Grand Challenge (2005); place-cell decoding in rats; forensic likelihood ratios as bans by another name. McGrayne's 2011 horizon is now the substrate of search, ads, robotics and much of machine learning.
Appendices
A. Dr Fisher's casebook: the doctor sees the light
Michael J. Campbell's satire casts statistical schools as religions. Frequentists are metaphorical Catholics: "significant" and "nonsignificant" replace mortal and venial sins, randomisation is grace, and the creed asks "Do you reject the null hypothesis?" Bayesians are born-again believers who can name the day Bayes entered their lives and ask strangers at conferences, "are you a Bayesian?" Sects include nonparametric agnostics, bootstrappers who see no need for God, the bell-curve cult and simulators.
Campbell calls himself an Anglican: he uses p-values and confidence intervals "but temper[s] them with prior beliefs," likes empirical Bayes, and prefers simply to be "a believer." The closing joke: monks as a vague prior (lost), a uniform prior (soldier), an informative prior (guide book) and an improper prior (tongue out).
B. Applying Bayes' rule to mammograms
A symptom-free 40-year-old gets an abnormal routine mammogram. What is the chance she has breast cancer? Quite low.
| Quantity | Value |
|---|---|
| Prior: have cancer at screening | 40 (0.4%) |
| Of those, test positive (about 80% sensitivity) | 32 |
| Healthy women falsely positive (about 10%) | 996 |
| All positives | 1,028 |
| P(cancer given positive) = 32 / 1,028 | about 3% |
When a condition is rare, most positives from a good test are still false. The prior (base rate) dominates. A woman who finds a lump has a much higher prior, so the same positive result means far more. Negative results also miss about 1 in 5 cancers. Recalculate whenever new data arrive.
The 2009 US Preventive Services Task Force advice against routine mammograms for most women in their forties was reversed: the April 2024 final USPSTF recommendation calls for screening every two years from age 40 to 74, citing rising incidence in younger women and higher risk for Black women. Digital breast tomosynthesis and AI-assisted reading (for example Sweden's MASAI trial, 2023 to 2025) aim to cut false positives and workload, which changes the likelihood terms in this calculation, not the logic.
CARTOON + MAMMOGRAM. Fisher's conversion joke; base rates that refuse to die.
Prevalence 1%, sensitivity 90%, false-positive rate 9%. After a positive mammogram, what is P(cancer | positive)? Do the natural-frequency version with 1,000 women before you touch the formula.
Quote a positive predictive value from a high-prevalence clinic and apply it to mass screening. That is how base-rate neglect kills. Appendix B exists to stop you.
Two afterthoughts that are not afterthoughts
Appendix A reprints the comic of Fisher seeing the light. Appendix B walks mammogram arithmetic until the posterior is felt in the gut. Together they close the book's loop: ideology versus calculation, and the base-rate lesson every screening programme relearns.
Synthesis
25 takeaways
- Bayes' rule is a learning loop: prior belief x likelihood of new evidence → posterior, which becomes the next prior.
- Laplace, not Bayes, made it a method, and applied it to births, astronomy and courts.
- Even a rough prior works if evidence keeps arriving; data swamp the starting point.
- The rule was attacked for subjective priors, not for its mathematics.
- Fisher and Neyman built a rival, prior-free statistics that dominated universities for 50 years.
- Jeffreys kept Bayes alive in geophysics almost alone.
- Turing used Bayes to break naval Enigma, measuring evidence in bans and decibans.
- Secrecy hurt the theory: wartime successes stayed classified for decades.
- Actuaries used Bayes (credibility theory) without the name; Bailey proved it.
- Savage and de Finetti gave subjective probability a rigorous decision-theoretic base.
- Cornfield used Bayesian reasoning to link smoking with lung cancer and to model heart-attack risk factors.
- Base rates matter: Rand's Madansky warned of nuclear accidents no one had yet seen.
- Shrinkage (empirical Bayes, James-Stein) beats treating each estimate alone.
- Decision analysis (Raiffa, Schlaifer, Howard) made Bayes a business tool.
- Mosteller and Wallace showed Madison wrote the disputed Federalist papers, the biggest Bayesian civilian study of its day.
- Tukey called US elections for NBC with Bayes-like methods while refusing the label.
- Expert-opinion priors powered the Rasmussen reactor report, vindicated by Three Mile Island.
- Bayesian search found the Palomares H-bomb and USS Scorpion and now guides Coast Guard rescue.
- Failure is evidence: an empty search cell raises the probability everywhere else.
- Computation was the real bottleneck; MCMC (1990) and BUGS (1991) removed it.
- Model comparison (Raftery's BIC) can test competing hypotheses frequentism handles badly.
- Rare conditions defeat good tests: most positive mammograms are false.
- The noisy channel equation of speech and translation is an ancestor of today's language models.
- Pragmatism won: Bayes spread because it worked, not because it won the argument.
- As of 2026 the war is mostly over: Bayesian and frequentist tools coexist, and regulators accept both.
Frameworks to reuse
| Framework | Use it when | Origin |
|---|---|---|
| Prior → likelihood → posterior | Any belief that should change with evidence | Bayes, Laplace (1-2) |
| Bans and decibans | Adding up independent clues | Turing (4) |
| Credibility weighting | Blending small own data with large pooled data | Bailey (6) |
| Shrinkage / empirical Bayes | Many similar noisy estimates | Stein, Efron, Morris (10) |
| Decision tree with expected value | Costly choices under uncertainty | Raiffa, Schlaifer (11) |
| Search probability map | Finding something lost; prioritising effort | Koopman, Craven, Stone (15) |
| Natural-frequency tree | Explaining test results | App B |
| Model comparison | Choosing between rival explanations | Raftery (16) |
A closer look
- Hero narrative. McGrayne writes as a partisan; frequentists appear mainly as obstacles. Fisher's and Neyman's real contributions (randomisation, error control) get short shrift.
- "Bayes" stretched. Many successes (actuarial credibility, Tukey's polling, Kalman filters, naive Bayes) used Bayes' rule without full subjective-prior inference. Some, like Tukey, rejected the label.
- Red October's origin, K-129 "never raised," Goldsboro bomb yields, Urban Challenge date.
- Thin on failure. Prior sensitivity, MCMC convergence problems and cases where Bayesian forecasts failed (such as some 2016 election models) receive little attention.
Who's who
| Person | Role |
|---|---|
| Thomas Bayes | Presbyterian minister; discovered the rule, 1740s |
| Richard Price | Edited and published Bayes' essay, 1763 |
| Pierre-Simon Laplace | Made Bayes a general method |
| Harold Jeffreys | Geophysicist; lone defender in the 1930s |
| R. A. Fisher, Jerzy Neyman | Built prior-free statistics |
| Alan Turing, Jack Good | Bletchley Park; bans, Good-Turing |
| Arthur Bailey | Actuary; credibility is Bayes |
| Person | Role |
|---|---|
| L. J. Savage, Dennis Lindley | Postwar champions |
| Jerome Cornfield | Smoking, Framingham |
| Albert Madansky | Rand nuclear accident study |
| Howard Raiffa, Robert Schlaifer | Decision analysis |
| Frederick Mosteller, David Wallace | The Federalist |
| John Tukey | NBC election calls |
| Norman Rasmussen | Reactor safety report |
| John Craven, Henry Richardson, Lawrence Stone | Navy searches |
| Alan Gelfand, Adrian Smith | MCMC synthesis |
| David Spiegelhalter | BUGS |
| Judea Pearl | Bayesian networks |
| Robert Mercer, Peter Brown | Noisy channel, RenTech |
Glossary
Terms below track McGrayne's story. Prefer the chapter TO REMEMBER boxes for recall; use this list when a word reappears chapters later.
| Term | Meaning |
|---|---|
| Bayes' rule | P(H given E) = P(E given H) x P(H) / P(E) |
| Prior | Probability of a hypothesis before new evidence |
| Likelihood | Probability of the evidence if the hypothesis is true |
| Posterior | Updated probability after the evidence |
| Inverse probability | Old name for Bayesian inference: from effects to causes |
| Frequentism | Probability as long-run frequency; no priors on hypotheses |
| Subjective probability | Probability as a rational degree of belief |
| Uniform prior | Every possibility equally likely to start |
| Conjugate prior | A prior whose posterior has the same form, easing calculation |
| Ban / deciban | Turing's units of evidence weight: a factor of 10 / one tenth of a ban |
| Term | Meaning |
|---|---|
| Credibility | Actuarial weight given to an insurer's own data versus pooled data |
| Empirical Bayes | Estimating the prior from the data themselves |
| Shrinkage | Pulling individual estimates toward a group average |
| Hierarchical model | Model with layers of parameters sharing information |
| Decision analysis | Choosing actions by expected utility under uncertainty |
| Likelihood ratio | How much more probable the evidence is under one hypothesis than another |
| Term | Meaning |
|---|---|
| Base rate | Background frequency of a condition in a population |
| False positive | A positive test in someone without the condition |
| Sensitivity | Share of true cases a test detects |
| Monte Carlo | Using random sampling to compute quantities |
| MCMC | Markov chain Monte Carlo: sampling a posterior by a chain of dependent draws |
| Term | Meaning |
|---|---|
| Gibbs sampler | MCMC method updating one variable at a time |
| Metropolis-Hastings | General MCMC accept/reject algorithm |
| BUGS | Early general-purpose Bayesian software (1991) |
| Bayesian network | Graph of variables with conditional probabilities |
| Naive Bayes | Classifier assuming features are independent |
| Kalman filter | Recursive Bayesian estimator for tracking |
| Noisy channel model | Recover the most probable message given a garbled signal |
| BIC | Bayesian information criterion for comparing models |
| Probabilistic risk assessment | Estimating system failure odds from component probabilities |
| Search effectiveness probability | Chance a search of a cell would find the target if it is there |
| In-context learning | An LLM adapting to examples in its prompt, often modelled as implicit Bayesian inference |
| PRA | Probabilistic risk assessment (event/fault trees + rates) |
About this book
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 by Sharon Bertsch McGrayne, 2011. This is a chapter-by-chapter study of the book, with summaries, maps, tables and diagrams; short quotations keep their page references. It is shared free for personal study, is not for sale, and is not endorsed by the author or publisher. Please read and buy the original.
Insight Editions · rajeshrmahapatra.com/insight-editions