The Theory That Would Not Die

by Sharon Bertsch McGrayne

  Max Newman, formerly Turing’s mathematics instructor at Cambridge, wanted to automate the British attack on Tunny-Lorenz’s codes, and he, Michie, and Good were already working on new machines to do it. Michie had refined Turingismus, but it soon became obvious that mechanical switches would be far too slow. The process would have to be electronic; engineer Thomas H. Flowers suggested using glass vacuum tubes because they could switch current on and off much faster. With backing from Newman, Flowers built the first Colossus at the Post Office Research Station, which ran Britain’s telephone system. Installed at Bletchley Park, Colossus decrypted its first message on February 5, 1944. Flowers’s car broke down that day but not his Colossus.

  Flowers had strict orders—no reasons given—to get a second, more advanced Colossus model operational no later than June 1. Working until they thought their eyes were dropping out, Flowers and his team had Colossus II ready on schedule.

  Almost as soon as it began operating, Hitler teletyped an encrypted message to his commanding officer in Normandy, Field Marshal Erwin Rommel. He ordered Rommel not to move his troops for five days after any invasion of Normandy. Hitler had decided it would be a diversionary feint to draw German troops away from the ports along the English Channel and that the real invasion would take place five days later. Colossus II decoded the message, and a courier raced a copy from Bletchley Park to Gen. Dwight “Ike” Eisenhower. As Ike and his staff were trying to decide when to launch the invasion of Normandy the courier handed him a sheet of paper containing Hitler’s order. Unable to tell his staff about Bletchley Park, Eisenhower simply returned the paper to the courier and announced, “We go tomorrow,” the morning of June 6.39 He later estimated that Bletchley Park’s decoders had shortened the war in Europe by at least two years.

  The Colossi became the world’s first large-scale electronic digital computers, built for a special purpose but capable of making other computations too. Flowers would build ten more models during the war. With the Germans introducing complexities that made manual decrypting methods useless, the Colossi replaced Turing’s pencil-and-paper Turingery in August 1944. As Michie reported, Turing’s Bayesian scoring system based on bans had started “first as a minor mental aid in a variety of jobs” but then turned into “a major aid in the [Colossi’s] wheel pattern breaking.”40 Turing’s method also contributed intellectually to the use of the Colossi and produced procedures that made the machines much more effective. Each new Colossus was an improvement over the previous one, and Michie believed the eleventh “nudged the design further in the direction of ‘programmability’ in the modern sense.”41

  By 1945 Turing had moved on to voice encryption at a nearby military installation at Hanslope Park. Late in the war, others at Bletchley Park, ignorant of Turing’s work on Enigma, decided to use Bayesian methods to try to break the Japanese naval codes in the Pacific. Japan’s main naval cipher, JN-25, was becoming increasingly complex, and Bletchley Park began working on some particularly difficult versions shortly after September 1943.

  A trio of British mathematicians was assigned to work in tandem with Washington. The three were Ian Cassels, later a professor at Cambridge; Jimmy Whitworth; and Edward Simpson, who had joined Bletchley Park in 1942, immediately after earning a mathematics degree at Queen’s University, Belfast, at the age of 19. Simpson had been working on Italian codes at Bletchley Park, but after Italy’s surrender he was switched to JN-25.

  “The unbelievably tight security ethos” at Bletchley Park prevented the group from getting advice from Turing or Good, Simpson explained in 2009 after his wartime work was revealed.42 As a result, the men adopted and developed Bayes on their own. It was a full year before they were able to speak with Turing’s colleague Alexander, who by that time had begun work on Japanese naval codes too.

  Japanese coding clerks who used the principal code, JN-25, transmitted their messages in blocks of five digits. The British mathematicians knew that each block was the result of adding a random five-digit group, called an additive, to a five-digit code group taken from the JN-25 codebook. In effect, British cryptanalysts had to perform the reverse operation—but without the JN code and additive books. First, they identified groups that might be additives. Then a team composed of civilians and Wrens who, despite being newcomers to cryptography, had to identify the most probable additives rapidly, objectively, and in a standardized manner. They could judge the plausibility of an additive according to the plausibility or probability of the deciphered code group produced by the additive. As a measure of their belief, team members assigned a Bayesian probability to each speculative code group according to how often it had occurred in already deciphered messages. The most probable blocks, as well as borderline or especially important cases, were studied further.

  “For practical purposes, it was not necessary to agonise over the prior odds to be assigned to the hypothesis that an additive was true,” Simpson explained. “Instead, the essential judgment to be made was whether the [weight of] collective evidence . . . was sufficiently convincing for it to be accepted as genuine . . . As always in cryptanalysis, the inspired hunch grounded in experience could sometimes make the most important contribution of all.”43

  After October 1944 Alexander, Bletchley Park’s finest Banburismus solver, developed an elaborate use of Bayes’ theorem and Turing’s decibans for the Japanese codes.

  By 1945 U.S. cryptanalysts were writing memos to one another about Bayes’ theorem. Whether the Americans learned about Bayes from Bletchley Park or discovered its usefulness on their own is not known; 65 years after the war, the British government still refuses to declassify many documents about wartime cryptography. A young American mathematician, Andrew Gleason, who was working on Japanese naval codes and who looked after Turing during his stay in Washington, almost certainly knew about Bayes during the war. He, Good, and Alexander continued to work on top-secret cryptography for decades after the war. Gleason helped establish a postwar curriculum for training cryptanalysts at the U.S. National Security Agency (NSA), taught mathematics at Harvard and NSA, and published a probability textbook that instructed a generation of NSA’s cryptanalysts in how to use Bayes’ theorem, Turing’s decibans and centibans, Bayesian inference, and hypothesis testing. Some 20 of his students became leaders in Soviet code breaking during the 1960s and 1970s. Gleason was deeply knowledgeable but pragmatic about Bayes; his textbook also discussed methods developed by Neyman, the arch anti-Bayesian.

  A few days after Germany’s surrender in May 1945 Churchill made a surprising and shocking move. He ordered the destruction of all evidence that decoding had helped win the Second World War. The fact that cryptography, Bletchley Park, Turing, Bayes’ rule, and the Colossi had contributed to victory was to be destroyed. Turing’s assistant Good complained later that everything about decryption and the U-boat fight “from Hollerith [punch] cards to sequential statistics, to empirical Bayes, to Markov chains, to decision theory, to electronic computers” was to remain ultraclassified.44 Most of the Colossi were dismantled and broken into unidentifiable pieces. Those who built the Colossi and broke Tunny were gagged by Britain’s Official Secrets Acts and the Cold War; they could not even say that the Colossi had existed. Books by British and U.S. participants in the U-boat war were almost immediately classified, confined to high-level military circles, and not published for years or in some cases decades. Even classified histories of the war excluded the decryption campaign against the U-boats. Only after 1973 did the story of Bayes, Bletchley Park, and Turing’s nation-saving efforts begin to emerge.

  Why was the story concealed for so long? The answer seems to be that the British did not want the Soviet government to know they could decrypt Tunny-Lorenz codes. The Russians had captured a number of Lorenz machines, and Britain used at least one of the two surviving Colossi to break Soviet codes during the Cold War. Only when the Soviets replaced their Lorenz machines with new cryptosystems was Bletchley Park’s story revealed.

  The secrecy had
tragic consequences. Family and friends of Bletchley Park employees went to their graves without ever knowing the contributions their loved ones had made during the war. Those connected with Colossus, the epitome of the British decryption effort, received little or no credit. Turing was given an Order of the British Empire (OBE), a routine award given to high civil servants. Newman was so angry at the government’s “derisory” lack of gratitude to Turing that he refused his own OBE.

  Britain’s science, technology, and economy were losers, too. The Colossi were built and operational years before the ENIAC in Pennsylvania and before John von Neumann’s computer at the Institute for Advance Study in Princeton, but for the next half century the world assumed that U.S. computers had come first.

  Obliterating all information about the decryption campaign distorted Cold War attitudes about the value of cryptanalysis and about antisubmarine warfare. The war replaced human spies with machines. Decryption was faster than spying and provided unfiltered knowledge of the enemy’s thinking in real time, yet the Cold War glamorized military hardware and the derring-do of spydom.

  The secrecy also had a catastrophic effect on Turing. At the end of the war he said he wanted “to build a brain.”45 To do so, he turned down a lectureship at Cambridge University and joined the National Physical Laboratory in London. Because of the Official Secrets Act he arrived as a nobody. Had he been knighted or otherwise honored he would surely have found it easier to get more than two engineers as support staff. Ignorant of Turing’s achievements, the director of the laboratory, Charles Galton Darwin, a grandson of Charles Darwin, repeatedly reprimanded Turing for morning tardiness after working late the night before. Once an afternoon committee meeting with Darwin and others stretched late in the day. At 5:30 p.m. Turing promptly stood up and announced to Darwin that he was leaving—“punctually.”46

  At the laboratory, Turing designed the first relatively complete electronic stored-program digital computer for code breaking in 1945. Darwin deemed it too ambitious, however, and after several years Turing left in disgust. When the laboratory finally built his design in 1950, it was the fastest computer in the world and, astonishingly, had the memory capacity of an early Macintosh built three decades later.

  Turing moved to the University of Manchester, where Newman was building the first electronic, stored-program digital computer for Britain’s atomic bomb. Working in Manchester, Turing pioneered the first computer software, gave the first lecture on computer intelligence, and devised his famous Turing Test: a computer is thinking if, after five minutes of questioning, a person cannot distinguish its responses from those of a human in the next room. Later, Turing became interested in physical chemistry and how huge biological molecules construct themselves into symmetrical shapes.

  A series of spectacular international events in 1949 and 1950 intruded on these productive years and precipitated a personal crisis for Turing: the Soviets surprised the West by detonating an atomic bomb; Communists gained control of mainland China; Alger Hiss, Klaus Fuchs, and Julius and Ethel Rosenberg were arrested for spying; and Sen. Joseph McCarthy of Wisconsin began brandishing his unsubstantiated list of so-called Communists in the U.S. State Department.

  Even worse, two upper-crust English spies—an openly promiscuous and alcoholic homosexual named Guy Burgess and his friend from Cambridge student days Donald Maclean—evaded arrest by fleeing to the USSR in 1950. The United States told British intelligence they had been tipped off by Anthony Blunt, another homosexual graduate of Cambridge, a leading art historian, and the queen’s surveyor of paintings. With both the British and American governments panicked by visions of a homosexual spy scandal, the number of men arrested for homosexuality in Britain spiked.

  On the first day of Queen Elizabeth II’s reign, February 7, 1952, Turing was arrested for homosexual activity conducted in the privacy of his home with a consenting adult. As Good protested later, “Fortunately, the authorities at Bletchley Park had no idea Turing was a homosexual; otherwise we might have lost the war.”47

  In the uproar over Burgess and Maclean, Turing was viewed not as the hero of his country but as yet another Cambridge homosexual privy to the most closely guarded state secrets. He had even worked on the computer involved in Britain’s atomic bomb test. As a result of his arrest, Britain’s leading cryptanalyst lost his security clearance and any chance to continue work on decoding. In addition, because the U.S. Congress had just banned gays from entering the country, he was unable to get a visa to travel or work in the United States.

  As the world lionized the Manhattan Project physicists who engineered the atomic and hydrogen bombs, as Nazi war criminals went free, and as the United States recruited German rocket experts, Turing was found guilty. Less than a decade after England fought a war against Nazis who had conducted medical experiments on their prisoners, an English judge forced Turing to choose between prison and chemical castration. He chose the estrogen injections. Over the next year he grew breasts. And on June 7, 1954, the day after the tenth anniversary of the Normandy invasion he helped make possible, Alan Turing committed suicide. Two years later the British government knighted Anthony Blunt, the spy who later admitted tipping off his friends Burgess and Maclean and precipitating the witch hunt against homosexuals. Even today, it is difficult to write—or read—about Turing’s end. In 2009, 55 years after Turing’s death, a British prime minister, Gordon Brown, finally apologized.

  Turing’s Bayesian work lived on in cryptography. Secretly for decades, an American colleague of Turing’s taught Bayes to NSA cryptographers. With Turing’s blessing, Good developed Bayesian methods and theory and became one of the world’s leading cryptanalysts and one of the three leaders in the Bayesian renaissance of the 1950s and 1960s. He wrote roughly 900 articles about Bayes’ rule and published most of them.

  Outside of cryptography, however, no one knew that some of the most brilliant thinkers of the mid-twentieth century had used Bayes to defend their countries during the Second World War. It emerged from the war as vilified as ever.

  5.

  dead and buried again

  With its wartime successes classified, Bayes’ rule emerged from the Second World War even more suspect than before. Statistics books and papers stressed repeatedly and self-righteously that they did not use the rule. When Jack Good discussed the method at the Royal Statistical Society, the next speaker’s opening words were, “After that nonsense . . .”1

  “Bayes” still meant equal priors and did not yet mean making inferences, conclusions, or predictions based on updating observational data. The National Bureau of Standards suppressed a report to Aberdeen Proving Ground, the U.S. Army’s weapons-testing center, during the 1950s because the study used subjective Bayesian methods. During Sen. Joseph McCarthy’s campaign against Communists, a bureau statistician half-jokingly called a colleague “un-American because [he] was Bayesian, and . . . undermining the United States Government.”2 Professors at Harvard Business School called their Bayesian colleagues “socialists and so-called scientists.”3

  “There still seems to remain in some quarters a lingering idea that there is something ‘not quite nice,’ something unsound, about the whole concept of inverse probability,” a prominent statistician wrote.4 Unless declared otherwise, a statistician was considered a frequentist.

  The Bayesian community was small and isolated, and its publications were well-nigh invisible. Prewar theory by Frank Ramsey, Harold Jeffreys, and Bruno de Finetti lay unread. Nearly all the papers published in the Annals of Mathematical Statistics concerned issues framed by Jerzy Neyman’s frequentist work from the 1930s. Thanks to Ronald Fisher’s genetics research and the powerful anti-Bayesian stance of an Iowa State University statistician named Oscar Kempthorne, agricultural studies at most land-grant institutions relied on frequentism. When Gertrude Cox, the president of the American Statistical Society in 1956, spoke about the future of statistics, she barely mentioned Bayes. The first practical article telling scientists how to use
Bayesian analysis would not appear until 1963.

  Not even civilian researchers for the military knew much about Bayes in 1950. When an economist was preparing a research budget for the U.S. Air Force at RAND, a California think tank, he asked visiting statistician David Blackwell how to assess the probability that a major war would occur within five years. Blackwell, who had not yet become a Bayesian, answered, “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 and said, “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.”5

  The Bayesian theorist Dennis V. Lindley concluded, “The upstart Bayesian movement is being contained, largely by being ignored.”6 Another statistician recalled, “A lot of us thought [Bayes] was dead and buried.”7

  part III

  the glorious revival

  6.

  arthur bailey

  After the Second World War the first public challenge to the anti-Bayesian status quo came not from the military or university mathematicians and statisticians but from a Bible-quoting business executive named Arthur L. Bailey.