The Weight of Evidence

Bayesian Updating and Decibans Inside Bletchley Park's Hut 8

Punched onto two-foot-wide sheets of heavy manila stationery manufactured in the Oxfordshire market town of Banbury, rows of twenty-six alphabet letters sat aligned over illuminated glass tables inside Hut 8 at Bletchley Park in the spring of 1941. Across the North Atlantic, Admiral Karl Dönitz's German U-boat wolfpacks were sinking more than 300,000 tons of Allied merchant shipping every month. Unlike the German Army and Luftwaffe, whose radio operators frequently repeated predictable weather headers that allowed British electromechanical Bombe machines to brute-force their Enigma cipher settings, the German Navy (Kriegsmarine) enforced strict cryptographic discipline. Naval operators selected three rotors from a set of eight (yielding $8 \times 7 \times 6 = 336$ possible wheel orders) and encrypted their three-letter message indicators using secret bigram tables that changed every forty-eight hours.

Testing all 336 wheel orders and 17,576 ring settings on Bletchley Park's scarce Bombe machines would have taken weeks per daily key, rendering decrypted U-boat positions useless for rerouting merchant convoys. Sitting in the drafty rooms of Hut 8 alongside Cambridge mathematician Irving John "Jack" Good, Alan Turing realized that Hut 8 did not need mathematical certainty before engaging the Bombes. They needed an accounting system to accumulate tiny probabilistic hints from encrypted intercepts until only two or three candidate wheel orders remained worth testing.

To build that engine—named Banburismus after the Banbury paper sheets—Turing and Good rescued an eighteenth-century theorem written by the English minister Thomas Bayes and published posthumously by Richard Price in the Philosophical Transactions of the Royal Society in 1763. Bayes' theorem specifies how a rational agent updates the probability of a hypothesis $H$ after observing new empirical evidence $E$:

Here $P(H)$ is the prior probability of the hypothesis before the evidence arrives; $P(E \mid H)$ is the likelihood of observing $E$ if the hypothesis is true; and $P(E)$ is the total baseline probability of observing $E$ under all possible hypotheses. While standard probabilities must be multiplied and re-normalized after every clue, Turing transformed Bayes' rule by dividing the probability that $H$ is true by the probability that its complement $\neg H$ is true, converting probabilities into odds $O(H) = P(H) / P(\neg H)$. When independent clues arrive sequentially, the posterior odds equal the prior odds multiplied by the likelihood ratio of each piece of evidence.

To make the calculations fast enough for young Women's Royal Naval Service (Wrens) operators to perform using paper strips and pencil addition, Turing took the base-10 logarithm of the likelihood ratio. Just as acoustical engineers at Bell Labs had defined the bel and decibel to measure logarithmic ratios of sound power, Turing defined the unit of rational belief update as the ban—named after Banbury—and its working unit as one-tenth of a ban, the deciban:

A single deciban of evidence increases the odds of a hypothesis by a factor of $10^{0.1} \approx 1.26$, a shift barely perceptible to human intuition. Ten decibans (one ban) multiply the odds by 10 to 1; twenty decibans multiply the odds by 100 to 1. Because logarithms convert multiplication into addition, a Hut 8 cryptanalyst could start with the baseline prior odds that two intercepted U-boat messages shared the same right-hand Enigma rotor turnover—initially 1 in 25, or roughly $-14$ decibans—and slide two punched Banbury sheets across each other to count letter repeats.

In pure random noise across a 26-letter alphabet, two ciphertexts overlaid at an arbitrary offset match at the same letter position 1 time in 26 (3.85 percent). If two German radio operators enciphered plaintexts at the exact same rotor step, however, the non-uniform frequency of German words like EINS (one) and X (space) caused identical ciphertext letters to collide 1 time in 17 (5.88 percent). A single four-letter repeat between two 200-character naval intercepts was inconclusive in isolation. But on Turing's scoring sheets, a four-letter repeat contributed $+9.2$ decibans of weight; a two-letter repeat contributed $+1.1$ decibans; a stretch with zero matches subtracted $-0.4$ decibans per column.

Every afternoon in Hut 8, Wrens slid dozens of punched Banbury strips over glass lightboxes, adding and subtracting deciban scores on chalkboards as fresh radio intercepts arrived from coastal listening stations at Scarborough. Whenever the cumulative score for an overlap crossed $+20$ decibans (100-to-1 odds in favor of a genuine rotor alignment), Turing's team locked in the relative rotor offsets, deduced the Kriegsmarine right-hand and middle wheels without touching a Bombe, and cut machine runtime from hundreds of hours to less than three. By June 1941, powered by deciban scoring tables, Hut 8 was reading Admiral Dönitz's daily U-boat traffic within thirty-six hours of transmission, enabling the British Admiralty's Submarine Tracking Room to steer North Atlantic convoys silently around waiting wolfpacks for the remainder of the year.