The Bayesian Quill

Frederick Mosteller, David Wallace, and the Disputed Federalist Papers

Two authors writing on the same constitutional crisis in the same year, using the same eighteenth-century vocabulary and averaging 34.55 and 34.59 words per sentence respectively, cannot be told apart by traditional literary criticism. For 175 years, that stylistic convergence deadlocked American historians over the authorship of twelve essays inside The Federalist. Published in New York newspapers between October 1787 and August 1788 under the shared Roman pseudonym Publius, the eighty-five essays urged voters to ratify the United States Constitution.

Because the three authors—Alexander Hamilton, James Madison, and John Jay—kept their assignments secret during the ratification debates, no public attribution list existed until July 1804. Two days before crossing the Hudson River to Weehawken, New Jersey, for his fatal duel with Vice President Aaron Burr, Alexander Hamilton slipped a handwritten list into a bookshelf inside the law office of Judge Egbert Benson. In it he claimed personal authorship of sixty-three of the eighty-five Federalist numbers. A decade later, upon retiring from the presidency, James Madison published his own annotated list in the 1818 Jacob Gideon edition of The Federalist. Both men agreed that John Jay wrote five essays, Madison wrote fourteen undisputed essays, and Hamilton wrote fifty-one undisputed essays (plus three joint papers). However, Madison claimed sole authorship of twelve essays that Hamilton had claimed before his death: Federalist Numbers 49 through 58, and Numbers 62 and 63.

For a century and a half, historians tried to adjudicate the twelve disputed papers by hunting for political ideas or structural statistics. Every traditional test failed. Both Hamilton and Madison were writing from the same Philadelphia convention notes under tight deadlines. When statisticians measured sentence lengths across all undisputed papers, Hamilton averaged 34.55 words per sentence (standard deviation 19.2) while Madison averaged 34.59 words per sentence (standard deviation 20.3). Looking at political nouns like executive, legislature, court, taxation, or war also proved useless. The frequency of content words depends on the topic of an essay rather than who wrote it.

In 1959, encouraged by historian Douglass Adair—who had noticed that Hamilton habitually wrote while whereas Madison wrote whilst—Harvard statistician Frederick Mosteller and University of Chicago statistician David L. Wallace launched a three-year forensic investigation. Using Harvard's IBM 7090 mainframe computer and student assistants punching word cards by hand, Mosteller and Wallace realized that a writer's true fingerprint lies in low-level, context-free function words. Prepositions, conjunctions, articles, and common modifiers (by, from, to, upon, whilst, enough, also, an) keep nearly invariant usage rates regardless of whether an essay discusses naval militias, judicial tenure, or Greek confederacies.

Screening 165 candidate markers across forty-eight known Hamilton essays (94,000 words) and fifty known Madison essays and external pamphlets (114,000 words), Mosteller and Wallace isolated thirty discriminator words. The sharpest separator was the preposition upon. Across forty-eight undisputed essays, Hamilton used upon an average of 3.24 times per thousand words and never wrote a thousand-word essay without it. Madison used upon only 0.23 times per thousand words, omitting it completely in over eighty percent of his papers. Similarly, Madison used by 11.4 times per thousand words against Hamilton's 7.3, while Hamilton used to 40.8 times per thousand words against Madison's 35.2.

Because writers repeat a word slightly more often once it enters short-term memory, word counts in real prose are mildly overdispersed compared to a memoryless Poisson process. Mosteller and Wallace modeled the probability density $f_M(x_i)$ and $f_H(x_i)$ of observing count $x_i$ of word $i$ under a negative-binomial distribution. They combined all $k = 30$ independent function words $\mathbf{x} = (x_1, x_2, \dots, x_k)$ via Bayes' theorem to compute the log posterior odds between Madison ($M$) and Hamilton ($H$):

Even when Mosteller and Wallace set the initial prior odds $\frac{P(M)}{P(H)}$ to 1 or handicapped Madison by ten thousand to one, the cumulative log-likelihood sum overwhelmed the prior. In every one of the twelve disputed essays, upon was virtually absent, whilst appeared repeatedly, and the prepositional ratios aligned with Madison's negative-binomial baseline. Only Federalist No. 55 showed slightly weaker odds (roughly 90 to 1 for Madison). Across the remaining eleven disputed essays, Mosteller and Wallace's likelihood ratios favored James Madison by odds exceeding millions to one, turning statistical stylometry into a standard forensic tool.