The Denominator Problem

A numerator without a denominator is a number with nothing under it, not a fraction. Before it is anything about statistics, it is a basic condition on any argument of the form “look how many matches I found.” Before “many” can mean anything, you have to know how many you would expect. Without that baseline, accumulation cannot generate probability, no matter how long the list grows.

The problem in everyday life

The denominator problem is not unique to the authorship debate. It is a persistent feature of human reasoning, and it shows up in ordinary life before it reaches the SAQ.

A financial pundit predicted the 2008 crash and has been cited as prescient ever since. The natural question (how many other crashes did he predict that never happened?) is almost never asked. If he called ten market collapses over twenty years and one arrived, that is the record you would expect from someone making frequent confident predictions about an uncertain system, impressive only until the misses are counted. The one hit is remembered while the misses vanish, and the denominator (the total number of predictions made) is what would tell you whether the one hit was skill or noise, but it is the one number almost never reported.

The same structure appears in medical testing. A diagnostic test that is 99 per cent accurate sounds nearly infallible. Apply it to a disease that affects one person in a thousand, and the picture changes. Screen a thousand people: roughly one has the disease, and the test will almost certainly find them. But the test will also return a false positive on roughly 1 per cent of the 999 healthy people, about ten false alarms. Of every eleven positive results the test produces, ten are wrong. The accuracy figure is real; the denominator (the prevalence of the condition in the population being tested) is what determines whether a positive result is meaningful, and it is the figure most commonly omitted when the test is being sold.

The friend who seems to call just when you are thinking of them is a gentler version of the same error. You remember those three occasions vividly because the coincidence was striking. You do not record the ten thousand times you thought of them and the phone stayed silent, because nothing happened and nothing demanded to be remembered. The hits enter memory and the misses leave none, and the denominator (the total number of times you thought of them) would dissolve the pattern entirely, but it is the number you never kept.

In the Stratford context

The denominator problem has a specific application to the question of Shakspere’s documentary record. The standard response to the absence of literary documents in that record is that records from the period are sparse and many have been lost: absence of evidence is not evidence of absence. This is a reasonable general principle. It fails here for a specific reason: Shakspere left over seventy surviving documents covering property transactions, litigation, tax assessments, business dealings, and personal affairs. The denominator (total surviving documents for this individual) is not small. It is large enough to make the numerator (literary documents: zero) meaningful. The same records that establish he existed establish that writing-related records are not what is missing from the archive. They are missing from his archive specifically, while present in the archives of his contemporaries. Diana Price’s comparison of Shakspere’s record against twenty-four other writers of the period makes the point quantitatively: he is the only one for whom the denominator is large and the literary numerator is zero. The argument from general record loss cannot explain a specific gap in a large surviving record.

In the correspondence-cataloguing context

The more pervasive form of the denominator problem operates across almost all advocacy writing, and it has four components that almost never appear together in a single critical treatment. Each compounds the others.

1. The denominator is rarely computed

Advocates are meticulous about their candidate’s matches: the number of characteristics their candidate shares with the implied author of the works. Legal expertise, continental travel, knowledge of a specific craft, aristocratic milieu, the right dates: each match is carefully documented. What is almost never computed is how many other Elizabethans of similar education, rank, and circumstance would match the same profile. Where an advocate does attempt this baseline, it helps only if the counting is real: a comparison class kept narrow, or one that ignores what every educated contemporary knew, looks like a denominator but has not actually counted how common the trait was. The characteristics that seem to point uniquely at one candidate are often characteristics of an entire class. Without the baseline (the expected rate for any comparably situated person), no hit rate can be calculated, and no correspondence can be assessed as remarkable rather than ordinary.

2. The numerator is curated, not calculated

There is a prior problem that compounds the first. The numerator itself is typically overstated, because the search for matches is one-directional: advocates look for biographical facts about their candidate that correspond to features of the works, and report the ones that fit. The non-matches (the biographical facts that cut against the case, the features of the works that find no correspondence in the candidate’s known life) are almost never tabulated alongside the hits. A genuine numerator would include both. A profile-matching exercise that records only the confirmations curates its numerator instead of counting it. The total number of proposed correspondences examined and rejected is never stated, so the selectivity of the list is invisible to the reader.

A worked example: the author who knew everything

John Michell’s Who Wrote Shakespeare? (1996) assembles a striking figure: thirty-one separate fields (law, medicine, music, falconry, seamanship, heraldry, Italian topography, Greek drama, statecraft, printing, and twenty more) in each of which a specialist has pronounced the author expert. As a list it looks overwhelming. But look at the shape of every entry: a specialist in one field, reading the works through that field alone, concludes that the author must have shared their own training. Thirty-one specialists reach thirty-one versions of the same conclusion, each pointing back at their own profession.

They cannot all be right, because no real person (not Shakspere, and not any alternative candidate) was at once a trained lawyer, physician, musician, falconer, sailor and printer. But the fields are not of one kind, and the honest response is to sort the list rather than wave it away. Many of the thirty-one thin out on inspection to knowledge any well-read man of the theatre could reach: the English history that tracks Holinshed page by page, the New World detail traceable to Strachey’s letter, the folklore, the printing-house terms, the almanac astronomy. A harder residue does not: the field-verified Italian topography (Roe), the untranslated classical sources, the accurate musical vocabulary, the insider court usage, and, the most argued-over of all, the legal knowledge. Here the specialists stand on firmer ground, and this site does not pretend otherwise.

But depth is only half the test. A competence identifies an author only if it is both demanding and unequally distributed, something the works show that most candidates could not have had. Sort the hard residue on that second question and it divides again. Some of it discriminates, and those are the lines carried in the Convergence Analysis, where each is scored candidate by candidate. Some of it does not, however deep it runs: the seamanship documented by a former Royal Navy commander, the heraldry analysed by a practising herald, the thousand biblical allusions catalogued by Naseeb Shaheen are all real and all beyond casual reading, yet a maritime nation, a heraldic literacy that had spread through the gentry, and a scriptural culture spread through church, sermons and print mean the knowledge points at no one in particular. Those lines were examined and set aside for exactly that reason, in the open, on a separate page: Convergence Lines Considered and Set Aside. “The author knew everything” becomes a finding about a person only after the list is sorted twice (for depth, then for discriminating power) and each surviving competence is checked against a documented life. Taken whole and unsorted, a list that points at everyone points at no one.

3. The profile was built with the candidate already in mind

Most advocates arrive at their candidate before constructing their case. This means the profile (the set of characteristics used to identify the author) is assembled in full knowledge of the candidate’s biography. A set of characteristics derived this way cannot then be tested against that candidate without circularity: the profile was built to fit, and a close match is guaranteed in advance. J. T. Looney, who originated the Oxford candidacy in 1920, was more rigorous than most: he claimed to have derived his eighteen characteristics from the works before searching for a candidate, and to have arrived at Oxford without prior knowledge of him. Whether his account of his own procedure is accurate is a separate question; most advocates who followed Looney do not share even his stated procedural discipline.

4. The evidentiary threshold shifts between candidacies

The fourth component operates between candidacies rather than within them. The same quality of evidence, a verbal parallel of a given closeness, is read as a mark of authorship when it links the works to the favoured candidate’s own writing, and as ordinary influence when it links them to a rival’s. The Marlovians read the dense parallels between Marlowe’s plays and the works as evidence that one hidden hand wrote both; the rest of the field, Oxfordians included, reads the same parallels as one contemporary’s influence on another, or at most a shared hand in a few early plays, not as proof that Marlowe was the hidden author. An advocate cannot do otherwise while he backs his man: to read a rival’s parallels as proof the rival wrote the works would destroy the advocate’s own case, so reclassification is the only move left. The move is forced by the position, not by any dishonesty. But the same bar is then not applied across the field, and when the reading of the evidence changes with the conclusion it must support, the conclusion is doing the sorting.

But the denominator cannot be computed

The commonest reply to all of this is that the demand is impossible: no one can list every educated Elizabethan, so no one can supply the denominator, and a figure no one can produce is no fair test. One half of that is right. No register of every man who could have written the works will ever be drawn up. But a denominator need not be an exact headcount; what matters is a defensible estimate of how common the trait was in the relevant population. The medical-testing example above did not count the healthy one by one; it worked from a known prevalence. You bound the rate from the records that survive: the print runs of the standard translations, the rolls of the peerage and gentry, the matriculation and admission registers. For most of the competences in question that is enough to fix an order of magnitude, and an order of magnitude is enough to sink “only my candidate could have known this.”

There is a stronger form of the reply. The advocate concedes each trait is common and rests his case on the rarity of the combination: court access, foreign travel, aristocratic culture, Italian settings, each ordinary alone, the intersection small. But the parts of that combination do not become separate proofs merely because they can be listed separately, and here they do not: they flow from one station and one education, so their correlation has to be allowed for before the combination can be said to narrow the field. Gentle birth carries court access, aristocratic culture, a genealogist’s habit and the leisure to travel; a humanist education carries the Latin and the classical reading. Multiplying the separate rates as though they were independent manufactures a tiny denominator out of a single circumstance, the same base-rate error one level up. The proper denominator counts the contemporaries who shared the actual combination, or gives a defensible bound on them, and the burden is on the advocate to show the traits stand on their own, not one station and one schooling entered several times. What survives once the shared social fact is subtracted is the short residue of independent competences, and sorting that residue for discriminating power is exactly what the worked example above already does.

Where even a bound is out of reach, the honest verdict is that the rate is unknown, and an unknown rate is not the same as a small one. When the background frequency of a trait cannot be estimated, the weight of a match cannot be estimated either: it may be suggestive, but it cannot be offered as a measured sign of rarity, and nothing lets the advocate call it large, which was his claim. Uncertainty is not a licence to set the rate at one. His burden is to show the resemblance is appreciably more likely under his candidate than under the plausible alternatives; short of that, the match points to no one in particular. That the sorting can be done, and done in the open, is shown above: the “author who knew everything” list is sorted twice, for depth and then for discriminating power, and what survives is the small set of competences that actually narrow the field.

What follows

These four components are a system, not independent errors that happen to co-occur. Because the profile was built with the candidate in mind, it is a description shaped around one person rather than a neutral question put to the population, and any frequency estimated from it has to allow for that hindsight. Because the numerator is curated rather than calculated, the denominator problem is invisible in the presentation: the list looks exhaustive when it is selective. And because the evidentiary threshold shifts between candidacies, the asymmetry never becomes apparent to the reader who only reads one candidate’s case.

The result is a list that looks like evidence without being it. However long the catalogue grows, its length alone cannot carry the case while the selection, the background frequency, the dependence among the clues, and a consistent threshold go unexamined. Probability requires a denominator. The structural fault is enough to stop a catalogue’s length from counting as force, but it does not exempt a critic from weighing any single correspondence claimed to be unusually precise, rare, and unavailable by other routes. Adding more correspondences only compounds the fault.

The feeling of inevitability that correspondence arguments generate is real, but it tracks the length of the list and how completely the four components above have operated without being named, rather than the discriminating power of the evidence.

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The denominator problem in its Stratford form is examined in full on the Evidentiary State of Play page. The Mirror Problem, Logic Error 12, and the Correspondences pages each apply the correspondence-cataloguing version to specific contexts. The Of Course It Was Him/Her page demonstrates the denominator problem concretely: the same correspondence method, applied honestly to two different candidates, produces equally compelling conviction for both. For a short, seven-question version of this same check, addressed to any position including the orthodox one, see The Self-Check.