The Die Known From One Coin

I.The First Known Thing — the mint whose dies were never counted

numismatics · the die study

An ancient coin was struck by hand between two dies, one set in an anvil and one on a punch, and each die was itself cut by hand. No two were alike, so that two coins from one die can be recognised as such by close comparison, and a die study does this for a whole issue: every specimen that can be traced in collections, hoards and sale catalogues is assembled and assigned to its dies. The mint’s own records do not survive. What the student wants is the number of dies the mint used, that being the nearest thing to a measure of the issue’s size that the coins afford, and what he has is a sample: so many coins, from so many dies, of which some dies are known from many coins and some from one. François de Callataÿ, reviewing the practice in 2011, counted 609 such studies of Greek coinages, in which 92,550 coins had been assigned to 21,973 obverse dies, a little over four coins to a die.

coverage, and the die seen once

The inference from the sample to the mint rests on a result published by I. J. Good in 1953, who credited the idea to Turing. If the items in a sample are independent draws from a population of kinds, the share of that population belonging to kinds not yet seen is estimated by the share of the sample made up of kinds seen exactly once. Warren Esty brought the result into numismatics in the 1980s and gave it its working form in 2006. The coverage of a sample is one, less the number of dies known from a single coin divided by the number of coins, and it is, in his words, the probability that the next coin discovered from the issue will be from a die already observed. The estimate asks nothing about how the mint’s output was divided among its dies. It follows that the coins which repeat a known die, and which add nothing to the catalogue of types, are the whole of the evidence. A sample in which every die is seen once has a coverage of nothing, and bounds the mint not at all.

from coverage to a number of dies

The number of dies is a further step and a weaker one. The dies seen, divided by the coverage, are the number there would have been had all dies struck equally, and since they did not, the figure is too low; Esty’s formula adds a correction that grows with the share of singletons, and G. F. Carter in 1983 had given a simpler rule that needs only the numbers of coins and of dies. Stewart Lyon in 1989 tried such rules on simulated samples drawn from populations of exactly one hundred dies, and found that where the dies were unequally represented a sample of a given size showed markedly fewer of them and the plainer estimates fell short. Esty observed that real samples hold more singletons than the usual models of unequal output can explain, which he took to mean a body of dies that broke early and struck little; these he calls essentially unknowable, and the estimate is of the dies that contributed materially to the issue. Where a check exists the method does tolerably. Certain issues of the Roman Republic carry a running numeral on each die, so that the highest numeral seen is a floor under the true count. Of eighteen such issues examined by Alice Sharpless from the records of the Roman Republican Die Project, the highest numeral fell inside Esty’s interval in six and a little below the estimate in eight, and in four the estimate was too low.

what the method cannot do, as its users state it

Its limits are stated by those who use it. The draws are not independent: Esty warns that samples of coins are unlikely to be random, and a hoard is a parcel withdrawn together in one place, whose coins are apt to share dies and so to overstate the coverage of a sample they dominate. The sample can be bent by the student, and Sharpless notes that one who goes looking for the missing numerals will be left with fewer singletons than chance would have given. The method belongs to an issue that is closed, where the mint has stopped and the sample grows only by discovery. And the count of dies is not a count of coins. T. V. Buttrey, in two papers of 1993 and 1994, the second subtitled why it cannot be done, held that the number of coins struck from a die is known for no ancient issue and varied widely, so that no multiplication of an estimated number of dies by an assumed output yields a production; de Callataÿ replied in 1995, seeking a balance, and has since proposed twenty thousand coins to an obverse die as a working average for large silver, within a range of ten to forty thousand. By his own account neither persuaded the other. The dispute was chiefly over the step from dies to coins.

II.The Second Known Thing — the pipeline that crowds

drug discovery · portfolio direction, programme termination, and herding

A development programme is aimed at a target, usually a protein whose modulation is expected to alter a disease, and the choice of target is the first and largest decision in the direction of a portfolio. Sponsors choose separately and in confidence, but they choose from much the same public stock: the same literature, the same genetic associations, the same judgements as to which proteins a drug can reach. Their programmes therefore repeat one another’s targets, and the industry has a word for it, which is herding.

the crowding, as measured

Christian Fougner and colleagues, in Nature Reviews Drug Discovery in 2023, took the pipelines of the ten largest companies and asked what share of them was aimed at targets with more than five assets in development. By their firm’s summary of the analysis it was 16 per cent in 2000 and 68 per cent in 2020, the number of assets per target having risen more than two and a half times over the twenty years, with oncology leading. L.E.K. Consulting, working from Citeline’s Pharmaprojects in January 2025, counted about 10,000 pipeline drugs with a known target, making some 13,600 pairs of drug and target. Thirty-seven targets, about 2 per cent of those in play, had fifty or more drugs each and held a quarter of the pipeline; the 83 per cent of targets with fewer than ten drugs held 43 per cent of it. Over the preceding decade the pipeline had nearly doubled, from about 11,000 programmes in 2015 to about 21,000 in 2024.

the novelty, as measured

A second series is kept beside the first. The same report found that targets entering the pipeline for the first time had fallen from about a hundred a year a decade earlier to thirty in 2024, and remarked that some 55 per cent of the 4,500 or so proteins judged druggable remained untouched; Finan and colleagues in 2017 had put the druggable genome at 4,479 genes. Among approved drugs the pace is slower still, Rask-Andersen and colleagues in 2011 having counted 435 human targets acted on by approved drugs and about four drugs a year approved against a target not exploited before. An earlier count had been kinder. Agarwal, Sanseau and Cardon in 2013, taking 1,251 human protein targets with an active project in the same database, found that more than half of the novel targets in pipelines were being pursued by one company alone, and the finding was taken as a sign of a diversity that the talk of duplication had missed.

how the field explains it

The two series are reported as two problems, and both are explained by conduct. Crowding is put down to imitation, to the lower risk of a target that another has validated, and to what investors will fund; the fall in novelty to a want of boldness, and the title of the L.E.K. report asks whether biopharma is doing enough to advance novel targets. The one careful test of imitation went against it. DiMasi and Faden in 2011, examining the development and patent histories of drug classes over five decades, concluded that the arrival of several drugs in a class was better described as a race among sponsors who had set out independently than as the copying of a success. What has not been done with these records, so far as a search can find, is what the numismatist does first with his. The share of targets pursued exactly once has been reported as a sign of health and has not been read as an estimate; and the crowding and the falling novelty have not been asked whether they are one measurement taken twice.

III.The Bridge

no paper on the estimation of dies cites a paper on the crowding of pipelines; both halls hold a sample of items struck from an unseen stock of types, and only one of them reads the repeats as evidence

The halls do not meet. The die study belongs to ancient numismatics and is argued over hoards, die life and the output of mints; herding belongs to the economics and management of pharmaceutical research and is argued over incentives, differentiation and the cost of duplicated trials. Neither cites the other and no review spans them. Three neighbours must be set aside before the proximity is worth anything. That Good’s result is used elsewhere, for species in a habitat or words in a vocabulary, is not the bridge; it is the instrument, and its use in other halls does not place it in this one, where what is taken from the numismatist is the reading of an apparatus of production through the things it struck, with the hoard, the closed issue and Buttrey’s refusal attached. The counting of the druggable genome is not the bridge either; that is a census of proteins by their structure, and the quantity here belongs to whatever nominates targets and not to the genome. And the economist’s question, whether the entrants to one class raced or copied, concerns the motives of those entrants, where the matter here is what the repeats across all classes say about the stock they were drawn from.

X: where items are struck from an unseen stock of types, the share of the sample that is of types seen once estimates the chance that the next item is of a type not yet seen; the repeats are the evidence, and the chance belongs to the stock and the sample and to no one’s choice

Call it repeat coverage. Where a sample consists of items each made from one of a stock of types, and the stock cannot be inspected, the chance that the next item is of a type not yet in the sample is estimated by the share of the sample made up of types seen exactly once. The estimate holds whatever the inequality among the types, on the sole condition that the items are independent draws on a stock that is not changing. Three things are contained in it. The repeats are not redundancy; they are the only evidence that the stock is finite. The rate at which new types appear is not a free quantity: it is fixed by the singletons, and it falls as the sample grows whether or not anything about the makers has altered. And the number of types in the stock, which is the thing wanted, is bounded below by the types seen divided by the coverage, and is a count of those types that are drawn on materially, the remainder being beyond the reach of any sample.

three consequences, not one said three times

Three things follow for the pipeline. The first is that a measure of crowding which uses a threshold has no zero. The share of programmes on targets with more than five assets must rise as the pipeline grows, with no change in how any sponsor chooses, exactly as a larger sample of one issue shows more duplicated dies and no one supposes the mint to have changed its habits. A stock drawn on twice as heavily shows more repeats on that account alone, and a rise from 16 to 68 per cent cannot be divided between conduct and arithmetic until the arithmetic has been done. The second is that the crowding and the scarcity of new targets are tied by an identity. Suppose, for illustration only, that the record of every programme ever entered holds 40,000 pairs of drug and target on 3,000 targets, of which 800 have had one programme and no more. The coverage is one less 800 over 40,000, or 0.98; the next programme has two chances in a hundred of opening a target; and a year that enters 4,000 programmes is expected to open eighty. If such a year opens eighty, the scarcity of new targets is the crowding seen from the other side and wants no further explanation. If it opens thirty, the fifty that are missing are the part that the conduct of sponsors has to account for, programmes now being drawn from a narrower stock than the record was, and herding has for the first time a size, in targets not opened. If it opens a hundred and fifty, types have entered the stock since the record was struck. The same figures give the least the stock can be, 3,000 divided by 0.98, or about 3,060, and more to the degree that the targets are unequally drawn on, as they are. The third is that the stock so measured is not the genome’s. It is the repertoire of whatever nominates targets, the published biology and the judgement of what is tractable taken together, and its distance from the count of druggable genes is the number that biology permits and the apparatus does not reach. Where coverage is high, to urge boldness is to ask sponsors for singletons from a stock that has few left to give, and the lever is the cutting of new dies. Buttrey’s refusal crosses with the method: a count of targets is not a count of medicines.

IV.The Question That Becomes Possible

neither discipline would ask this alone

The numismatist has no reason to ask it. His mint is closed, his sample grows only as coins are found, and no one decides from which die the next find shall come. The portfolio analyst has no reason either. He counts programmes per target to describe competition and new targets per year to describe innovation, each series with an explanation of its own, and a target pursued once is to him a novelty and not a datum about the remainder. Held together, the halls pose a question that is neither’s: whether the rate at which new targets enter the pipeline is the rate that the repeats already in the record predict.

the covered repertoire hypothesis

Concretely. Take a record of development programmes in which each has a date of first entry and an identified target. Fix in advance what counts as a target — the gene of the protein acted on, a drug with several targets counting once for each — and what counts as entry. For each year form the cumulative record to the start of that year: the pairs of drug and target, the targets, and the targets that have had exactly one programme. The claim has three parts. First, that the share of the year’s entering programmes that fall on targets not in the record equals the singleton share of the record, within its sampling error, over any run of years in which the stock was not being enlarged. Second, that the number of targets in the stock, estimated from the repeats by a rule fixed beforehand, is stable as the record grows, later and larger samples revising it by no more than their error allows, and that it is materially smaller than the count of druggable genes. Third, that the departures from the forecast are signed and can be dated: deficits after conspicuous successes, when programmes crowd; excesses after a new instrument of nomination comes into use; and deficits that shrink when the count is confined to programmes entered before any clinical result on their target was public. The first part decides whether the instrument can see anything. The third is the finding.

why it would matter

Because the field is at present debating its own courage on the strength of two series that may be one. If the forecast holds, the fall in novel targets is what a doubled pipeline drawing on an unchanged stock was bound to produce, and a sponsor directed to seek novel targets is directed at a remainder whose size can be computed before the strategy is adopted; the remedy lies upstream of every portfolio, in whatever enlarges the stock. If it fails by deficit, herding acquires a magnitude with a null beneath it, comparable from one therapeutic area and one decade to another, which a share above a threshold is not. In either case a committee weighing a crowded target against an empty one would know what part of the reachable stock in its area the field has already opened, which is a different thing from how many competitors stand on one target.

V.How It Could Be Proven — or Broken

a candidate must be breakable, or it is merely poetry

An intercognix candidate earns nothing until it names its own test. This one needs no experiment. It needs a dated record of programmes and their targets, which exists commercially and in part publicly, and it can be run backwards, every past year being a forecast whose outcome is already known.

the honest status

Note the narrowness. The claim concerns the entry of targets into pipelines and says nothing as to whether the programmes that crowd a target are wasted; several molecules against one target may each be warranted. It does not say that sponsors do not imitate. A steady habit of joining targets in proportion to the following they already have produces the same record as independent draws on an unequal stock, and counts cannot tell the two apart; only a departure in time from the forecast is visible, so the deficit measures that part of herding which narrows the stock from what it was, and a habit that has not changed is already inside the stock. The estimated stock is a lower bound and a count of targets drawn on materially, as the estimated dies are. A new kind of drug against an old target is not a new target in this reckoning, and would need a second count, of pairs, as the reverse die needs one beside the obverse. The worked figures above are invented to show the arithmetic and carry no evidential weight. A count of targets is not a count of medicines, the yield of a target being as little known and as unequal as the output of a die. And where a field’s stock is in truth being enlarged every year the proposal can say nothing of its size, and ought to report that it cannot, which is how it should behave.