The Width of the Silence: Selective Survival, Lost Sources, and the Limits of Inference from What Remains

1. Introduction

Every historical statement about a population begins from a remnant. Marc Bloch described knowledge of the past as knowledge by way of tracks, and the tracks that reach the historian are those that were made, kept and not destroyed (Bloch 1953). Arnold Esch gave the problem its classic formulation when he separated the Überlieferungs-Chance, the systematic chance that a kind of source has of being transmitted, from the Überlieferungs-Zufall, the accident that decides the fate of a particular item (Esch 1985). His examples remain instructive. A bill of exchange that was honoured left fewer traces than one that ended before a court; whatever generated further records, such as a dispute, an expense that needed approval or an act of repression, stood a better chance of reaching us; a rising count of peasant disturbances may therefore register a change in record making and not a change in unrest. Michel-Rolph Trouillot made a parallel argument for the archives of slavery and revolution: silences enter the historical record when sources are made, when archives are assembled, when narratives are composed and when significance is assigned in retrospect (Trouillot 1995). Historians, in short, have long known that survival is not blind. What they have lacked is a way of saying how much this matters for a given claim.

A growing body of quantitative work addresses one half of the problem, namely how much has been lost. Cisne (2005) treated the copies of a medieval text as a population with births and deaths, following the stochastic models of manuscript traditions introduced by Weitzman (1987). Egghe and Proot (2007) estimated the number of lost printed programmes of Jesuit plays from the numbers surviving in one and in two copies, and Burrell (2008) showed that their model is a variant of the unseen species problem familiar from ecology and from the study of vocabulary (Good 1953; Efron and Thisted 1976; Chao 1984). Green et al. (2011) modelled the highly dispersed survival of incunable editions and concluded that lost editions were far more numerous than had been supposed. Buringh (2011) combined medieval library catalogues with surviving manuscripts to estimate loss rates and production in the Latin West. Most visibly, Kestemont et al. (2022) applied unseen species estimators to 3648 surviving documents that transmit 799 works of medieval heroic and chivalric narrative in six vernaculars and concluded that 68.3 per cent of the works and 9.0 per cent of the documents survive, with large differences between traditions (see also Kestemont and Karsdorp 2020). Camps et al. (2025) have since replaced the static ecological analogy with a dynamic birth-death model of copying and loss, calibrated on Old French chivalric narrative, and obtained still heavier losses. Recent syntheses for a wider readership tell the same story of destruction at length (Ovenden 2020; Bartlett 2024).

The other half of the problem has received far less formal attention. Most claims that historians draw from surviving material are not claims about how many items once existed. They are claims about composition: what share of a literature was written in a given language or form, what proportion of a playwright’s output belonged to a given genre, how many of the charters issued in a county concerned lay persons, how the share of women among testators changed between two decades. For claims of this kind the number of lost items is only half of what matters. The other half is whether loss was selective with respect to the trait in question, and here the evidence is sobering. Milne (2025), in the first systematic history of manuscript destruction in England, observes that scholars agree that some sorts of manuscripts were disproportionately destroyed but do not agree on which sorts. The documented magnitudes are large. For Venetian editions with known print runs, the evidence assembled by Camps et al. (2025) from Trovato (2014) shows copies of a legal text printed on parchment surviving at about 27 per cent while copies of popular chivalric literature survive at about 0.7 per cent, a difference of a factor of almost forty. The same authors report studies of late medieval chanceries in which originals of imperial and royal acts survive at 10 to 20 per cent while the judgments of the Parlement of Louis X of France survive at about 1 per cent. That smaller incunabula survive less well than larger ones is a recurrent finding (Green et al. 2011; Milne 2025). Camps et al. (2025) accordingly list the representativeness of what survives among the questions that remain open.

This article treats representativeness as a problem of identification and not of estimation. The distinction, fundamental in econometrics, separates what could be learned from an unlimited supply of observations of the available kind from what sampling error obscures in a finite supply (Manski 2007). More survivors do not help if survivors are the wrong witnesses. Charles Manski showed that when data are missing through an unknown selection mechanism, the quantities of interest are typically confined to an interval that can be computed from the data and from whatever assumptions the analyst is prepared to defend (Manski 1989, 2003; Horowitz and Manski 1995; Tamer 2010). A related tradition in epidemiology and statistics asks how strong an unobserved bias would have to be in order to overturn a conclusion (Cornfield et al. 1959; Rosenbaum 2002), a question that Masten and Poirier (2020) formalise as a breakdown frontier. Economic historians have met the same difficulty in another guise: the heights of soldiers, prisoners and students are selected samples, and Bodenhorn et al. (2017) showed that a celebrated decline in stature during industrialisation appears chiefly in the samples most exposed to selection (see also Zimran 2019). Parametric corrections of the kind proposed by Heckman (1979) require a model of the selection mechanism and a variable that affects selection but not the outcome, and historical transmission rarely supplies either. The familiar parable of Abraham Wald and the returning aircraft makes the point in miniature: the damage visible on the survivors describes where an aircraft can be hit and still return (Mangel and Samaniego 1984).

The contribution of the article is a framework that turns Esch’s distinction into quantities that can be reported, together with five results, two empirical illustrations and a simulation. The framework rests on one accounting identity, which holds without any probabilistic assumption: the odds of a trait among survivors equal the odds of the trait in the original population multiplied by the relative survival rate of the items that carry it. Esch’s Chance is this relative survival rate; his Zufall is the sampling variation around it. The first result is that the composition of the survivors alone places no restriction whatever on the original composition. The second is that knowledge of the overall loss rate confines any original share to an interval whose width is exactly the loss rate, which I call the width of the silence. The third is that a defensible bound on relative survival, a factor Γ in the manner of Rosenbaum, yields closed-form intervals and a breakdown value for every compositional claim, that is, the smallest degree of selectivity that would overturn it. The fourth is constructive: when two contexts passed through the same selection regime, the odds ratio between them is identified exactly although neither level is, so that comparisons across time and space are more robust than the levels on which they are built. The fifth concerns the counting estimators themselves: the sum of unseen species estimates computed separately for any partition of the survivors is never smaller than the estimate computed on the pooled material, with equality only when the groups share the same survival chances.

The illustrations show that these results bite. For the corpus of Kestemont et al. (2022), estimating the six traditions separately implies that 61.9 and not 68.3 per cent of works survive, a difference of about 120 works, and moves the English share of the six-tradition corpus from 8.6 to 13.8 per cent. A birth-death simulation calibrated on the orders of magnitude of Camps et al. (2025) shows that loss amplifies selection: a disadvantage of 20 per cent in the rate of copying becomes a disadvantage of 50 per cent in the chance that a work survives at realistic depths of loss, and of more than 80 per cent when loss is very deep. The two transmissions of Euripides, one through a school selection and one through the accident of an alphabetical volume, provide a natural experiment in which the original composition is approximately known from the organisation of the dramatic festivals, and imply a relative survival of satyr plays between one third and one sixth of that of tragedies.

The framework does not recover what was lost, and it does not replace the interpretive work that scholars of archival silence have shown to be necessary (Hartman 2008; Fuentes 2016). It measures the region that inference from survivors cannot enter, and it makes the assumptions under which that region shrinks explicit enough to be argued about in the ordinary terms of source criticism. Section 2 sets out the framework, the transmission model, the estimators and the data. Section 3 presents the results. Section 4 discusses how a bound on selectivity can be defended, proposes a reporting convention, relates the argument to the literature on archives and power, and states the limitations. Section 5 concludes.

2. Materials and Methods

2.1. Transmission as a Selection Process

Consider a historical population of N units. The units may be works of literature, editions, charters, letters, wills, transactions, persons or events; all that is required is that the historian can say what would count as a member. Each unit either carries a binary trait or does not. The trait may be a language, a genre, a format, the sex or status of a person, the outcome of a transaction, or any other property that can be observed on the units that survive. A unit survives if it is available to the historian today, in whatever sense is relevant: a work survives if at least one witness of its text is extant, a transaction survives if a record of it was made and has been kept.

Two filters therefore separate the population from the evidence. The first is the making of records, which Esch stressed and which is itself selective: events that cost money, caused disputes or provoked repression were written down more often than those that did not. The second is the survival of what was written, which depends on material, format, custody, use and later esteem. For texts that circulated in copies the second filter has a demographic structure, since copying multiplies witnesses and loss removes them, so that the number of extant witnesses of a work is the outcome of a process of births and deaths (Weitzman 1987; Cisne 2005; Camps et al. 2025). The framework below treats the two filters together. It asks only what fraction of the units with the trait, and what fraction of the units without it, have reached the historian. The demographic model of Section 2.4 is used to show how differences in copying and loss translate into differences in these fractions; none of the identification results depends on it.

2.2. Notation and the Accounting Identity

Let N₁ units carry the trait and N₀ = N − N₁ units lack it, and write π = N₁/N for the original share of the trait. This is the quantity of interest. Of the units with the trait, n₁ survive; of those without it, n₀ survive; the historian observes n = n₁ + n₀ survivors and the share of the trait among them, p = n₁/n. Define the realised survival rates σ₁ = n₁/N₁ and σ₀ = n₀/N₀, the overall survival rate σ = n/N, the loss rate 1 − σ, and the relative survival rate ρ = σ₁/σ₀. By construction,

$$p={\displaystyle \frac{\pi \hspace{0.17em}{\sigma }_1}{\sigma }}, \sigma =\pi \hspace{0.17em}{\sigma }_1+\left(1-\pi \right)\hspace{0.17em}{\sigma }_0,$$

(1)

and dividing the expression for p by the corresponding expression for 1 − p gives the identity on which the article rests:

$${\displaystyle \frac{p}{1-p}}=\rho \hspace{0.17em}{\displaystyle \frac{\pi }{1-\pi }}.$$

(2)

The odds of the trait among survivors equal its odds in the original population multiplied by the relative survival rate. Equations (1) and (2) are accounting identities. They involve no probability model and hold whatever the heterogeneity of survival chances within each group, because only the group averages enter. In Esch’s terms, ρ is the Überlieferungs-Chance of units that carry the trait relative to those that do not, and the difference between the realised ρ and its expectation under any particular loss process is the Überlieferungs-Zufall. Survival is blind with respect to the trait when ρ = 1, and only then does the survivor share equal the original share.

2.3. Information Regimes, Identified Sets and Breakdown Values

The identified set for π is the set of values of the original share that are compatible with what the historian observes and with the assumptions that the historian maintains. It is the object that would remain uncertain even if sampling error were absent. I consider four regimes of information. In the first, only the survivors are available. In the second, the overall survival rate σ is also known, from an inventory, a register, a catalogue of lost works or a counting estimator. In the third, the historian is prepared to defend a bound on selectivity of the form 1/Γ ≤ ρ ≤ Γ for some Γ ≥ 1, so that items with the trait were at most Γ times more likely, and at most Γ times less likely, to survive than items without it. The parameter plays the role that the bound on the odds of treatment plays in Rosenbaum’s sensitivity analysis of observational studies (Rosenbaum 2002), with the difference that it bounds a ratio of survival rates, which is the quantity about which historians of transmission have evidence. Where the direction of selection is known, the bound becomes one-sided, in the spirit of the monotonicity restrictions of Manski and Pepper (2000). In the fourth regime two contexts A and B, such as two periods documented by the same archive, are compared, and the historian bounds the difference between their selection regimes, Λ⁻¹ ≤ ρ_B/ρ_A ≤ Λ, without bounding either regime itself.

For a claim of the form “the original share of the trait was below c”, made when the survivor share is below c, the breakdown value Γ* is the smallest Γ for which the identified set contains values that contradict the claim. A claim with a breakdown value of 1.2 is overturned by a difference of one fifth in survival chances; a claim with a breakdown value of 20 survives any selectivity short of a twentyfold difference. The breakdown value translates a compositional claim into a statement about transmission that can be assessed with the ordinary instruments of source criticism.

2.4. A Birth-Death Model of Copying and Loss

To show how selectivity arises and how it interacts with the depth of loss, I use the linear birth-death process that Weitzman (1987), Cisne (2005) and Camps et al. (2025) apply to manuscript traditions. Each work begins as a single copy. During an active phase of length T₁ every extant copy is copied at rate λ and destroyed at rate μ. During a subsequent passive phase copying has ceased and each copy that reached the end of the active phase survives to the present independently with probability s. At the end of the active phase the lineage of a work is extinct with probability α, and the number of copies of a work that is not extinct is geometric with parameter β (Kendall 1948), where

$$\alpha ={\displaystyle \frac{\mu \hspace{0.17em}\left(E-1\right)}{\mathit{\lambda E}-\mu }}, \beta ={\displaystyle \frac{\lambda \hspace{0.17em}\left(E-1\right)}{\mathit{\lambda E}-\mu }}, E=e^{\left(\lambda -\mu \right)T_1}.$$

(3)

Thinning the geometric distribution by s gives the probability that a work survives to the present (Appendix B):

$$\sigma \left(\lambda ,\mu ,T_1,s\right)={\displaystyle \frac{\left(1-\alpha \right)\hspace{0.17em}s}{1-\beta \hspace{0.17em}\left(1-s\right)}}.$$

(4)

Works with the trait are copied at rate κλ, with 0 < κ ≤ 1, and are otherwise treated identically, so that 1 − κ is their copying disadvantage and the relative survival rate is ρ = σ(κλ, μ, T₁, s)/σ(λ, μ, T₁, s). I take the orders of magnitude from the region of best fit reported by Camps et al. (2025) for Old French chivalric narrative, expressed per century: λ = 3.4, μ = 1.52, an active phase of two and a half centuries and a passive phase of the same length during which copies are destroyed at the same rate, so that s = exp(−1.52 × 2.5) = 0.022. Large-population values are computed from Equations (3) and (4). Sampling variation is assessed by Monte Carlo simulation of populations of 1000 works, of which 30 per cent carry the trait, with 2000 replications for each value of κ.

2.5. Counting Estimators

When units survive in countable witnesses, the numbers of units represented by exactly one and exactly two witnesses, f₁ and f₂, carry information about the number of units represented by none. The estimator of Chao (1984), in the bias-corrected form used by Kestemont et al. (2022) and implemented in their copia package, is

$${\widehat{f}}_0={\displaystyle \frac{n_w-1}{n_w}}\hspace{0.17em}{\displaystyle \frac{f_1^2}{2f_2}}, \widehat{N}=S+{\widehat{f}}_0, \widehat{\sigma }={\displaystyle \frac{S}{\widehat{N}}},$$

(5)

where S is the number of surviving units and n_w the number of surviving witnesses. When the numbers of witnesses per unit follow Poisson distributions with unit-specific means, the estimator is a lower bound on the number of lost units in expectation, and the bound is approached only when all units share the same mean (Chao 1984; Chao and Jost 2012). The survival rate that it implies is therefore an upper bound. I compute the estimator for the pooled material, as Kestemont et al. (2022) do, and separately for groups defined by an observed trait; in the latter case the ratio of the group-specific survival rates is an estimate of ρ and the ratio of the group-specific totals an estimate of the original composition. Confidence intervals follow the bootstrap of the copia package, in which the unseen units are added to the resampled assemblage with their estimated relative abundances, with 10,000 replications. For the difference between the group-wise and the pooled estimates I use a coherent version in which the groups are resampled jointly and the pooled estimator is recomputed on the union of the same resamples.

2.6. Data

The empirical illustration uses the open data that accompany Kestemont et al. (2022), released under a Creative Commons Attribution-ShareAlike 4.0 licence (Kestemont et al. 2021). For each of six vernacular traditions of medieval heroic and chivalric narrative (Dutch, English, French, German, Icelandic and Irish) the files list every surviving document together with the work that it transmits. Counting documents per work reproduces exactly the numbers of works, documents, singletons and doubletons published by the authors (Table 2). The Icelandic file labels each work by genre, which permits a second illustration within one tradition: prose sagas and the narrative verse cycles known as rímur, each on chivalric matter (riddarasögur, riddararímur) or legendary matter (fornaldarsögur, fornaldarrímur). Both forms continued to be copied by hand in Iceland long after the Reformation (Driscoll 1997). The evidence on the transmission of Euripides is taken from the survey by Mastronarde (2017), supplemented by the standard accounts of Zuntz (1965) and Reynolds and Wilson (2013).

2.7. Software

All computations were carried out in Python 3.12 with NumPy, pandas, Matplotlib and copia 0.1.4 (Kestemont and Karsdorp 2021). The scripts that load the data, compute the estimates, run the simulations and draw the figures are provided as Supplementary Material.

3. Results

3.1. Survivors Alone Identify No Level

Proposition 1 (Non-identification from survivors). 

Suppose that only the survivors are observed and that 0 < p < 1. Then for every value π′ strictly between 0 and 1 there is a relative survival rate ρ > 0 under which an original share π′ produces exactly the observed survivor share p. The identified set for π is the open interval (0, 1). If p = 0, the identified set is the interval [0, 1).

The proof is immediate from Equation (2) and is given, with the other proofs, in Appendix A. Its content is that the composition of the survivors is whatever it is for any original composition whatsoever, provided that selection was strong enough, and the survivors themselves say nothing about how strong it was. If 30 per cent of the wills that survive from a jurisdiction were made by women, the original share may have been 10 per cent, if women’s wills were about 3.9 times more likely to be preserved than men’s, or 60 per cent, if they were 3.5 times less likely. No increase in the number of surviving wills discriminates between these possibilities, because a larger n reduces the sampling error around p and leaves ρ untouched. The proposition gives formal expression to the historian’s habitual caution. It also shows that the caution, consistently applied, would forbid every compositional statement, which is not how historians proceed. They proceed by bringing knowledge of transmission to bear, and the remaining results describe what different kinds of such knowledge deliver.

Remark 1 (Total silence). 

The case p = 0 deserves separate notice. A category that has left no survivors may have been empty, or it may have been large and entirely lost. For such a category no finite bound on selectivity helps: under 1/Γ ≤ ρ ≤ Γ with finite Γ, a survivor share of zero implies an original share of zero. Every bound of the kind used below presupposes that each category under discussion has left at least some survivors, and the framework is silent, as the evidence is, about categories that have left none.

3.2. The Width of the Silence

Proposition 2 (The width of the silence). 

Suppose that the survivor share p and the overall survival rate σ are known and that nothing is assumed about ρ. Then the identified set for π is the interval

p σ ≤ π ≤ p σ + (1 − σ).

(6)

The bounds are sharp, the interval always contains p, and its width equals the loss rate 1 − σ, whatever the values of p and n.

The lower bound is reached when every lost unit lacked the trait and the upper bound when every lost unit carried it; nothing in the observed material excludes either extreme. The width of the interval is a property of the transmission and not of the sample. For the works of medieval narrative studied by Kestemont et al. (2022), with a survival rate of between 0.62 and 0.68 (Section 3.7), any original share is known to within an interval of width 0.32 to 0.38. For the documents that carried those works, of which about 9 per cent survive, the width is 0.91. For the plays of Euripides, of which roughly one in five survives (Section 3.8), it is 0.79. A historian who knows that nine in ten items have perished, and nothing about how they perished, knows almost nothing about the composition of what once existed. Since survival rates obtained from counting estimators are upper bounds (Section 2.5), widths computed from them are lower bounds.

The result extends to any number of categories. If the survivors fall into K categories with shares collected in the vector p, the original shares satisfy

π = σ p + (1 − σ) q,  q ∈ Δ_K−1,

(7)

where q, the unknown composition of the lost units, may be any point of the simplex of K shares. The identified set is a copy of the whole simplex, shrunk by the factor 1 − σ and attached to the point σp.

Corollary 1 (Attested losses). 

Suppose that of the L = N − n lost units, m are attested well enough to be classified, by a title, a catalogue entry, a quotation or a fragment, and that m₁ of them carry the trait. Then

$${\displaystyle \frac{n_1+m_1}{N}}\hspace{0.33em}\le \hspace{0.33em}\pi \hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{n_1+m_1}{N}}+{\displaystyle \frac{L-m}{N}},$$

(8)

and the width of the identified set is the share of unattested losses in the original population.

Every attested loss narrows the silence. The patient collection of testimonies to lost works, of which Wilson (1952) is the classic instance for medieval England, and the study of booklists, inventories and advertisements as witnesses to lost editions (Bruni and Pettegree 2016) thus have a precise inferential value: they convert unattested losses, about which nothing can be said, into attested ones, which enter the numerator. Table 1 summarises the identified sets under the successive regimes of information.

3.3. Bounded Selectivity and Breakdown Values

Proposition 3 (Bounded selectivity). 

Suppose that 1/Γ ≤ ρ ≤ Γ for some Γ ≥ 1. Then the identified set for π is the interval

$${\displaystyle \frac{p}{p+\Gamma \hspace{0.17em}\left(1-p\right)}}\hspace{0.33em}\le \hspace{0.33em}\pi \hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{\Gamma \hspace{0.17em}p}{\Gamma \hspace{0.17em}p+\left(1-p\right)}}.$$

(9)

If σ is also known, the identified set is the intersection of this interval with that of Proposition 2. Both sets are sharp. If it is known in addition that units with the trait survived no better than the others (ρ ≤ 1), the lower bound becomes p; if they survived no worse (ρ ≥ 1), the upper bound becomes p.

Figure 1a shows the identified sets for a survivor share of 0.30. When the survival rate alone is known, the set is the grey wedge of Proposition 2, which closes only as survival approaches completeness. A bound on selectivity cuts a horizontal band through the wedge: for Γ = 1.5 the original share lies between 0.22 and 0.39, for Γ = 2 between 0.18 and 0.46, and for Γ = 3 between 0.13 and 0.56. At the low survival rates typical of pre-modern material the band is far narrower than the wedge, so that almost all of the information comes from what the historian is able to assert about selectivity, and almost none from the loss rate. The width of the band is greatest for survivor shares near one half, where it equals (Γ − 1)/(Γ + 1), and it shrinks towards zero as p approaches zero or one.

For a claim that the original share lay on the same side of a threshold c as the survivor share, the breakdown value follows from Equation (9):

$${\Gamma }^{*}=\exp \left(\left|\mathrm{logit}\left(p\right)-\mathrm{logit}\left(c\right)\right|\right), \mathrm{logit}\left(x\right)=\ln {\displaystyle \frac{x}{1-x}}.$$

(10)

Figure 1b plots the breakdown value against the survivor share for three thresholds. A claim that a trait found in 30 per cent of survivors characterised a minority of the original population breaks down at Γ* = 2.33: it fails if items with the trait were less than 43 per cent as likely to survive as the others. Given the magnitudes documented in the Introduction, a breakdown value below two marks a claim as fragile, and the shaded band of the figure shows how much of the range of survivor shares produces fragile claims about any given threshold.

3.4. Comparisons Survive Selection

Proposition 4 (Comparisons). 

Let two contexts A and B have survivor shares p_A and p_B strictly between 0 and 1, original shares π_A and π_B, and relative survival rates ρ_A and ρ_B. Write OR_π for the ratio of the odds of the trait in B to its odds in A in the original populations, and OR_p for the same ratio among survivors. Then

$$\ln {\mathrm{OR}}_{\pi }=\ln {\mathrm{OR}}_p-\ln {\displaystyle \frac{{\rho }_B}{{\rho }_A}}.$$

(11)

If ρ_A = ρ_B, the odds ratio is identified exactly, whatever the common value of ρ and whatever the two loss rates, and π_B − π_A has the sign of p_B − p_A. If 1/Λ ≤ ρ_B/ρ_A ≤ Λ, the identified set for OR_π is the interval from OR_p/Λ to Λ OR_p, and the direction of the difference between the two contexts is identified if and only if |ln OR_p| > ln Λ.

The logic is that of the case-control study, in which sampling on the outcome distorts the composition of both groups and leaves their odds ratio intact (Cornfield 1951). Selective survival distorts the level of a share in each context, possibly severely, but when the distortion is common to the two contexts it cancels from their comparison. For a sequence of periods that passed through one selection regime, Equation (11) implies that the whole trajectory of the log-odds of the trait is identified up to a constant vertical shift: trends are identified where levels are not. The practical question is when a common regime is plausible. It is most plausible when the contexts compared are documented by the same series in the same custody, such as successive decades of one court’s registers, since whatever favoured one kind of entry over another in later centuries is then likely to have acted on all decades alike. It is least plausible when the contexts differ in medium, institution or later history. Overall loss may differ arbitrarily between the contexts, as it typically does when older material has suffered more; what must be shared is relative survival. Section 3.6 shows that this is a real restriction, because the depth of loss itself affects relative survival, and gives a sense of the values of Λ that differences in depth require.

3.5. Counting Under Selectivity

Proposition 5 (Disaggregation). 

Let the surviving units be partitioned into G groups, with f_1g > 0 units represented by one witness and f_2g > 0 units represented by two witnesses in group g. Then

$${\displaystyle \sum _{g=1}^G}{\displaystyle \frac{f_{1g}^2}{2f_{2g}}}\hspace{0.33em}\ge \hspace{0.33em}{\displaystyle \frac{{\left({\displaystyle {\sum }_{g=1}^G}{f}_{1g}\right)}^2}{2{\displaystyle {\sum }_{g=1}^G}{f}_{2g}}},$$

(12)

with equality if and only if the ratio f_1g/f_2g is the same in every group.

The left-hand side is the number of lost units obtained by applying the estimator of Chao (1984) to each group and adding the results; the right-hand side is the number obtained from the pooled material. The inequality is a form of the Cauchy-Schwarz inequality and holds for every partition. Under Poisson sightings with mean λ_g in group g, the expected ratio of singletons to doubletons is 2/λ_g, so that equality corresponds to equal mean numbers of witnesses, and hence equal survival chances, across groups. The difference between the two sides is therefore a non-parametric index of selectivity between the groups. Two consequences follow. Since each side is a lower bound on the number of lost units, the larger is the more informative, and pooled estimates understate loss whenever survival differs between groups that the historian is able to distinguish. And since any further partition can only raise the estimate, every dimension of heterogeneity that can be named, whether language, genre, format or region, increases the estimated loss. The bias-corrected estimator of Equation (5) differs from the classic form by factors close to one, and the inequality holds for it in every application reported below.

3.6. Simulated Transmission: Loss Amplifies Selection

Figure 2 reports the behaviour of the birth-death model of Section 2.4. In panel (a) the true share of the trait is 30 per cent. With no copying disadvantage the survivors reproduce it. With a disadvantage of 20 per cent in the rate of copying (κ = 0.8), works with the trait are half as likely to survive as the others (ρ = 0.50) and make up 17.7 per cent of the survivors. The group-wise counting correction of Section 2.5 moves the estimate in the right direction, to 22.7 per cent in large populations (mean of 23.1 per cent over the Monte Carlo replications, of which 95 per cent lie between 17.4 and 30.5 per cent), but it closes less than half of the gap. The reason is that survival chances also differ within each group. In a birth-death world between 44 and 55 per cent of lineages become extinct during the copying phase itself, and these early deaths leave no trace in the numbers of witnesses of the works that survive. The same mechanism makes the counting estimators loose as bounds on the overall loss: at κ = 0.8 the true loss of works is 61.3 per cent, while the pooled and the group-wise estimators return 14.8 and 15.6 per cent in large populations. Camps et al. (2025) report the same discrepancy for their simulated traditions. The group-wise estimate exceeds the pooled one throughout, as Proposition 5 requires. An interval with Γ = 2 just contains the truth, since its upper bound is 30.1 per cent; no point estimate based on the survivors does.

Panel (b) shows the central result of the simulation. For a given copying disadvantage, the relative survival rate depends strongly on the depth of loss. With κ = 0.8, works with the trait are 82 per cent as likely to survive as the others when no copy is lost after the copying phase, 50 per cent as likely at the calibrated depth of loss, at which about one copy in forty-five survives, and 18 per cent as likely in the limit of very deep loss, for which Appendix B gives

$$\underset{s\to 0}{\lim }\rho =e^{-\left(1-\kappa \right)\lambda T_1}.$$

(13)

Loss amplifies selection. Copying compounds, so that a modest difference in the rate of copying becomes a large difference in the number of copies that exist when copying ceases (on average 110 against 20 in the example), and when loss is deep the chance that a work survives at all is nearly proportional to the number of its copies. Relative survival rates of one half to one sixth, such as those inferred for Euripides in Section 3.8, therefore do not require dramatic differences in contemporary esteem. A disadvantage of 20 to 30 per cent in the rate of copying, sustained over two centuries and followed by heavy loss, is enough.

Panel (c) qualifies Proposition 4 accordingly. If two contexts share the copying disadvantage but differ in the depth of loss, their relative survival rates differ and the odds ratio computed from survivors is in error by ln(ρ_B/ρ_A). The error vanishes when the depths are equal and grows smoothly with their ratio. With κ = 0.8, a context whose copies survived at half the rate of the other requires Λ = 1.27 and one whose copies survived at a quarter of the rate requires Λ = 1.62. In a worked example with true shares of 20 and 40 per cent, the survivors show 11.2 and 25.1 per cent when the depths are equal, so that both levels are badly wrong while their odds ratio is exactly the true value of 2.67. When the second context has lost twice and four times as deeply, the survivor odds ratios are 2.10 and 1.64. The direction of the difference is preserved in every case, and its magnitude is understated.

3.7. Six Medieval Literatures

Table 2 reproduces the counts of Kestemont et al. (2022) and adds the ratio of singletons to doubletons, which varies from 1.28 in the Irish material to 5.25 in the English. By Proposition 5 this variation guarantees that the six tradition-specific estimates of lost works sum to more than the pooled estimate, and the difference is substantial: 493.8 against 371.6 lost works with the classic estimator and 492.3 against 371.5 with the bias-corrected one. The pooled analysis, which is the basis of the published figure, implies an original corpus of 1170 works and a survival rate of 68.3 per cent (95 per cent interval 63.0 to 73.5). Summing the six traditions implies 1291 works (1169 to 1457) and a survival rate of 61.9 per cent (54.8 to 68.4). The difference of 121 works has a coherent bootstrap interval of 68 to 224 and was positive in all 10,000 replications. Both figures are lower bounds on the original number of works; the group-wise figure is the tighter of the two, and further disaggregation within traditions can only raise it.

Table 2. Surviving works and documents of medieval heroic and chivalric narrative in six vernaculars, with estimates of the number of lost works.

Tradition	Works S	Documents nw	f1	f2	f1/f2	Lost works	Original works (95% interval)	Survival rate
Dutch	75	167	45	13	3.46	77.4	152.4 (109.6–224.9)	0.492
English	69	176	42	8	5.25	109.6	178.6 (121.0–295.4)	0.386
French	222	1473	90	21	4.29	192.7	414.7 (346.5–508.8)	0.535
German	128	1088	36	19	1.89	34.1	162.1 (139.9–194.9)	0.790
Icelandic	117	295	44	28	1.57	34.5	151.5 (128.7–183.6)	0.773
Irish	188	449	69	54	1.28	44.0	232.0 (205.3–267.0)	0.810
Pooled material	799	3648	326	143	2.28	371.5	1170.5 (1087.4–1267.4)	0.683
Sum of the six traditions	799	3648				492.3	1291.3 (1168.5–1456.9)	0.619

f₁ and f₂ are the numbers of works transmitted by exactly one and exactly two documents. Lost works, original works and survival rates follow Equation (5); intervals are based on 10,000 bootstrap replications. The row for the pooled material reproduces the estimate of Kestemont et al. (2022); the last row adds the six tradition-specific estimates (Proposition 5). Data: Kestemont et al. (2021).

Table 2. Surviving works and documents of medieval heroic and chivalric narrative in six vernaculars, with estimates of the number of lost works.

Tradition	Works S	Documents nw	f1	f2	f1/f2	Lost works	Original works (95% interval)	Survival rate
Dutch	75	167	45	13	3.46	77.4	152.4 (109.6–224.9)	0.492
English	69	176	42	8	5.25	109.6	178.6 (121.0–295.4)	0.386
French	222	1473	90	21	4.29	192.7	414.7 (346.5–508.8)	0.535
German	128	1088	36	19	1.89	34.1	162.1 (139.9–194.9)	0.790
Icelandic	117	295	44	28	1.57	34.5	151.5 (128.7–183.6)	0.773
Irish	188	449	69	54	1.28	44.0	232.0 (205.3–267.0)	0.810
Pooled material	799	3648	326	143	2.28	371.5	1170.5 (1087.4–1267.4)	0.683
Sum of the six traditions	799	3648				492.3	1291.3 (1168.5–1456.9)	0.619

f₁ and f₂ are the numbers of works transmitted by exactly one and exactly two documents. Lost works, original works and survival rates follow Equation (5); intervals are based on 10,000 bootstrap replications. The row for the pooled material reproduces the estimate of Kestemont et al. (2022); the last row adds the six tradition-specific estimates (Proposition 5). Data: Kestemont et al. (2021).

The same estimates imply a correction to the composition of the corpus (Table 3 and Figure 3). English works make up 8.6 per cent of the surviving works and an estimated 13.8 per cent (9.7 to 21.2) of the original corpus; Irish works fall from 23.5 to 18.0 per cent (15.4 to 20.9). Taken together, Dutch, English and French account for 45.8 per cent of the surviving works and 57.7 per cent of the estimated original corpus, while the Icelandic and Irish traditions, which Kestemont et al. (2022) single out for their high survival, fall from 38.2 to 29.7 per cent. The relative survival rate of each tradition against the other five ranges from 0.59 for English (0.35 to 0.88) to 1.40 for Irish (1.17 to 1.69), and the largest pairwise contrast, between English and Irish, is a factor of 2.1. A bound of Γ = 2 on the relative survival of each tradition against the rest contains all six corrected shares; a bound of Γ = 1.5 does not, since it would confine the English share below 12.4 per cent. The sets that follow from the survival rate alone are, as Proposition 2 requires, 38 percentage points wide for every tradition.

Two cautions apply. The corrected shares are point estimates only under the assumption that survival chances are homogeneous within each tradition, and the simulation of Section 3.6 shows that corrections of this kind fall short when they are not. And the population to which the shares refer, the narrative works of these six vernaculars, is an artefact of the design of the original study and is used here only to illustrate the method.

Table 3. The composition of the six-tradition corpus: shares among surviving works, corrected shares, relative survival rates and identified sets.

Tradition	Share of surviving works p	Corrected share (95% interval)	Relative survival (95% interval)	Set from the loss rate alone	Set for Γ = 2
Dutch	0.094	0.118 (0.086–0.167)	0.77 (0.52–1.10)	0.058–0.439	0.049–0.172
English	0.086	0.138 (0.097–0.212)	0.59 (0.35–0.88)	0.053–0.435	0.045–0.159
French	0.278	0.321 (0.272–0.375)	0.81 (0.64–1.03)	0.172–0.553	0.161–0.435
German	0.160	0.126 (0.105–0.150)	1.33 (1.08–1.63)	0.099–0.480	0.087–0.276
Icelandic	0.146	0.117 (0.097–0.142)	1.29 (1.04–1.60)	0.091–0.472	0.079–0.255
Irish	0.235	0.180 (0.154–0.209)	1.40 (1.17–1.69)	0.146–0.527	0.133–0.381

Corrected shares are the ratios of the tradition-specific estimates of original works in Table 2 to their sum. Relative survival is the estimated survival rate of a tradition divided by that of the other five taken together. The set from the loss rate alone applies Proposition 2 with σ = 0.619; the set for Γ = 2 applies Proposition 3 to each tradition against the rest.

Table 3. The composition of the six-tradition corpus: shares among surviving works, corrected shares, relative survival rates and identified sets.

Tradition	Share of surviving works p	Corrected share (95% interval)	Relative survival (95% interval)	Set from the loss rate alone	Set for Γ = 2
Dutch	0.094	0.118 (0.086–0.167)	0.77 (0.52–1.10)	0.058–0.439	0.049–0.172
English	0.086	0.138 (0.097–0.212)	0.59 (0.35–0.88)	0.053–0.435	0.045–0.159
French	0.278	0.321 (0.272–0.375)	0.81 (0.64–1.03)	0.172–0.553	0.161–0.435
German	0.160	0.126 (0.105–0.150)	1.33 (1.08–1.63)	0.099–0.480	0.087–0.276
Icelandic	0.146	0.117 (0.097–0.142)	1.29 (1.04–1.60)	0.091–0.472	0.079–0.255
Irish	0.235	0.180 (0.154–0.209)	1.40 (1.17–1.69)	0.146–0.527	0.133–0.381

Corrected shares are the ratios of the tradition-specific estimates of original works in Table 2 to their sum. Relative survival is the estimated survival rate of a tradition divided by that of the other five taken together. The set from the loss rate alone applies Proposition 2 with σ = 0.619; the set for Γ = 2 applies Proposition 3 to each tradition against the rest.

The Icelandic material permits an illustration within a single tradition (Table 4). The verse cycles make up 34.2 per cent of the 117 surviving works. Their estimated survival rate is 66.9 per cent against 83.3 per cent for the prose sagas, a relative survival rate of 0.80, and their corrected share is 39.2 per cent (29.5 to 52.1). The bootstrap interval for the relative survival rate, 0.43 to 1.24, includes one, so that the data do not establish that the verse cycles survived less well. The claim that they were a minority of the Icelandic works on these matters has a breakdown value of 1.93 (Figure 1b). The point estimate of selectivity, a factor of 1.24, leaves the claim standing; the lower end of the interval, a factor of 2.3, would overturn it. Sampling uncertainty thus compounds the uncertainty of identification, and a complete report would combine them (Imbens and Manski 2004). Works on legendary matter make up 40.2 per cent of the survivors, with a relative survival rate of 0.89 (0.52 to 1.32) and a breakdown value of 1.49 for the claim that they were a minority. Proposition 5 can be observed at small scale: the classic estimate of lost Icelandic works is 34.6 from the pooled material, 34.9 when the works are divided by matter, 35.6 when they are divided by form, and 36.7 when all four genres are distinguished.

3.8. A Natural Experiment: The Two Transmissions of Euripides

Natural experiments in transmission are rare, because they require that some part of a population should have survived for reasons unrelated to its content. The plays of Euripides offer one. Ancient sources credit him with roughly ninety plays. Nineteen survive under his name, and about seventy never reached the medieval manuscript tradition (Mastronarde 2017). The nineteen reached us by two routes. Ten are the select plays, which by late antiquity dominated the school curriculum and the book trade; Mastronarde attributes the selection to a combination of popularity in performance and in the schools, usefulness for rhetorical training, mythographic interest and the availability of commentaries. The other nine have titles beginning with the Greek letters epsilon, eta, iota and kappa. They descend from part of a collected edition arranged alphabetically, perhaps the contents of two cases of five rolls (Snell 1935), which was copied into a codex and survived to be transcribed around 1300 in the circle of Demetrius Triclinius into the manuscript L, from which the only other medieval witness of these plays was copied (Zuntz 1965; Mastronarde 2017). Esch’s two categories thus stand side by side in the transmission of one author. The select plays survived by Chance, because they were the kind of play that was read, taught and recopied; the alphabetic plays survived by Zufall, because their titles happened to stand next to one another.

The strength of the selection can be gauged from the surviving fragments of late antique codices: of the 29 or 30 codices of Euripides published so far, 26 or 27 contain select plays and only three contain any of the other titles, nearly eighty in number, although readers in Ptolemaic and early Roman Egypt had still had access to about three dozen titles (Mastronarde 2017). Per title, the disparity in copying is of the order of seventy to one. The transition from the roll to the codex, which favoured works gathered into collections and exposed the rest, did the remainder (Reynolds and Wilson 2013; Mastronarde 2017). Bagnall (2002) has made the general point that ancient literature perished less through spectacular destruction than through the failure to recopy it.

What the accident preserved shows what the selection removed. The alphabetic plays include the Cyclops, the only satyr play of any author to survive complete; among the ten select plays there is none, as there is none among the seven plays each of Aeschylus and Sophocles, which descend from comparable selections. Satyr plays are therefore 1 in 19 of the surviving plays of Euripides, a share of 5.3 per cent. With a survival rate of about 0.21, Proposition 2 confines the original share to the interval from 1.1 to 80 per cent, which is uninformative, as it must be. In this case, however, the original composition is approximately known from outside the transmission. Tragedians competed at the City Dionysia with three tragedies and a satyr play, and the loss of satyr plays began early: the satyr play that accompanied the Medea in 431 BCE was already unavailable to the Alexandrian scholars (Mastronarde 2017). Since Euripides is known to have put a play without satyrs, the Alcestis, in the fourth place on at least one occasion, a deliberately wide bracket for the satyric share of his output is 15 to 25 per cent. Equation (2) then implies a relative survival rate of satyr plays between 0.17 and 0.31, that is, a value of Γ between about 3 and 6.

The example shows the framework working in both directions. The claim that satyr plays were a minority of the output has a breakdown value of 18 and is secure. The claim that they were less than a tenth of it, which is what the survivors suggest, has a breakdown value of 2.0 and is false. And had only the select plays survived, as is the case for Aeschylus and Sophocles, the survivor share would be zero and no finite bound on selectivity could have revealed that the genre existed (Remark 1); only external testimony could. Table 5 sets these magnitudes beside the others encountered in this article.

4. Discussion

4.1. From Source Criticism to a Number

The parameter Γ is not a nuisance to be estimated away. It is a compact statement of what the historian knows about transmission, and the results of Section 3 show that at pre-modern rates of loss it carries almost all of the information that can be had about original composition. The question is how a value can be defended, and the illustrations suggest four kinds of evidence. The first is internal: where units survive in countable witnesses, the numbers of witnesses per unit reveal differences in survival chances between groups that the historian can observe, as they do between the six medieval traditions, and Proposition 5 turns those differences into a tighter bound on loss. This evidence is confined to material transmitted in copies, and it is blind to differences within groups. The second is the natural experiment, in which part of a population survived for reasons unrelated to content, as the alphabetic plays of Euripides did. The third is the external register: print runs, chancery registers, booklists and inventories record what once existed independently of what survives, and the studies gathered by Bruni and Pettegree (2016) and by Camps et al. (2025) show how much can be extracted from them. The fourth is a model of the process, which shows, as in Section 3.6, how plausible differences in copying and handling translate into relative survival at a given depth of loss.

Table 5 indicates the range within which such arguments move. Between broadly comparable bodies of vernacular narrative, relative survival differs by factors of up to about two. Between a genre that a school canon taught and one that it neglected, it differs by factors of three to six. Between media, formats and documentary genres, it differs by factors of ten to forty. These magnitudes are large in relation to the breakdown values of ordinary claims: by Equation (10), any statement that a category found in between one third and two thirds of survivors formed a majority, or a minority, of the original population breaks down at a Γ of two or less.

The direction of selection matters as much as its size, and it varies with the medium. In a manuscript culture esteem multiplies copies and therefore protects works, so that what was most read is over-represented among survivors. In print the relation can be reversed: heavily used books were read to pieces, the regularity that Egghe and Proot (2007) call the book historical law holds that the chance of losing a copy rises with the size of the edition, and the Venetian evidence in Table 5 sets a legal text on parchment against popular romance. In documentary archives legal value and institutional continuity protect, which is why the records of ecclesiastical lordship dominate the early medieval evidence and why monastic communities were able to reshape their documentary past by selective copying and discarding (Esch 1985; Geary 1994; Clanchy 2013). Milne (2025) adds a temporal regularity, which she calls age without vintage: the manuscripts most at risk were those too old to be current and too recent to be antique. Where the direction of selection can be argued, the one-sided sets of Proposition 3 remove one side of the interval, which on the scale of log-odds is exactly half of it. Where it cannot, as in the dispute that Milne describes over which sorts of English manuscripts were disproportionately destroyed, the two-sided set is the honest report.

4.2. A Reporting Convention

The results suggest a convention for compositional statements based on surviving material, which can be stated briefly. The population and its units should be defined, since every quantity above is relative to them. The survivor share should be reported with its sampling interval, as it usually is. The regime of information should then be stated: whether the loss rate is known, from what source, and whether it is a point value or, as with counting estimators, a bound. Where a loss rate is available, the width of the silence should be reported, because it tells the reader at once how much of the conclusion will have to come from assumptions about selectivity. Each headline claim should be accompanied by its breakdown value, together with the reasons, drawn from the history of the material, for thinking that selectivity of that strength and direction did or did not occur. Comparisons should be preferred to levels wherever the contexts compared passed through one regime of transmission, and the argument for a common regime, or the value of Λ allowed for departures from it, should be given. Finally, counts of lost units should be computed for the finest partition that the material supports. None of this requires more than arithmetic, and all of it is more informative than the customary sentence that warns of the accidents of survival and is then set aside.

The convention applies beyond texts. The identity of Equation (2) holds for any population known through records, and Esch’s first filter, the making of records, enters through the same relative rate as the second. If transactions that ended in litigation were recorded and preserved ten times as often as those that did not, a survivor share of 50 per cent disputed transactions corresponds to an original share of 9 per cent. The historian of credit who infers the prevalence of default from surviving instruments, the historian of crime who infers the composition of offences from surviving indictments and the economic historian who infers a wealth distribution from surviving inventories face the same identity, with different evidence about ρ.

4.3. Silence, Measured and Unmeasured

A large literature has shown that archives are instruments of power as well as deposits of evidence, that what they hold reflects the purposes of those who made and kept them, and that their silences are structured (Trouillot 1995; Steedman 2001; Schwartz and Cook 2002; Stoler 2009; Farge 2013). The framework of this article formalises one part of that insight and marks the boundary of another. It formalises the claim that survival is differential, by giving differential survival a parameter and showing what follows from any stated value of it. It marks a boundary in Remark 1. Persons and practices that left no records of their own, and that appear in the archive only through the records of those who held power over them, belong to categories with no survivors in the relevant sense, and no finite Γ reaches them. Hartman (2008) and Fuentes (2016) have shown what responsible work at that boundary requires, and nothing in an interval replaces it. What the interval can do is guard against the opposite error, which is to read the absence of evidence as evidence of absence. An identified set that runs from 1 to 80 per cent is a statement in its own right: it says that the survivors do not decide the matter. Perreault (2019) has made the corresponding argument for archaeology, where the quality of the record, and not the ingenuity of the analyst, sets the limit to the questions that can be answered.

4.4. Limitations and Extensions

Several limitations should be stated. The trait has been treated as binary and as observed without error, whereas fragments and testimonies often permit only a probable classification; misclassification widens the sets in ways that can be computed but have not been computed here. The units have been taken as given, although deciding when two versions of a story are one work or two is a substantial philological judgment, for which Camps et al. (2025) set out explicit criteria. Survival events are not independent when several works travel in one codex, which leaves the identities intact but makes sampling intervals too narrow. The parameter Γ bounds an average relative survival rate over the whole history of transmission and is silent about when and how selection operated. The loss rate is itself only bounded when it comes from a counting estimator, and the simulation of Section 3.6 shows that the bound can be loose. The empirical illustration rests on a corpus whose boundaries were drawn for another purpose. And sampling uncertainty has been reported separately from the uncertainty of identification, whereas confidence intervals for partially identified parameters would combine the two (Imbens and Manski 2004).

Each limitation indicates an extension. Shares generalise to means and distributions of bounded quantities by the methods of Horowitz and Manski (1995), which would cover the wealth recorded in surviving inventories or the size of surviving holdings. The framework is an instance of what the statistical literature on missing data calls a pattern-mixture model, in which the distribution of the unobserved is left free and indexed by sensitivity parameters (Little and Rubin 2019), and the tools of that literature apply. Most promising is the combination of the identification results with dynamic models of transmission. Camps et al. (2025) note that their birth-death model with a single pair of rates does not reproduce the heavy-tailed distribution of witnesses per work and call for models that include selection. Dated witnesses carry information about the rate of copying of different kinds of work at different times, and a model with trait-specific rates, estimated from such data, would replace the bound Γ by an estimate of ρ whose uncertainty could be propagated through Equation (2).

5. Conclusions

Historians have always known that what survives is not a fair sample of what existed, and the recent application of unseen species models to cultural material has given them estimates of how much is gone. This article has argued that the more consequential question for most historical inference is what kind of thing is gone, and that the question has a precise structure. The odds of a trait among survivors are its original odds multiplied by a relative survival rate. From this identity it follows that survivors alone identify no original share; that a known loss rate confines a share to an interval exactly as wide as the loss; that a defensible bound on relative survival yields usable intervals and a breakdown value for every claim; that comparisons within one regime of transmission are identified when levels are not; and that disaggregated counts of the lost always exceed pooled ones.

The illustrations show that the structure has consequences for published results and familiar cases. Estimating six medieval literatures separately lowers the estimated survival of their works from 68.3 to 61.9 per cent and raises the English share of the corpus by more than half. A birth-death model shows that loss amplifies selection, so that a disadvantage of one fifth in the rate of copying halves the chance of survival at realistic depths of loss. The double transmission of Euripides implies that a school canon reduced the survival of an entire dramatic genre to between a third and a sixth of that of tragedy, and that a second accident of the same kind was needed for the genre to be known from a complete play at all.

The practical proposal is modest. A compositional claim based on surviving sources should be accompanied by the width of the silence, where a loss rate is known, and by the breakdown value of the claim, together with the historian’s reasons for believing that selection of that strength did or did not occur. Esch asked historians to keep the chance and the accident of transmission in mind. The quantities proposed here allow them to be kept on the page.