--- title: "Finite-Sample Signal Uncertainty and Sharp Partial Identification of Diversification Histories" date: 2026-08-21 version: "0.3.0-candidate-r2" doi: 10.5281/zenodo.22041054 pdf: https://github.com/ipitchford/affine-diversification-fibres/releases/download/v0.3.0-candidate-r2/finite-sample-affine-diversification-0.3.0-candidate-r2.pdf repository: https://github.com/ipitchford/affine-diversification-fibres archive: https://zenodo.org/records/22041054 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Finite-Sample Signal Uncertainty and Sharp Partial Identification of Diversification Histories ## Plain-English summary A reconstructed family tree does not preserve a unique history of speciation and extinction. Many different histories can produce the same observable signal. This creates a practical risk: one fitted curve or a large finite cloud can look decisive even when other histories supported by the same tree reverse the conclusion. This candidate supplies a finite-sample safeguard against that false confidence. Under one exact fixed-stem, stem-survival, homogeneous time-varying model, it builds a confidence set for the entire pulled scale $F$ from the observed number of tips and unordered node ages. It then carries every signal in that set through the affine fibre and reports one of three conclusions for a registered target: 1. **certified incompatible** for every signal in the confidence set; 2. **compatible throughout the band**; or 3. **unresolved** because the decision changes across the band. This matters because it distinguishes conclusions that survive finite-sample uncertainty from conclusions that only look secure on one estimated signal. In the frozen 22-tip example, the method certifies one turnover cap as incompatible across the whole confidence-containing set. For another cap, it withdraws a plug-in "compatible" answer and reports **unresolved**. The method therefore changes the kind of scientific conclusion that the evidence permits, not merely the width of an error bar. The separate CRABS benchmark produced two results that must be read together: - every one of the 240 returned primary clouds missed at least one sharp endpoint beyond tolerance, with 60 additional structural censors; and - exact certification changed 532 of 3,840 registered decision statuses, or 13.854%, below the preregistered 20% H4 gate. The first result is a concrete warning: in this benchmark, a finite random cloud was not a certificate of the sharp limits. The second limits the breadth of the claim: the evidence does not show that exact certification is already a must-have or broadly useful complement to CRABS. Failing that ambitious utility gate does not erase the finite-sample method or the endpoint diagnostic; it states honestly how much decision impact the registered benchmark measured. > **Status:** anonymous, unrefereed theorem-and-method candidate published with > notes. Producer-side replay passes locally and on public Linux CI. No > unaffiliated reproduction, external specialist review, proof-assistant > formalisation, editorial peer review, empirical validation, or broad CRABS > utility is claimed. ## The statistical object Let $T$ be the fixed stem age. Write the pulled scale as $$ F(\tau)=\exp\!\left\{\int_0^\tau \lambda_p(s)\,\mathrm ds\right\}, \qquad F(0)=1. $$ Conditional on the stem lineage leaving sampled descendants, the tip count $N$ has a geometric law with parameter $$ p=\frac{1}{F(T)}. $$ Conditional on $N$, the $N-1$ unordered internal node ages have a known distribution whose cumulative probability is a monotone transformation of $F$. These two facts separate the finite-sample problem into two exact pieces: - invert the geometric count tails to obtain an interval for $F(T)$; and - use a Dvoretzky--Kiefer--Wolfowitz--Massart band for the transformed node-age distribution. The candidate combines the two pieces by Bonferroni propagation. Under the declared fixed-stem law, the resulting set contains the full pulled-scale trajectory with at least the stated nominal probability. This guarantee is conditional and model-specific. It is not a confidence set for an estimated topology, uncertain node dates, a smoothed derivative, a lineage-dependent process, or a misspecified biological model. ## From a signal band to an identified set For a fixed signal $F$, the compatible homogeneous histories are represented by a nondecreasing cumulative-loss coordinate $A$ below the survival barrier: $$ 0\le A(0)<1, \qquad A'\ge0, \qquad A - Louca, S., and Pennell, M. W. (2020). Extant timetrees are consistent with a myriad of diversification histories. Nature, 580, 502-505. - Höhna, S., Kopperud, B. T., and Magee, A. F. (2022). CRABS: Congruent rate analyses in birth-death scenarios. Methods in Ecology and Evolution, 13, 2709-2718. - Andréoletti, J., and Morlon, H. (2023). Exploring congruent diversification histories with flexibility and parsimony. Methods in Ecology and Evolution. - Massart, P. (1990). The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. The Annals of Probability, 18(3), 1269-1283. - Manski, C. F. (2003). Partial Identification of Probability Distributions. Springer.