Silent coverage failures in rare-event searches and a degeneracy index that predicts them

Davide Pagno

REVIEW 2 major objections 4 minor 33 references

Silent coverage failures in rare-event searches and a degeneracy index that predicts them

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Two numbers predict silent coverage failures in rare-event searches.

desk verdict A genuinely useful screening diagnostic for coverage risk under model misspecification, with a real boundary-identifiability gap that needs a convention; deserves a serious referee, not a desk reject. read the letter →

arxiv 2608.09203 v1 pith:ZSIGOGNS submitted 2026-08-10 hep-ex stat.ME

Davide Pagno This is my paper · ORCID

classification hep-exstat.ME

keywords confidenceintervalscoveragemodelmisspecificationPoisson–Fisherdegeneracyindexrare-eventsearchesdarkmatterneutrinolessdouble-betadecaygoodness-of-fit

topics Dark Matter

open problems Dark Matter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that rare-event search intervals can silently miss their nominal frequentist coverage: the interval fails while the accompanying goodness-of-fit test stays at its nominal false-positive rate. It introduces the Poisson–Fisher degeneracy index, a pair $(\beta,\gamma)$ computed from expected bin counts and local model derivatives, and argues that the sign of $\beta$ identifies which endpoint of the confidence interval is threatened, while $\gamma$ orders how detectable the model error is by a saturated-Poisson goodness-of-fit test at fixed 5\% type-I error. The claim is tested on controlled model deformations in an exact Poisson counting experiment, a dark-matter recoil spectrum, and a neutrinoless-double-$\beta$-decay peak search, where positive signal-like bias degrades discovery-side coverage and negative signal bias from overestimated efficiency makes the upper endpoint undercover. If the claim is right, analyses can triage candidate model deformations before toy calibration, promoting large-$|\beta|$, small-$\gamma$ deformations to constrained nuisances with auxiliary information. The paper states that Eq. (8) is only a local reference, not a finite-sample coverage calibration.

What carries the argument

The load-bearing object is Eq. (7), the Poisson–Fisher degeneracy index. With the Poisson–Fisher inner product $\langle u,v\rangle_W=\sum_i u_i v_i/\nu_{0,i}$ and the tangent matrix $T$ whose columns are the local derivatives $\partial\nu/\partial\vartheta_a$, the fitted displacement is $\Delta\hat\vartheta=F^{-1}T^T W\,\delta\nu$ with $F=T^T W T$; then $\beta=\Delta\hat\mu/\sqrt{(F^{-1})_{\mu\mu}}$ and $\gamma^2=(\delta\nu-T\Delta\hat\vartheta)^T W(\delta\nu-T\Delta\hat\vartheta)$. The paper connects $\beta$ to endpoint coverage through the local Gaussian reference $c_L\simeq\Phi(z_{1-\alpha/2}-\beta)$ and $c_U\simeq\Phi(z_{1-\alpha/2}+\beta)$, and uses $\gamma$ to order the rejection probability of the saturated-Poisson deviance test. A constrained-nuisance extension adds auxiliary Fisher information $F_{\mathrm{aux}}$ to the total information and a tension term $\Delta\hat\vartheta^T F_{\mathrm{aux}}\Delta\hat\vartheta$ to the residual. Projection onto the full fitted tangent space is what makes the diagnostic nuisance-aware: a deformation parallel to a background or mass derivative is absorbed there, changing both the predicted damage and the residual available to a goodness-of-fit test.

What would settle it

A decisive test is a single-bin Poisson experiment with fixed background and an exactly signal-collinear positive excess scanned over amplitude: the index predicts $\beta>0$, $\gamma=0$, monotone decrease of lower-endpoint coverage in $\beta$, and goodness-of-fit rejection at the 5\% floor for every amplitude. If toy coverage at $\mu=0$ is non-monotone in $\beta$, or if the goodness-of-fit rejection probability rises above the floor for any of these deformations, the sign and detectability ordering is falsified; the paper's own reported miss at $\beta=4.2$ (predicted 0.005, measured 0.20) already shows that the Gaussian reference can fail without falsifying the qualitative ordering.

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Extended reading notes

Core claim

The central discovery is the Poisson–Fisher degeneracy index $\mathcal{I}_{\mathrm{PF}}(\delta\nu;\vartheta_0)=(\beta,\gamma)$, a two-coordinate screening diagnostic for a candidate model deformation. Given a deformation $\delta\nu$ of the nominal expected-count vector, one projects it onto the complete fitted tangent space using the local Poisson–Fisher metric $W=\mathrm{diag}(1/\nu_{0,i})$; $\beta$ is the resulting signed shift in the parameter of interest in profiled standard-error units, and $\gamma$ is the Poisson–Fisher norm of the residual outside all fitted directions. The paper shows empirically that the sign of $\beta$ tells which interval endpoint loses coverage—positive $\beta$ threatens the lower (discovery) endpoint, negative $\beta$ threatens the upper (exclusion) endpoint—while $\gamma$ tracks the power of a saturated-Poisson deviance test. Across the counting, dark-matter, and $0\nu\beta\beta$ settings, a deformation with large $|\beta|$ and small $\gamma$ is the dangerous case: the interval can move while the fitted spectrum looks fine. The paper also shows that calibration under the nominal simulator does not protect against misspecification of that simulator, and that modelling an exactly collinear deformation as a constrained nuisance restores coverage at evaluated points, at a sensitivity cost quantified by a factor up to 1.6.

Load-bearing premise

The diagnostic assumes that a first-order local projection computed at a reference point remains informative for coverage behavior at the boundary $\mu=0$ and for nonlocal deformations, where profile-likelihood asymptotics are nonstandard and parameters such as the WIMP mass are unidentified.

Editorial extensions

If this is right

Robustness studies should report lower-endpoint, upper-endpoint, and whole-interval coverage separately; whole-interval coverage at $\mu=0$ tests only the lower endpoint and cannot validate a pure upper limit.
Positive signal-like contamination degrades discovery-side coverage while making upper limits conservative, whereas negative signal bias from overestimated efficiency can make the upper endpoint undercover, so both deformation signs should be tested.
Calibration under the nominal simulator, including a simulation-based method's internal coverage check, does not detect misspecification of that simulator relative to the data-generating process; a separate data-based model check is needed.
A plausible deformation with large $|\beta|$ and small $\gamma$ should be promoted to a nuisance constrained by auxiliary data or a declared envelope; in the exactly collinear benchmark this restores coverage at evaluated grid points at up to a factor-1.6 sensitivity cost.
The index can serve as pre-toy triage, but Eq. (8) is only a local reference; final coverage statements still require toy calibration, especially at boundaries and for nonlocal deformations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension is to compute the same projection with an observed-information metric for unbinned or learned likelihoods, giving $\beta$ and $\gamma$ for simulation-based inference pipelines without toy calibration.
Because $\gamma$ orders the power of a residual check, it could guide test construction: binning and region-of-interest choices that increase $\gamma$ for a candidate deformation family should also increase the sensitivity of a goodness-of-fit test.
The large-$|\beta|$, small-$\gamma$ corner is effectively an identifiability statement: near collinearity, no finite data-driven interval exists without external information, so the practical boundary is set by the strength of auxiliary constraints rather than by sample size.
One testable extension is to turn the index into a calibration target, choosing deformation severities that reach a specified $\beta$ and using toy coverage to set analysis-specific 'large' and 'small' cutoffs, which the paper leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this paper gives the field a cheap pre-toy triage tool for coverage risk under model misspecification, and it validates the qualitative claims carefully. The main caveats are the missing code, the acknowledged quantitative failure of Eq. (8) at large deformation, and a boundary identifiability issue that makes the dark-matter validation at mu=0 convention-dependent.

The new construction is I_PF = (beta, gamma): project a specified deformation onto the full fitted tangent space, take the signed standardized shift of the signal parameter, and pair it with the orthogonal Poisson–Fisher residual. That combination is not in the cited literature, and it is practically motivated. The paper deserves credit for the endpoint-separated stress-test protocol and for demonstrating the directionality: positive signal-like contamination threatens the lower endpoint, negative signal bias can make upper limits anti-conservative, and the constrained-nuisance remedy restores coverage at the evaluated points at a measurable sensitivity cost.

The paper is also honest. It states explicitly that Eq. (8) is a local reference, not a finite-sample calibration, and reports the beta=4.2 case where the formula gives 0.005 but the measured coverage is 0.20. Its own limitations section lists the restricted dimensionality and the grid-points-only nuisance calibration. That honesty makes the read easier.

The soft spots are real but not disqualifying. No code or data are shipped, which undercuts the "screening tool" pitch. The quantitative prediction is only reliable in the small-deformation regime, and the paper's own validation says so. The stress-test concern about the boundary is legitimate: at mu=0 in the dark-matter spectral analysis, the WIMP mass column of the tangent matrix is zero, F is singular, and the text's "best-fitting branch" convention is not well defined because all mass branches give identical expected counts under the null. The boundary beta values in Fig. 9 left therefore depend on an arbitrary branch choice unless the paper specifies a rule. The counting and 0nu-beta-beta examples are unaffected because they have no unidentified shape parameter, so the qualitative screening claim likely survives once the convention is fixed.

Who is this for? Experimental statisticians in rare-event searches who need to decide whether a candidate deformation can be ignored or must be propagated into a nuisance. This deserves peer review; the right outcome is a major revision asking for code and data, an explicit branch-selection convention at non-identifiable boundaries, and a more modest presentation of the quantitative coverage-prediction claims. I would not desk-reject.

Referee Report

2 major / 4 minor

Summary. The paper studies how controlled model misspecification changes the frequentist coverage of intervals for a non-negative signal strength in low-count searches, using an exact Poisson counting benchmark, a dark-matter recoil spectrum, and a 0νββ peak search. It introduces the Poisson–Fisher degeneracy index I_PF = (β, γ): after projecting a candidate expected-count deformation onto the full fitted tangent space, β is the signed fitted signal shift in profiled standard-error units and γ is the Poisson–Fisher norm of the residual. The authors claim that locally the sign of β identifies which interval endpoint is threatened and that γ orders the detectability of the deformation by a saturated-Poisson goodness-of-fit test. They validate the qualitative predictions with toy Monte Carlo, show that calibration under the nominal simulator does not protect against misspecification, and demonstrate in an exactly collinear benchmark that promoting the deformation to a constrained nuisance restores coverage at the evaluated grid points at a quantifiable sensitivity cost.

Significance. If the screening diagnostic works as claimed, it would give experimental analyses a cheap, principled way to triage candidate model deformations before expensive toy calibration: large-|β|, small-γ directions are the dangerous ones and should be promoted to constrained nuisances. The paper's strengths include an explicit and simple formula (Eq. 7) for the index; a transparent controlled-severity protocol with stated sample sizes and binomial errors; exact Poisson summation in the counting benchmark; independent toy validation of the direction and rough severity of endpoint failures; and unusually honest reporting of the limitations of Eq. (8), including the large-β failure. The central qualitative claims—signed endpoint vulnerability and the importance of the full fitted tangent space—are supported by the clean counting and 0νββ examples as well as by the interior spectral tests. The main unresolved issue is the definition of the index at the boundary in settings with an unidentified secondary shape parameter, which affects part of the validation.

major comments (2)

[Section 4, Eq. (7); Section 7, Fig. 9] The definition of β (and γ) at μ=0 for the spectral dark-matter analysis is not well-posed. At μ=0 the WIMP mass mχ is unidentified: the mχ column of the tangent matrix T vanishes, so F is singular, and the statement in Section 4 that "the projection is evaluated separately in each candidate local tangent space and the best-fitting branch is used" does not select a unique branch, because under the null all mχ branches give identical expected counts. The DM-tail points in Fig. 9 left, and the corresponding residual in Fig. 9 right, are therefore convention-dependent unless a branch-selection rule is specified (for example, a fixed reference mass, or the branch that maximizes |β|). This matters because the screening workflow is most valuable precisely at μ=0 and because the DM-tail point is used as boundary validation. Please specify the convention and recompute the affected points; if no unique convention is intended, state that boundary β is defined only up to branch and restrict the empirical boundary validation to settings with identified secondary parameters (counting and 0νββ).
[Section 7, first paragraph; Eq. (8)] The paper honestly reports the failure of Eq. (8) at β=4.2 (predicted coverage ≈0.005, measured 0.20), but the assessment plot in Fig. 9 left still draws the Gaussian reference curve through the full range of plotted points. This curve is not a predictive relation in the large-deformation regime, and its presence could visually overstate the quantitative support for the index. Since the paper's central claim is that β is a screening coordinate rather than a finite-sample calibration, please either remove the dashed curve from the coverage-vs-β plot, restrict it to a clearly marked local regime, or add an explicit caption note that the curve is not a calibration and is known to fail at large β. The qualitative ordering claim is acceptable if framed in this way.

minor comments (4)

[Section 2.1 and Fig. 1] The spectral model description would benefit from explicitly listing which parameters are fitted (μ, mχ and, if any, background scale or shape parameters), so that the "complete fitted tangent space" in Eq. (7) is unambiguous.
[Reference [1]] Reference [1] contains the typo "tonne−Years" and the misspelling "lux-zeplin"; these should be corrected to "tonne-years" and "LUX-ZEPLIN".
[Section 6.3, Fig. 3] The legend entry "CLs / Bayesian upper limit: upper edge" may confuse readers because both constructions are pure upper limits whose lower-endpoint coverage is trivially 1 at all severities; please clarify this in the caption.
[Section 7, right panel] The sentence "the right-panel tail and line residuals use the same reference points" is ambiguous: please specify that the DM-tail residual is evaluated at the boundary branch used for β, and the line residuals at the Q-value, to avoid confusion with the interior halo point.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Poisson–Fisher index is an independent diagnostic validated against toy-calibrated coverage; its collinear γ=0 property is a mathematical consequence, not a fitted prediction.

full rationale

The central claim is that the sign of β and the size of γ, computed from Eq. (7) using only nominal model derivatives and a specified deformation, predict endpoint coverage and goodness-of-fit detectability. These predictions are validated against independent toy-calibrated coverage and saturated-Poisson deviance rejection probabilities in Figures 3, 6, and 9. No parameter is fitted to the validation data; Eq. (8) is explicitly labeled a local reference, and the paper reports a large quantitative miss at β=4.2 (predicted coverage 0.005, measured 0.20), which would be impossible if the validation were forced by construction. The observation that a deformation exactly collinear with the signal template has γ=0 is a direct mathematical consequence of γ being defined as the residual after projection; the goodness-of-fit test has no sensitivity to that component because the fit absorbs it. This is a property the index is designed to quantify, not a circular renaming. The paper contains no load-bearing self-citations: all cited statistical and experimental works are external and used as background or as benchmarks. A genuine limitation, honestly stated, is that at μ=0 the WIMP mass mχ is unidentified, making the mχ column of the tangent matrix zero and F singular; the paper's solution of evaluating each candidate local tangent space and using the 'best-fitting branch' does not uniquely define β at the boundary, since all branches give identical expected counts under the null. This is a definitional/correctness concern about boundary identifiability, not circularity, because the coverage measurements are still independent of the index definition. The paper accordingly advises toy calibration at boundaries and does not present Eq. (8) as a finite-sample guarantee. Overall, the derivation chain is self-contained and the predictions are externally tested rather than reduced to their inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central diagnostic has no fitted constants; its empirical support depends on hand-chosen stress-test deformations and on regularity assumptions that are stated but not proven beyond the local approximation.

free parameters (3)

Stress-test severity endpoints (s=1) for each deformation
Hand-chosen endpoint amplitudes (e.g., up to six background events, halo v0 220 to 420 km/s, 50% or 80% efficiency loss, threshold shift to 9 keV) define the controlled departure scale; these are disclosed as stress tests, not calibrated uncertainty envelopes (Section 3).
Nuisance calibration grid and finite-calibration guard = η in [-0.20, 0.20], five points, 90.5% quantile
The five-point grid and the 0.5 percentage-point guard are analyst choices that affect the coverage-restoration results in Section 6.4.
Goodness-of-fit type-I error = 5% marginal prior-predictive
Standard convention; the paper notes this is marginal over the prior, not uniform point-wise control.

assumptions (5)

domain assumption Data follow a Poisson likelihood n ~ Poisson(κμ + b)
Used throughout, Eq. (1); exact for the toy experiments by construction.
domain assumption Misspecification is representable as an additive deformation of expected counts δν
Eq. (5); excludes non-additive or stochastic-model departures.
domain assumption Local identifiability and positive-definite Fisher information F at the reference point
Needed for Eq. (7); the paper notes the degenerate limit when profiled information on μ vanishes.
ad hoc to paper Local Gaussian approximation relating β to endpoint coverage (Eq. 8)
Introduced specifically to connect β to cL and cU; the paper labels it a local reference and shows it fails at large β.
ad hoc to paper Saturated Poisson deviance is approximately quadratic with noncentrality γ²
Used to link γ to GoF power; the paper's GoF test uses the deviance (Eq. 10), not the exact quadratic form.

invented entities (1)

Poisson-Fisher degeneracy index I_PF = (β, γ) independent evidence
purpose: Statistical diagnostic to screen endpoint coverage risk and GoF detectability before toy calibration
Statistical construct, not a physical entity; the paper validates it against independent toy-calibrated coverage frequencies not used to fit the index.

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Cite this review

Pith. "Pith review of Silent coverage failures in rare-event searches and a degeneracy index that predicts them." pith.science (2026). https://pith.science/paper/ZSIGOGNS

@misc{pith2026260809203,
  author       = {Pith},
  title        = {Pith review of: Silent coverage failures in rare-event searches and a degeneracy index that predicts them},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSIGOGNS}},
  note         = {Machine review of arXiv:2608.09203}
}

abstract

Searches for new physics in low-background experiments infer a non-negative signal strength from few events and often report an upper limit. Nominal frequentist coverage requires both a valid interval construction and an adequate data model. We study how controlled model departures affect lower- and upper-endpoint coverage for six interval procedures in an exact Poisson counting experiment, dark-matter recoil spectra, and a neutrinoless-double-beta-decay peak search. We introduce the Poisson--Fisher degeneracy index $\mathcal I_{\mathrm{PF}}(\delta\nu;\vartheta_0)=(\beta,\gamma)$, which maps a specified expected-count deformation, after projection onto the complete fitted tangent space, to $\beta$, the signed fitted signal shift in profiled standard-error units, and $\gamma$, the Poisson--Fisher norm of the unabsorbed residual. Locally, the sign of $\beta$ identifies the threatened endpoint, while larger $\gamma$ implies greater detectability by the saturated-Poisson goodness-of-fit test used here at a fixed $5\%$ type-I error rate. Across the studied deformations, positive signal-like bias degrades discovery-side coverage while making upper limits conservative; negative signal bias from overestimated signal efficiency can make the upper endpoint undercover. Calibration under the nominal simulator does not protect against misspecification of that simulator relative to the data-generating process. A plausible deformation with large $|\beta|$ and small $\gamma$ may therefore evade diagnosis and should be represented by a nuisance constrained with auxiliary information or included in a defensible envelope. In the exactly collinear constrained-nuisance benchmark, modelling the deformation restores coverage at the evaluated grid points, at a quantifiable cost in interval sensitivity.

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Reviewed August 11, 2026 · model on record in the stance chip above.

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