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Modelling Under-Reported Data: Pitfalls of Na\"ive Approaches and a New Statistical Framework for Epidemic Curve Reconstruction

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper proves that ignoring under-reporting in Poisson autoregressions biases reproduction numbers and offers a normal-normal approximation to recover true counts.

desk verdict Useful new results on the pitfalls of ignoring under-reporting in Poisson autoregressions, but the key approximation claim rests on an unvalidated dependence assumption and a few loose ends. read the letter →

arxiv 2509.10668 v1 pith:L7QVK3ZF submitted 2025-09-12 stat.AP

Justin J. Slater , Sindi Bebeziqi This is my paper · ORCID
classification stat.AP
keywords under-reportingepidemiccurvereconstructionPoissonautoregressionbinomialthinninglatentGaussiantransformationnormal-normalapproximationstate-spacemodelsBayesianinference
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

The paper establishes that fitting Poisson autoregressions to under-reported case counts systematically biases the reproduction number downward and often biases the exogenous component upward, and quantifies exactly when each bias occurs. To correct this, it introduces a Bayesian framework that replaces the binomially thinned Poisson autoregression with a normal-normal approximation of matching conditional moments, then maps the continuous latent states back to integers via the inverse-CDF transform X_t = F^{-1}(Φ(Z*_t)). The authors argue that this yields nearly the same posterior distribution for parameters and epidemic curve reconstructions as the exact thinned model, while being fast enough for standard gradient-based Bayesian software. If correct, this gives practitioners a general, software-friendly way to estimate hidden incidence while preserving the mechanistic interpretation of thinned autoregressions.

What carries the argument

The central mechanism is the latent Gaussian transformation X_t = F^{-1}_{X_t}(Φ(Z*_t)), applied to standardized latent variables Z*_t from a normal-normal approximation to a thinned count autoregression. The approximation replaces the binomial thinning Y_t | X_t ~ Bin(X_t, π) and the Poisson autoregression X_t | X_{t-1} ~ Pois(λ_t) with normal distributions that match conditional means and variances, yielding a continuous latent process Z_t that is easy to sample with gradient-based Bayesian software. The inverse-CDF transform then maps each posterior draw of Z*_t back to integer counts, and the paper asserts that the joint distribution of these transformed integers is close to the posterio

What would settle it

Run a long simulation (T ≥ 200) from a binomially thinned Poisson autoregression with small mean counts (e.g., about 3–5) and strong autocorrelation (ϕ ≈ 0.8). If the approximate-model credible intervals for the true epidemic curve have coverage far below nominal, or the lag-1 autocorrelation of Z*_t deviates materially from that of the true latent process, the central claim fails.

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Extended reading notes

Core claim

The paper's central claim is that the posterior distribution of the parameters and latent true counts from a binomially thinned Poisson autoregression can be accurately approximated by fitting a normal-normal approximation—replacing each Poisson with a normal of matching conditional mean and variance, and each binomial thinning with a normal of matching conditional mean and variance—and then transforming each posterior draw of the standardized latent variable Z*_t via X_t = F^{-1}_{X_t}(Φ(Z*_t)). This transformation is key: it converts the continuous approximate latent process into integer counts with the correct marginal distributions while preserving the serial dependence structure, so the

Load-bearing premise

The method's central premise is that the standardized latent variable from the normal approximation carries essentially the same serial dependence as the latent Gaussian that would generate the true integer counts, an assumption the paper checks only in simulations with moderately large counts (means 6.7–40, series length 50) and explicitly concedes breaks down for very small counts.

Editorial extensions

If this is right

  • Epidemic curve reconstructions can be computed with standard gradient-based Bayesian software instead of custom particle filters or MCMC samplers over discrete latent states, enabling faster and more scalable analyses.
  • Published reproduction numbers from Poisson autoregressions fit to under-reported counts are likely to be systematically underestimated unless under-reporting is explicitly modeled.
  • The framework extends to multivariate spatiotemporal settings with covariates and additional data sources, as demonstrated by the England conurbation analysis that integrated random PCR testing data.
  • The method is designed for series with reasonably large counts; the paper notes the approximation degrades when the mean count falls below about 5.
  • Moment-based estimators for the thinned model are too variable for practical use, supporting the need for full posterior inference.

Reading between the lines

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

  • If the normal-normal approximation with the latent Gaussian transform is as accurate as claimed for larger counts, it could serve as a fast surrogate for inference in other latent count models, such as ecology n-mixture or capture-recapture models, where discrete latent states create computational bottlenecks.
  • The theoretical result that under-reporting biases ϕ downward implies that apparent declines in reproduction numbers over time may be confounded with improvements in reporting coverage; a testable implication is that regions with better reporting should show higher estimated ϕ from naive fits.
  • The paper validates the approximation only for moderately large counts and a single series length; a formal error bound on the Gaussian approximation to the autocorrelation of Z*_t would strengthen the generalizability of the claim.
  • The authors' framework naturally supports time-varying reporting probabilities driven by covariates, as they demonstrate for Covid-19; validating reconstructions against independent serosurveys, as they do with REACT data, could become a standard model-checking step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies count-valued autoregressions with under-reporting. Section 2 derives moment-based results characterizing how ignoring binomial thinning biases estimates of the autoregressive parameter φ and the exogenous component ν, with two propositions on when ν is overestimated. Section 3 proposes an approximate Bayesian framework: replace a binomially thinned Poisson autoregression with a normal-normal model, standardize the latent process as Z*, and reconstruct integer counts via the latent Gaussian transformation X_t = F^{-1}_{X_t}(Φ(Z*_t)). The approximation is evaluated in a simulation study and applied to rotavirus data in Saarland and COVID-19 data in England. The paper argues that the approach retains the mechanistic appeal of thinned autoregressions while substantially simplifying inference.

Significance. If the central approximation is valid, this is a practically useful contribution: it offers a software-friendly route to epidemic curve reconstruction in multivariate, covariate-adjusted settings where exact MCMC is infeasible. The Section 2 results on the direction of bias from ignoring under-reporting are also of independent interest and extend earlier moment-matching work. Strengths include the reproducible analysis scripts, the two real-data applications, and the explicit connection to the latent Gaussian count time series construction of Jia et al. (2023). However, the paper's central claim that the approximate posterior is 'virtually identical' to the exact posterior is currently supported mainly by pointwise comparisons, and several implementation details of the approximate model are underspecified.

major comments (4)
  1. [Section 2 / Appendix A.1] Proposition 1 states a result for 'any consistent estimator' of ν in the misspecified model, but the proof only uses the moment equation \hatν = (1-\hatϕ)\tilde μ. Under misspecification, different estimators (e.g., MLE vs. method of moments) converge to different pseudo-true values, and the Poisson AR MLE need not satisfy this moment equation asymptotically. The claim is therefore overbroad. Please restrict the proposition to estimators based on the moment equations, or prove that all consistent estimators share the same pseudo-true value. The same issue propagates to Proposition 2.
  2. [Eq. (9) and Eqs. (11)-(12)] The normal-normal model uses variances sqrt(π(1−π)Z_t) and sqrt(λ_t), and model (12) uses a binomial probability of sum_{k=0}^{13} Z_{i,d-k}/pop_i. Since Z_t is Gaussian, it is negative with positive probability, making these expressions undefined or out of range. No positivity or truncation constraint is stated. This is not cosmetic: the approximate posterior is not well defined for negative Z. Please specify how the implementation handles negative Z (truncation, abs(), softplus, or similar) or reparameterize to guarantee positivity.
  3. [Section 3.3 and Section 4] The central claim Z*_{1:t} ≈ Z^X_{1:t} is validated only by pointwise agreement. Table 1's 'perfect match rate' and Figure 4 compare posterior medians and credible intervals time point by time point; they do not compare the joint serial dependence of the reconstructed X_t (e.g., ACF, partial ACF, or joint distributions of (X_t, X_{t-1})). Since epidemic curve reconstruction is about a coherent trajectory, a distortion of the lag structure would bias reconstructed epidemics even if marginal quantiles line up. Please add diagnostics for joint/autocorrelation structure.
  4. [Section 3.3 and Section 5.1] The reconstruction algorithm is not fully specified. The paper recommends computing X_t = F^{-1}_{X_t}(Φ(Z*_t)) with Z* the standardized innovations, but the rotavirus application writes X_t = F^{-1}_{X_t}(Φ(Z_t)). Also, F_{X_t} should be the conditional CDF of X_t given the relevant past and λ_t; this dependence is not stated. Please clarify the exact mapping used in the code and in each application.
minor comments (4)
  1. [Abstract] The sentence 'maps accurately maps this continuous process back to the integers' contains a duplicated phrase.
  2. [Section 1] Typo: 'thehhhmodel' should be 'the hhh model'.
  3. [Section 3.2] The sentence about 'attenuate the correlation between X and π' should presumably read 'between Z and π'.
  4. [Section 4] Minor typos: 'from from' in the Figure 3 caption, and 'mispecified' in Section 5.1. Also, the text says simulations with Rhat>1.05 were removed, but Table 1 says Rhat<1.01 was required; please make the filtering criterion precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the approximation is validated against the true model, and the latent transform is attributed to external work.

full rationale

The derivation chain is not circular. Section 2 derives the under-reporting consequences from the moment equations of Bracher and Held (2021), an independent external source, and the paper's Propositions 1 and 2 are algebraic consequences proven in Appendix A; no self-citation is used to establish them. The approximate model (9) is explicitly constructed by matching the first two conditional moments of the binomially thinned Poisson autoregression, and the paper does not claim this moment-matching is a prediction. The latent Gaussian transformation X_t = F^{-1}_{X_t}(Phi(Z*_t)) is explicitly borrowed from Jia et al. (2023), an external construction, and its marginal-matching property is a standard probability integral transform, not the paper's own derived output. The load-bearing approximation Z*_{1:t} approximately equal to Z^X_{1:t} is not established by definition; Section 4 tests it by comparing approximate-model reconstructions with MCMC samples from the true data-generating model across simulation scenarios, an external benchmark. No parameter is fitted to a subset and then reported as a prediction, no uniqueness theorem from the authors' own prior work is invoked, and no ansatz is hidden behind a citation. The only self-citation (Slater et al., 2025, Section 5.2.2) is an incidental remark about estimated serial intervals and is not load-bearing for the central claims. The reviewer concern that the simulation validation focuses on pointwise interval agreement rather than joint serial dependence is a genuine robustness/evidence limitation, but it is not circularity by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The framework does not introduce new physical or conceptual entities; it uses standard statistical latent variables (Z_t, X_t) and borrows the latent Gaussian transform from Jia et al. (2023). The main extra assumptions are the fidelity of the normal-normal approximation and the inverse-CDF mapping, both of which are empirically validated rather than derived.

free parameters (3)
  • reporting probability pi = 0.274 (rotavirus case study)
    Estimated from data in the framework; the central claim that pi can be inferred from the series' mean, variance, and autocorrelation depends on this parameter being identifiable and estimable.
  • autoregressive/transmission parameter phi = varies by model and time (e.g., seasonal in rotavirus)
    Central to the thinning model; the framework's posterior approximation is evaluated for phi in {0.4, 0.6, 0.8} in simulations.
  • exogenous mean nu = varies by model and time (e.g., seasonal in rotavirus)
    Estimated from data; the bias formulas in Section 2 explicitly describe how nu is mis-estimated when under-reporting is ignored.
assumptions (4)
  • domain assumption The true data-generating process is a binomially thinned Poisson autoregression with equations (1) and (2).
    All theoretical results in Section 2 and the simulation study in Section 4 rest on this model. It is a standard mechanistic assumption for infectious disease counts, but it is an assumption about the real-world process.
  • standard math The process is stationary with finite moments, requiring phi < 1 and suitable initial conditions.
    The moment equations (3)-(4) and Proposition 1/2 assume stationarity and finite moments of the count process. This is conventional in autoregressive time series.
  • ad hoc to paper The normal-normal approximation preserves the posterior distribution of the true model to a practically useful degree.
    This is the core assumption of the framework, stated in Section 3.2 and tested only by simulation in Section 4. No theoretical bound is given for the approximation error.
  • ad hoc to paper The latent Gaussian transformation X_t = F^{-1}(Phi(Z*_t)) from Jia et al. (2023) produces a joint posterior of X_t similar to the true integer-valued process.
    This assumption is introduced in Section 3.3 and justified empirically in Section 4, but only for the specific simulated settings. It is not a known theorem for this posterior reconstruction context.

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Cite this review

Pith. "Pith review of Modelling Under-Reported Data: Pitfalls of Na\"ive Approaches and a New Statistical Framework for Epidemic Curve Reconstruction." pith.science (2026). https://pith.science/paper/L7QVK3ZF

@misc{pith2026250910668,
  author       = {Pith},
  title        = {Pith review of: Modelling Under-Reported Data: Pitfalls of Na\"ive Approaches and a New Statistical Framework for Epidemic Curve Reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L7QVK3ZF}},
  note         = {Machine review of arXiv:2509.10668}
}
read the original abstract

Count-valued autoregressions are widely used to analyse time-series of reported infectious-disease cases because of their close connection with discrete-time transmission models. However, when such models are applied directly to under-reported case counts, their mechanistic interpretation can break down. We establish new theoretical results quantifying the consequences of ignoring under-reporting in these models. To address this issue, reported cases are often modelled as a binomially thinned version of an underlying count process, but such models are difficult to fit because the unobserved true counts are serially correlated and integer-valued. We develop a new statistical framework for under-reported infectious-disease data that uses a normal-normal approximation to a broad class of thinned count autoregressions and then accurately maps this continuous process back to the integers. Through simulations and applications to rotavirus incidence in a German state and Covid-19 incidence in English conurbations, we demonstrate that our approach both retains the mechanistic appeal of thinned autoregressions and substantially simplifies inference.

Figures

Figures reproduced from arXiv: 2509.10668 by the authors.

Figure 1
Figure 1. Consider observed data generated by a binomially thinned Poisson autoregression [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. 200 data points are simulated from a Poisson autoregression. The series is then [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. For each combination of ν = 10, π = 0.4, 0.6, 0.8, and ϕ = 0.4, 0.6, 0.8 (9 scenarios), 50 times series of length T=50 were simulated from from a binomially thinned Poisson autoregression with the respective parameters. The combination of parameters induces a specific mean of the time series which is noted in the top left corner. MCMC was used to compute the posterior of (10) (red), and the normal-normal approximati… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Examples of simulations with different perfect match rates between the recon [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: a) shows the reconstructed incident cases in Saarland, Germany, computed as the [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Incident cases from Pillar 2 of the United Kingdom’s Covid-19 surveillance program [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Reconstructed incidence and prevalence based on the model described by (11), [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Reconstructed prevalence in 9 English Conurbations based on the model repre [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Each plot corresponds to a simulation scenario with different [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]

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