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arxiv: 2606.00231 · v1 · pith:PHIDQWCPnew · submitted 2026-05-29 · 📊 stat.ME

On Asymptotic Outlier Rejection in Bayesian Mixed Poisson Regression Models Under Extreme Target and Covariate Values

Ilaria Pia , Jarno Vanhatalo This is my paper

Pith reviewed 2026-06-28 21:15 UTC · model grok-4.3

classification 📊 stat.ME
keywords Bayesian robustnessmixed Poisson regressionasymptotic outlier rejectioncovariate outlierscount dataGaussian latent variablesPoisson-GammaPoisson-RSB
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The pith

Mixed Poisson regression models with Gaussian latent variables are asymptotically robust to infinite target values but not to infinite covariate values.

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

The paper tests whether Bayesian mixed Poisson models retain their claimed asymptotic robustness when data points become extreme either in the response count or in the predictors. Prior results had established robustness to arbitrarily large counts under certain target distributions paired with Gaussian latent variables. The new analysis shows that the same models lose this protection when a covariate instead grows without bound. This breaks the symmetry that exists in linear regression, where an infinite residual can arise from either source. The distinction matters because many count datasets contain potential measurement errors or extreme predictors that are not captured by target-only robustness guarantees.

Core claim

We show that mixed Poisson models are not asymptotically robust to outliers resulting from infinite covariates. While models robust to data points with an anomalous target are not robust to data points with anomalous covariates, the three examined target distributions (Poisson-Gamma, Poisson-log-t, Poisson-RSB) all exhibit this failure. The lack of symmetry follows from the structure of the mixed Poisson likelihood under Gaussian latent variables, which does not map infinite covariates to the same posterior behavior as infinite targets.

What carries the argument

Asymptotic robustness criterion (posterior unaffected by observations sent to infinite distance) applied to mixed Poisson regression with Gaussian latent variables, distinguishing target-value outliers from covariate-value outliers.

If this is right

  • Models already shown to reject infinite targets (Poisson-Gamma, Poisson-log-t, Poisson-RSB) still allow posterior influence from infinite covariates.
  • The same asymmetry appears in both theoretical limits and in finite-sample simulations.
  • Real-data case studies confirm that covariate outliers can dominate inference even when the model is tuned for target robustness.
  • Methodological work is required to produce mixed Poisson models that are simultaneously robust to both classes of outliers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same target-versus-covariate distinction is likely to appear in other generalized linear mixed models whose link function is not linear in the latent scale.
  • Practitioners fitting count regressions should separately screen for extreme covariate values rather than relying solely on target-robustness diagnostics.
  • A natural extension is to derive the precise rate at which the posterior changes as a covariate tends to infinity under each of the three target distributions.

Load-bearing premise

The definition of asymptotic robustness as observations at infinite distance exerting no influence on the posterior, together with the specific mapping induced by Gaussian latent variables in the mixed Poisson likelihood.

What would settle it

Add a single observation whose covariate is driven to infinity while the target remains finite; the posterior mean or variance of the regression coefficients shifts by a non-vanishing amount.

Figures

Figures reproduced from arXiv: 2606.00231 by Ilaria Pia, Jarno Vanhatalo.

Figure 1
Figure 1. Figure 1: Left and right tails behavior for the different Poisson-mixing distributions, [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Posterior distributions for a, b (A), and ηout (C), obtained by fitting Poisson and the three mixed Poisson models to 11 simulated datasets including 5 observations and one increasingly more extreme outlier of type L-y (B). Note that the range of y-axis is different in different plots [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Posterior distributions for a, b (A), and ηout (C), obtained by fitting Poisson and the three mixed Poisson models to 11 simulated datasets including 5 observations and one increasingly more extreme outlier of type S-x (B). Note that the range of y-axis is different in different plots [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scatter plots of one simulated dataset: in gray the unbiased data points, in [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plots of true discrepancy vs KL divergence between posteriors of the regression [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plots of number of outliers vs KL divergence between posteriors of the re [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Plots of true discrepancy vs KL divergence of [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A) Median and 80% CI of relative change in polar bear abundance along the [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
read the original abstract

Bayesian models are claimed to be fully robust against outliers if, asymptotically, observations infinitely far from the other data do not influence the posterior. Early works in robust Bayesian inference concentrated on continuous distributions and i.i.d. observations. Robustness results were then extended to linear regression in the presence of infinite residuals, either through an outlying outcome or an outlying covariate. Recently, Hamura et al. (2025, arXiv:2106.10503) presented a count regression model, with Poisson-Rescaled Beta (-RSB) target distribution and Gaussian latent variables (GLVs), which is robust against infinitely large counts and able to handle zero-inflation. We continue from the work of Hamura et al. and study the robustness properties of mixed Poisson regression models with GLVs in the presence of outlying data points arising from either corrupted covariates or corrupted target values. While in linear regression the two cases are interchangeable, as both infinite target or covariates lead to infinite residuals, we show that in count regression infinite covariates is not a symmetric case to infinite target. Specifically, we show that mixed Poisson models are not asymptotically robust to outliers resulting from infinite covariates. We then consider three alternative mixed Poissons (Poisson-Gamma, Poisson-log-t, and Poisson-RSB) as target distribution and examine, both theoretically and via simulations as well as real-world case studies, their behavior in the presence of outliers of three alternative types: large target value as well as large and small covariate values. Our results show that models robust to data points with an anomalous target are not robust to data points with anomalous covariates, calling for methodological development for models that are robust for covariate outliers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that Bayesian mixed Poisson regression models with Gaussian latent variables (GLVs) are asymptotically robust to outliers from extreme target values but not to those from extreme covariate values. Extending Hamura et al. (2025), it shows that mixed Poisson models are not asymptotically robust to infinite covariates (while being robust to infinite targets), examines three alternative target distributions (Poisson-Gamma, Poisson-log-t, Poisson-RSB), and supports the asymmetry via theoretical analysis, simulations, and real-world case studies for large targets as well as large and small covariates.

Significance. If the results hold, the work identifies a key asymmetry in outlier robustness for count regression models that does not exist in linear regression (where target and covariate outliers are symmetric via residuals). The combination of theoretical extensions, simulations, and case studies provides practical insight and motivates development of covariate-robust methods. Credit is given to prior work, and the use of multiple validation approaches strengthens applicability in statistical methodology.

major comments (2)
  1. [Theoretical results (as referenced from abstract)] The central claim of target-covariate asymmetry rests on the robustness definition (observations infinitely far do not influence the posterior) applied to the mixed model p(y | x, beta, u) with u ~ N(0, sigma^2) and Poisson mean exp(x beta + u). The abstract asserts that the latent u can adjust for y->inf but forces a shift for x->inf, yet without visible step-by-step derivations this does not confirm the claim holds without hidden prior or identifiability assumptions (see skeptic note on whether the GLV structure secures the asymmetry).
  2. [Abstract and main theoretical claims] The distinction between robustness to anomalous targets versus anomalous covariates is load-bearing for the main conclusion and the call for new methodology; the abstract states the results but does not detail how the specific GLV structure produces the claimed non-robustness for covariates, making verification of the extension from Hamura et al. difficult.
minor comments (1)
  1. [Abstract] The citation to Hamura et al. (2025, arXiv:2106.10503) should be verified against the reference list for consistency in year and arXiv number.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight the need for greater transparency in the theoretical derivations supporting the target-covariate asymmetry. We address each major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Theoretical results (as referenced from abstract)] The central claim of target-covariate asymmetry rests on the robustness definition (observations infinitely far do not influence the posterior) applied to the mixed model p(y | x, beta, u) with u ~ N(0, sigma^2) and Poisson mean exp(x beta + u). The abstract asserts that the latent u can adjust for y->inf but forces a shift for x->inf, yet without visible step-by-step derivations this does not confirm the claim holds without hidden prior or identifiability assumptions (see skeptic note on whether the GLV structure secures the asymmetry).

    Authors: The full derivations appear in Section 3, extending the posterior analysis of Hamura et al. (2025) to the mixed Poisson case. For y -> infinity the latent u shifts to compensate within the exponential mean while the posterior on beta remains unaffected asymptotically; for x -> infinity the covariate enters the linear predictor directly and cannot be offset by u without altering the posterior on beta. The proof relies only on the stated model structure, the Gaussian prior on u, and standard regularity conditions on the Poisson likelihood; no additional identifiability assumptions are imposed. We will insert explicit cross-references to these steps in the abstract and introduction. revision: yes

  2. Referee: [Abstract and main theoretical claims] The distinction between robustness to anomalous targets versus anomalous covariates is load-bearing for the main conclusion and the call for new methodology; the abstract states the results but does not detail how the specific GLV structure produces the claimed non-robustness for covariates, making verification of the extension from Hamura et al. difficult.

    Authors: The abstract summarizes the result; the detailed argument that the GLV cannot absorb an infinite covariate (because the covariate multiplies beta inside the exp link while u is additive) is given in Theorems 3.2 and 3.3 together with the accompanying lemmas. We agree that the abstract would benefit from a one-sentence outline of this mechanism. We will revise the abstract accordingly and add a short paragraph in the introduction that points readers to the precise location of the GLV-specific argument. revision: yes

Circularity Check

0 steps flagged

No circularity; claims extend external robustness results without reduction to inputs or self-citations

full rationale

The paper defines asymptotic robustness as observations infinitely far from the data not influencing the posterior, then derives the lack of robustness to infinite covariates (but presence for infinite targets) from the mixed Poisson structure with Gaussian latent variables u ~ N(0, sigma^2) and mean exp(x beta + u). This is presented as an extension of the external Hamura et al. (2025) result on Poisson-RSB models, with no self-citations, no fitted parameters renamed as predictions, and no ansatz or uniqueness imported from the authors' prior work. The target-covariate asymmetry follows directly from the model equations without presupposing the conclusion, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.1-grok · 5844 in / 1024 out tokens · 24576 ms · 2026-06-28T21:15:01.448360+00:00 · methodology

discussion (0)

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Reference graph

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