On Asymptotic Outlier Rejection in Bayesian Mixed Poisson Regression Models Under Extreme Target and Covariate Values
Abstract
Bayesian models are defined to be fully robust against outliers if observations infinitely far from the other data do not influence the posterior. In regression models, this entails a need to consider outliers in both target and covariate values. While in linear regression these cases are interchangeable, as both lead to anomalously large residuals, this symmetry does not apply to generalized linear models. Importantly, Hamura et al. (2025, arXiv:2106.10503) presented sufficient conditions for mixed Poisson count regression models to be robust against infinitely large target values and proposed a mixed Poisson-Rescaled Beta model fulfilling these conditions. We continue from their work and study the robustness properties of mixed Poisson regression models with Gaussian latent variables in the presence of outliers in covariates. We show that in count regression the symmetry between covariate and target outliers breaks: mixed Poisson models are not robust to outlier covariates even if they were robust to target outliers. Furthermore, we show that, as a covariate gets infinitely large, the corresponding regression coefficient posterior collapses to a point-mass distribution concentrated around zero. We hence introduce a novel robustified mixed Poisson model which we denote as x-outlier rejective (x-OR) mixed Poisson, and demonstrate its theoretical and practical ability to accommodate outliers in the covariates. We investigate robustness properties of alternative (x-OR) mixed Poisson models in the presence of moderate outliers with simulations and a real world case study and show that the x-OR log-t and Rescaled Beta mixed Poisson models reject both types of outliers, leading to improved inference.
Related
- Support Vector Machines for Classification, Regression, and Outlier Detection: A Tutorial
- Three-group variance-ratio tests for heteroscedasticity in linear regression: an exact null distribution and an outlier-resistant version
- Conformalized Quantile Regression and Minimax Limits of Fixed-Score Calibration under Known Covariate Shift
- Flexible covariate adjustments in regression discontinuity designs
- Covariate-dependent Nonparametric g-modeling for regression via infinite Mixture-of-Expertizing class
BibTeX
@misc{pia2026asymptotic,
title = {{On Asymptotic Outlier Rejection in Bayesian Mixed Poisson Regression Models Under Extreme Target and Covariate Values}},
author = {Ilaria Pia and Jarno Vanhatalo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.00231},
eprint = {2606.00231},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.00231},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On Asymptotic Outlier Rejection in Bayesian Mixed Poisson Regression Models Under Extreme Target and Covariate Values
AU - Ilaria Pia
AU - Jarno Vanhatalo
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2606.00231
UR - https://arxiv.org/abs/2606.00231
AB - Bayesian models are defined to be fully robust against outliers if observations infinitely far from the other data do not influence the posterior. In regression models, this entails a need to consider outliers in both target and covariate values. While in linear regression these cases are interchangeable, as both lead to anomalously large residuals, this symmetry does not apply to generalized linear models. Importantly, Hamura et al. (2025, arXiv:2106.10503) presented sufficient conditions for mixed Poisson count regression models to be robust against infinitely large target values and proposed a mixed Poisson-Rescaled Beta model fulfilling these conditions. We continue from their work and study the robustness properties of mixed Poisson regression models with Gaussian latent variables in the presence of outliers in covariates. We show that in count regression the symmetry between covariate and target outliers breaks: mixed Poisson models are not robust to outlier covariates even if they were robust to target outliers. Furthermore, we show that, as a covariate gets infinitely large, the corresponding regression coefficient posterior collapses to a point-mass distribution concentrated around zero. We hence introduce a novel robustified mixed Poisson model which we denote as x-outlier rejective (x-OR) mixed Poisson, and demonstrate its theoretical and practical ability to accommodate outliers in the covariates. We investigate robustness properties of alternative (x-OR) mixed Poisson models in the presence of moderate outliers with simulations and a real world case study and show that the x-OR log-t and Rescaled Beta mixed Poisson models reject both types of outliers, leading to improved inference.
ER -