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, it has remained unclear whether this symmetry applies to generalized linear models. Moreover, only recently, the theoretical understanding of generalized linear models' robustness to outliers in target values has progressed significantly. 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 then summarize the theoretical asymptotic outlier rejection properties of Gamma, log-Student's-$t$, and Rescaled Beta mixed Poisson models in the presence of outliers in either target or covariate values. We also study the properties of these three mixed Poisson models in the presence of moderate outliers with simulations and a real world case study. Experimental results indicate that all mixed Poisson models are less sensitive to moderate outliers than (non-mixed) Poisson.