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.
From: Ilaria Pia [view email]
[v1]
Fri, 29 May 2026 18:06:11 UTC (1,873 KB)
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