Robust Bayesian exceedance screening for count data

Abstract

Rare-event screening in count data often concerns latent upper-tail risk rather than large observed counts alone. We develop a robust Bayesian procedure whose parameter-conditional tail functional is θi(ϑ)=Prϑ(Yirep>τi∣xi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _i(\vartheta ) =\Pr _{\vartheta }(Y_i^{\textrm{rep}}>\tau _i\mid x_i)$$\end{document}, where xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i$$\end{document} is the covariate vector; posterior averaging yields the posterior predictive exceedance probability θ¯i=E{θi(ϑ)∣Dn}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{\theta }_i =\mathbb {E}\{\theta _i(\vartheta )\mid \mathcal {D}_n\}$$\end{document}. The working model is negative binomial, with latent-scale robustification and bounded-influence computation used to reduce the impact of excess zeros, isolated large counts, and multiplicative contamination on fitted tail probabilities. The analysis separates posterior tail-risk ranking from formal discovery. Ranking is based on posterior mean exceedance probabilities, whereas multiplicity-controlled selection is obtained through an e-value layer. Under the Bayes-marginal null induced by the same hierarchical model, posterior odds are Bayes factors and hence valid e-values for e-BH false-discovery-rate control under arbitrary dependence. Under a working-model interpretation, the same quantities are treated as evidence scores unless calibrated, and the theory is formulated around a Kullback–Leibler pseudo-true exceedance target. Simulations show improved tail-probability calibration and fewer contamination-driven false discoveries under misspecification, with the expected efficiency loss under clean data. An application to the RAND Health Insurance Experiment illustrates high-utilization screening and identifies a subgroup enriched for large outpatient-visit counts.