Robust Bayesian exceedance screening for count data

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 θ ᵢ(ϑ ) = ϑ(Yᵢʳᵉᵖ>τ ᵢ xᵢ) θ i ( ϑ ) = Pr ϑ ( Y i rep > τ i ∣ x i ) , where xᵢ x i is the covariate vector; posterior averaging yields the posterior predictive exceedance probability θ ̄ᵢ =E\θ ᵢ(ϑ ) Dₙ\ θ ¯ i = E { θ i ( ϑ ) ∣ D n } . 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.