Fast Bayesian model selection for tail-regime Poisson mixtures

Poisson mixture models provide a principled approach to overdispersed count data by representing latent heterogeneity through a random intensity. Beyond variance inflation, the mixing distribution governs extremal behavior, yet the tail implications of common observation mechanisms and mixture-selection uncertainty are often left implicit in applied analyses. Building on recent asymptotic tail classification results for Poisson mixtures and fast Bayesian model selection for finite mixtures, we develop a tail-aware inference framework for counts observed under imperfect detection and related recording constraints. We show how binomial thinning, zero truncation, and right censoring transform discrete tail functionals, yielding transparent links between mixing-tail classes and the observed extremes. Efficient posterior computation is achieved through collapsed sampling strategies that learn mixture complexity and regime structure. Simulations confirm the theoretical tail predictions across representative mixing families, and an analysis of replicated wildlife survey counts illustrates how the approach delivers interpretable tail diagnostics and predictive summaries for rare large counts.