Bayesian Tail Index Learning for Poisson–Tweedie Counts via a Poisson-GIG Bridge

Environmental count series often exhibit long runs of zeros punctuated by sporadic, operationally critical bursts. We develop a distribution-first Bayesian framework that learns the Poisson–Tweedie (PT) tail index while shrinking toward the Poisson-Inverse Gaussian (PIG) corner when evidence supports heavier extremes. The likelihood is implemented via a Poisson-Generalized Inverse Gaussian (Poisson-GIG; Sichel) bridge, yielding closed-form marginal probability mass functions (pmfs) and a continuous path between negative binomial (NB) and PIG behavior. We establish identifiability through a monotone mapping from a one-dimensional GIG tail parameter to the PT index p∈[2,3], give minimal conditions for posterior propriety under weak priors, and develop computation based on latent-rate augmentation (with exact conditionals at the inverse-Gaussian (IG) corner), a stable one-dimensional tail update, and a fast variational surrogate. Posterior prediction targets decision-relevant functionals: high-threshold exceedance probabilities, a threshold-weighted continuous ranked probability score (CRPS), exceedance probability integral transform (PIT) diagnostics, and a zero-mass check orthogonalized via an optional mechanistic hurdle for structural zeros. Simulations confirm accurate recovery of the tail index and superior calibration of far-right probabilities when extremes matter. In an application to US wildfire event frequencies from the Storm Events archive, empirical exceedances—after matching dispersion—often align more closely with the heavy-tailed PIG corner, and the proposed shrinkage yields calibrated high-threshold predictions without ad hoc zero inflation. The framework remains compatible with mean/dispersion regression and spatial-temporal random effects, providing a scalable, interpretable basis for exceedance-oriented risk assessment.