Bayesian credible and prediction intervals for the two-parameter negative binomial (NB2) model

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Abstract

Overdispersed count data are common in many applications, where the variance exceeds the mean, and the Poisson model is inadequate. The two-parameter negative binomial (NB2) model addresses this limitation through a dispersion parameter and provides a standard framework for modeling overdispersed counts. In this paper, we evaluated Bayesian credible intervals for the NB2 mean and Bayesian posterior predictive intervals for a single future count and for the mean of a future sample in a controlled one-sample, intercept-only setting. Posterior inference was obtained using weakly informative priors and Hamiltonian Monte Carlo with the No-U-Turn Sampler. The Bayesian intervals were assessed through simulation using operational coverage probability and expected width, and were compared with Wald, MLE-based normal, and parametric bootstrap confidence and prediction intervals. Prior-family sensitivity was examined using calibrated lognormal, Gamma, and log-Student-t priors. Robustness was also evaluated under zero-inflated NB2, structured-zero, and Poisson-lognormal data-generating mechanisms while fitting the NB2 model. Two real-data examples, involving Quine school-absence counts and epilepsy baseline seizure counts, illustrated the practical implementation of the methods.

Keywords

Bayesian inference / coverage probability / credible interval / Markov chain Monte Carlo (MCMC) / overdispersed count data / posterior predictive interval / prediction of future counts / two-parameter negative binomial distribution (NB2)

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