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

Md Mahadi Hasan; Md Monzur Murshed; Md Nahid Hasan; Md Mahadi Hasan; Md Monzur Murshed; Md Nahid Hasan

doi:10.3934/era.2026302

Electronic Research Archive

2026, Volume 34, Issue 9: 6957-6998. doi: 10.3934/era.2026302

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

1.
Department of Mathematics and Statistics, Murray State University, Murray, KY 42071, USA
2.
Department of Mathematics and Statistics, Minnesota State University, Mankato, MN 56001, USA
3.
Department of Mathematics, East Texas A & M University, Commerce, TX 75428, USA
4.
Department of Mathematics, Augusta University, Augusta, GA 30912, USA

Received: 27 May 2026 Revised: 05 August 2026 Accepted: 06 August 2026 Published: 14 August 2026

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.
Citation: Md Mahadi Hasan, Md Monzur Murshed, Md Nahid Hasan. Bayesian credible and prediction intervals for the two-parameter negative binomial (NB2) model[J]. Electronic Research Archive, 2026, 34(9): 6957-6998. doi: 10.3934/era.2026302

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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.

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