Negative-binomial model

Description

The negative-binomial model is a model of counts that allows for overdispersion.

Definition

For discrete positive outcome \(y\) and predictors \(x\), the model is:

\[ \begin{align} y_i \sim \text{NegBinomial}(\exp{(\eta_i)}, \phi) \\ \eta_i = \alpha + \beta \cdot x_i \end{align} \]

Parameters needing priors

• \(\alpha\) (intercept)
• \(\beta\) (predictor weights)
• \(\phi\) (overdispersion parameter)

Prior for \(\phi\)

Weakly informative inverse gamma prior

Vehtari (2024) suggested an inverse gamma prior as an approximation of the penalized complexity prior:

\(\phi \sim \text{InvGamma}(0.4, 0.3)\)

Penalized complexity prior

Simpson et al. (2017) suggested a penalized complexity prior

See also

• Distribution explorer
• PreliZ
• Aki Vehtari’s case study
• Stan Functions Reference
• bambi example

References

Simpson, Daniel, Håvard Rue, Andrea Riebler, Thiago G. Martins, and Sigrunn H. Sørbye. 2017. “Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors.” Statistical Science 32 (1): 1–28. https://doi.org/10.1214/16-STS576.

Vehtari, Aki. 2024. “Default Prior for Negative-Binomial Shape Parameter.” https://users.aalto.fi/~ave/casestudies/Priors/negbinomial_shape_prior.html.