A gamma approximation to the Bayesian posterior distribution of a discrete parameter of the Generalized Poisson model

TL;DR

This paper demonstrates that the Bayesian posterior distribution of a discrete parameter in the Generalized Poisson model can be effectively approximated by a gamma distribution when the mean parameter is small, simplifying inference.

Contribution

It introduces a gamma approximation for the posterior of a discrete parameter in the Generalized Poisson model under an uninformative prior, providing a practical analytical tool.

Findings

Gamma approximation closely matches the true posterior for small b

Simplifies Bayesian inference for the discrete parameter

Applicable when the mean parameter b is small

Abstract

Let $X$ have a Generalized Poisson distribution with mean $kb$, where $b$ is a known constant in the unit interval and $k$ is a discrete, non-negative parameter. We show that if an uninformative uniform prior for $k$ is assumed, then the posterior distribution of $k$ can be approximated using the gamma distribution when $b$ is small.