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

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.