Sampling from an Unnormalised Discrete Distribution: Applied to COM-Poisson

The Conway–Maxwell–Poisson (COM-Poisson) distribution is a flexible generalisation of the Poisson model that can accommodate both over-dispersion and under-dispersion in count data. However, the absence of a closed-form expression for its normalising constant complicates the generation of random samples, which is essential for Bayesian inference methods. In this work, we investigate sampling from the COM-Poisson distribution by comparing three approaches: Ratio-of-Uniforms, rejection sampling with tailored envelope distributions, and an exact sampling algorithm based on adaptive truncation and ratio-bounding pairs. Through a simulation study covering a wide range of parameter values, we evaluate these methods in terms of computational efficiency and statistical accuracy. The results show that the rejection sampling approach achieves the best computational performance, while the exact sampling method provides comparable statistical accuracy with the advantage of theoretical exactness.