Gibbs Sampling for Bayesian Generalized Poisson Matrix Factorization

Abstract

Generalized Poisson matrix factorization (GPMF) is a matrix factorization method for count data with overdispersion based on the generalized Poisson distribution. While GPMF provides point estimates of the model parameters through maximum likelihood estimation, it does not quantify estimation uncertainty. In this paper, we propose a Bayesian extension of GPMF, referred to as Bayesian GPMF, and develop a Gibbs sampler for posterior inference. The proposed method is based on a compound Poisson representation of the generalized Poisson distribution, which introduces latent variables and yields closed-form full conditional posterior distributions for all model parameters. To efficiently sample the overdispersion parameter, we derive an infinite mixture representation of the exponentially tilted beta (EBeta) distribution and develop a finite approximation based on this representation. Simulation studies demonstrate that the proposed Bayesian GPMF achieves smaller mean squared errors than the conventional GPMF while providing credible intervals with empirical coverage probabilities close to the nominal level. Furthermore, the proposed finite approximation attains estimation accuracy comparable to Sampling/Importance Resampling (SIR) while requiring less computation. An application to football event data further illustrates the usefulness of Bayesian GPMF for extracting interpretable latent structures and quantifying estimation uncertainty. These results demonstrate that the proposed framework provides an effective Bayesian approach to generalized Poisson matrix factorization.