发表机构
Organization for the Establishment of the Faculty of Sports Data Science and Business, Chuo University; School of Pharmaceutical Sciences, Wakayama Medical University（中央大学体育数据科学与商业学部设立机构; 和歌山医科大学药学部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出贝叶斯广义泊松矩阵分解（Bayesian GPMF），通过复合泊松表示和吉布斯采样实现后验推断，量化不确定性，模拟和足球数据实验验证其精度优于传统GPMF。
AI 中文摘要
广义泊松矩阵分解（GPMF）是一种基于广义泊松分布、针对具有过度离散性的计数数据的矩阵分解方法。虽然GPMF通过最大似然估计提供模型参数的点估计，但它并未量化估计不确定性。在本文中，我们提出了GPMF的贝叶斯扩展，称为贝叶斯GPMF，并开发了一种用于后验推断的吉布斯采样器。所提出的方法基于广义泊松分布的复合泊松表示，该表示引入潜在变量，并为所有模型参数产生闭式全条件后验分布。为了高效地对过度离散参数进行采样，我们推导了指数倾斜贝塔（EBeta）分布的无限混合表示，并基于该表示开发了一种有限近似。模拟研究表明，所提出的贝叶斯GPMF相比传统GPMF实现了更小的均方误差，同时提供的可信区间具有接近名义水平的经验覆盖率。此外，所提出的有限近似达到了与采样/重要性重采样（SIR）相当的估计精度，同时所需计算量更少。对足球事件数据的应用进一步说明了贝叶斯GPMF在提取可解释潜在结构和量化估计不确定性方面的实用性。这些结果表明，所提出的框架为广义泊松矩阵分解提供了一种有效的贝叶斯方法。
英文摘要
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.
