AI 中文总结
针对足球比分等离散计数数据，提出结合CMP似然与SAS先验的贝叶斯框架建模，通过定制抽样器进行后验推断，经模拟和英超数据验证，能捕捉非等离散，揭示球队离散异质性，提升模型拟合与预测性能。
AI 中文摘要
足球比赛进球数的统计建模通常使用泊松分布及其变体。本文提出了一个贝叶斯框架，通过将康威 - 麦克斯韦 - 泊松（CMP）似然与单位特定离散参数上的尖峰和平板（SAS）先验相结合，对计数数据中的欠离散和过离散进行建模。该方法将基于泊松的计数数据模型进行了推广，将等离散作为明确基线，对偏离该状态进行概率量化并估计其大小。通过定制的吉布斯抽样器进行后验推断，处理双重难处理的似然并有效探索后验。用模拟数据检验新方法捕捉非等离散的能力，并应用于英超联赛数据。结果揭示了英超联赛中各队特定离散的异质性，且在模型拟合和预测性能方面相对于标准泊松模型有改进。
英文摘要
Statistical modeling for goals scored in football is typically achieved using the Poisson distribution and its variants. Here we propose a Bayesian framework for modeling under- and over-dispersion in count data by combining the Conway-Maxwell-Poisson (CMP) likelihood with a spikeand-slab (SAS) prior on unit-specific dispersion parameters. The proposed methodology generalizes Poisson-based count data models by treating equidispersion as an explicit baseline, and offering probabilistic quantification of departures from this regime, while simultaneously estimating their magnitude. Posterior inference is performed through a tailored Metropolis-within-Gibbs sampler that handles the doubly-intractable likelihood and provides efficient posterior exploration. The new method is examined using simulated data to confirm its ability to capture non-equidispersion, and applied to English Premier League (EPL) data. Dispersion is modeled at the team level and linked to goal-scoring behavior, and allows for thresholding mechanisms to distinguish teams based on their posterior probability of non-equidispersion. The results reveal heterogeneities in team-specific dispersion in the EPL, and demonstrate improvements in both model fit and predictive performance with respect to the standard Poisson model.
