AI 中文总结
该研究在现有图形多项、负二项模型基础上补充超几何、负超几何分布，构建多元计数数据的图形建模统一参数框架，开发贝叶斯层级并结合里德伯原子数据验证其应用潜力。
AI 中文摘要
经典的多项分布、负二项分布、超几何分布和负超几何分布可根据抽样方案的两个特征自然归类：有放回或无放回抽样，以及抽取固定数量或固定失败次数后停止。我们在Danielewska等人（2025）提出的图形多项模型和图形负二项模型基础上，添加图形超几何分布和图形负超几何分布，完成了该抽样方案针对可分解图的图形类似物。由此得到的四个族为多元计数数据的图形建模提供了统一的参数框架，其中依赖性和可容许构型由图编码。它们在空图对应的单变量分布乘积与完全图对应的对应经典多元分布之间插值，同时保留显式马尔可夫分解和易处理的抽样表示。我们进一步开发了基于图形狄利克雷型分布的统一贝叶斯层级，得到显式后验和预测定律。该框架对受排除或不相容约束产生的计数数据特别适用，我们讨论了若干此类应用，并利用里德伯原子激发数据说明其实际潜力。
英文摘要
The classical multinomial, negative multinomial, hypergeometric, and negative hypergeometric distributions are naturally organized by two features of the sampling scheme: sampling with or without replacement and stopping after a fixed number of draws or a fixed number of failures. We complete the graphical analogue of this scheme for decomposable graphs by adding graphical hypergeometric and graphical negative hypergeometric distributions to the previously introduced graphical multinomial and graphical negative multinomial models in Danielewska et al. (2025). The resulting four families provide a unified parametric framework for graphical modeling of multivariate count data, in which dependence and admissible configurations are encoded by a graph. They interpolate between products of univariate distributions for the empty graph and the corresponding classical multivariate distributions for the complete graph, while retaining explicit Markov factorizations and tractable sampling representations. We further develop a unified Bayesian hierarchy based on graphical Dirichlet-type distributions, obtaining explicit posterior and predictive laws. The framework is particularly natural for count data arising under exclusion or incompatibility constraints. We discuss several such applications and illustrate its practical potential using Rydberg-atom excitation data.
