Finite-dimensional Discrete Random Structures and Bayesian Clustering

Discrete random probability measures stand out as effective tools for Bayesian clustering. The investigation in the area has been very lively, with a strong emphasis on nonparametric procedures based on either the Dirichlet process or on more flexible generalizations, such as the normalized random measures with independent increments (NRMI). The literature on finite-dimensional discrete priors is much more limited and mostly confined to the standard Dirichlet-multinomial model. While such a specification may be attractive due to conjugacy, it suffers from considerable limitations when it comes to addressing clustering problems. In order to overcome these, we introduce a novel class of priors that arise as the hierarchical compositions of finite-dimensional random discrete structures. Despite the analytical hurdles such a construction entails, we are able to characterize the induced random partition and determine explicit expressions of the associated urn scheme and of the posterior distribution. A detailed comparison with (infinite-dimensional) NRMIs is also provided: indeed, informative bounds for the discrepancy between the partition laws are obtained. Finally, the performance of our proposal over existing methods is assessed on a real application where we study a publicly available dataset from the Italian education system comprising the scores of a mandatory nationwide test.

离散随机概率测度（Discrete random probability measures）是贝叶斯聚类（Bayesian clustering）的有效工具。该领域的研究一直十分活跃，重点关注基于狄利克雷过程（Dirichlet Process）或更灵活的推广形式（如带独立增量的归一化随机测度（normalized random measures with independent increments, NRMI））的非参数方法。针对有限维离散先验的研究文献则相对匮乏，且大多局限于标准狄利克雷-多项分布模型（Dirichlet-Multinomial Model）。尽管这类先验设定因共轭性而颇具吸引力，但在解决聚类问题时存在诸多显著局限。为克服上述缺陷，我们提出了一类新颖的先验分布，其构造源于有限维随机离散结构的分层复合。尽管该构造会带来分析层面的诸多难点，但我们仍得以刻画其诱导的随机划分，并推导得到了对应瓮模型（urn scheme）与后验分布的显式表达式。此外，我们还与（无限维）NRMIs展开了详尽的对比分析，成功得到了两类划分法则间差异度的有效界值。最后，我们通过一项真实应用场景评估了所提方法相较于现有方法的性能：该实验采用了意大利教育系统公开的全国强制统一考试分数数据集。