Learning the structure of dependence relations between variables is a pervasive issue in the statistical literature. A directed acyclic graph (DAG) can represent a set of conditional independencies, but different DAGs may encode the same set of relations and are indistinguishable using observational data. Equivalent DAGs can be collected into classes, each represented by a partially directed graph known as essential graph (EG). Structure learning directly conducted on the EG space, rather than on the allied space of DAGs, leads to theoretical and computational benefits. Still, the majority of efforts has been dedicated to Gaussian data, with less attention to methods designed for multivariate categorical data. A Bayesian methodology for structure learning of categorical EGs is then proposed. Combining a constructive parameter prior elicitation with a graph-driven likelihood decomposition, a closed-form expression for the marginal likelihood of a categorical EG model is derived. Asymptotic properties are studied, and an MCMC sampler scheme developed for approximate posterior inference. The methodology is evaluated on both simulated scenarios and real data, with appreciable performance in comparison with state-of-the-art methods.

学习变量之间依赖关系的结构是统计文献中普遍存在的问题。有向无环图 (DAG) 可以表示一组条件独立性，但不同的 DAG 可能编码相同的一组关系，并且使用观察数据无法区分。等效的 DAG 可以收集到类中，每个类都由称为基本图 (EG) 的部分有向图表示。直接在 EG 空间而不是 DAG 的联合空间上进行的结构学习会带来理论和计算上的好处。尽管如此，大部分工作都致力于高斯数据，而较少关注为多变量分类数据设计的方法。然后提出了一种用于分类 EG 结构学习的贝叶斯方法。将建设性参数先验引出与图驱动的似然分解相结合，导出了分类 EG 模型的边际似然的封闭形式表达式。研究了渐近特性，并为近似后验推断开发了 MCMC 采样器方案。该方法在模拟场景和真实数据上进行了评估，与最先进的方法相比具有可观的性能。