During the last years, graphical models have become a popular tool to represent dependencies among variables in many scientific areas. Typically, the objective is to discover dependence relationships that can be represented through a directed acyclic graph (DAG). The set of all conditional independencies encoded by a DAG determines its Markov property. In general, DAGs encoding the same conditional independencies are not distinguishable from observational data and can be collected into equivalence classes, each one represented by a chain graph called essential graph (EG). However, both the DAG and EG space grow super exponentially in the number of variables, and so, graph structural learning requires the adoption of Markov chain Monte Carlo (MCMC) techniques. In this paper, we review some recent results on Bayesian model selection of Gaussian DAG models under a unified framework. These results are based on closed‐form expressions for the marginal likelihood of a DAG and EG structure, which is obtained from a few suitable assumptions on the prior for model parameters. We then introduce a general MCMC scheme that can be adopted both for model selection of DAGs and EGs together with a couple of applications on real data sets.

中文翻译：

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高斯有向无环图结构的贝叶斯模型选择

在过去的几年中，图形模型已成为一种流行的工具，用于表示许多科学领域中变量之间的依存关系。通常，目标是发现依赖关系可以通过有向无环图（DAG）表示。DAG编码的所有条件独立性的集合确定其Markov属性。通常，编码相同条件独立性的DAG不能与观测数据区分开，可以收集到等效类中，每个等效类由称为基本图（EG）的链图表示。但是，DAG和EG空间的变量数量都呈指数增长，因此，图结构学习需要采用马尔可夫链蒙特卡洛（MCMC）技术。在本文中，我们回顾了在统一框架下关于高斯DAG模型的贝叶斯模型选择的一些最新结果。这些结果基于DAG和EG结构的边际可能性的封闭形式表达式，它是从模型参数先验的一些合适假设中获得的。然后，我们介绍一种通用的MCMC方案，该方案可以同时用于DAG和EG的模型选择，以及在实际数据集上的几个应用程序。