The imsets of Studen\'y (2005) are an algebraic method for representing conditional independence models. They have many attractive properties when applied to such models, and they are particularly nice for working with directed acyclic graph (DAG) models. In particular, the 'standard' imset for a DAG is in one-to-one correspondence with the independences it induces, and hence is a label for its Markov equivalence class. We first present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, using the parameterizing set representation of Hu and Evans (2020). In these cases the imset provides a scoring criteria by measuring the discrepancy for a list of independences that define the model; this gives an alternative to the usual BIC score that is also consistent, and much easier to compute. We also show that, of independence models that do represent the MAG, the imset we give is minimal. Unfortunately, for some graphs the representation does not represent all the independences in the model, and in certain cases does not represent any at all. For these general MAGs, we refine the reduced ordered local Markov property Richardson (2003) by a novel graphical tool called _power DAGs_, and this results in an imset that induces the correct model and which, under a mild condition, can be constructed in polynomial time.

翻译：Studený (2005) 提出的imsets是一种表示条件独立模型的代数方法。该方法应用于此类模型时具有诸多优良性质，尤其适用于有向无环图（DAG）模型。具体而言，DAG的"标准"imset与其所诱导的独立性关系一一对应，因此可作为其马尔可夫等价类的标签。我们首先基于Hu和Evans (2020)的参数化集合表示，提出将标准imset扩展至最大祖先图（MAG）模型。在此类情形中，imset通过测量定义模型的独立性列表的偏差提供评分准则；这提供了比常规BIC评分更易计算且同样相合的新选择。我们同时证明，在表示MAG的独立性模型中，我们给出的imset具有极小性。遗憾的是，对于某些图，该表示无法刻画模型中的所有独立性关系，在特定情况下甚至无法表征任何独立性关系。针对这些一般性MAG，我们通过名为_power DAGs_的创新图形工具精炼了Richardson (2003)的简化有序局部马尔可夫性质，由此得到能诱导正确模型的imset，并且在温和条件下可在多项式时间内构造完成。