The imsets of \citet{studeny2006probabilistic} 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 \citet{hu2020faster}. We show that for many such graphs our proposed imset is \emph{perfectly Markovian} with respect to the graph, including \emph{simple} MAGs, as well as for a large class of purely bidirected models. Thus providing 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 much easier to compute. We also show that, of independence models that do represent the MAG, the one we give is the simplest possible, in a manner we make precise. 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 \citep{richardlocalmarkov} by a novel graphical tool called \emph{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：\citet{studeny2006probabilistic}提出的imset是一种用于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许多优良性质，尤其在处理有向无环图（DAG）模型时表现突出。具体而言，DAG的“标准”imset与其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\citet{hu2020faster}的参数化集表示，提出针对最大祖先图（MAG）模型的标准imset扩展方案。研究表明，对于包括简单MAG在内的多类图结构，以及一大类纯双向模型，我们提出的imset具有关于该图的完美马尔可夫性。由此，通过衡量定义模型的一组独立性关系的差异，可建立评分准则；这提供了一种比传统BIC评分更易计算的替代方案。我们还证明，在能够代表该MAG的独立性模型中，我们给出的表示是（按精确界定的意义）最简单的。然而，对于某些图，该表示无法涵盖模型中的所有独立性关系，在极端情况下甚至无法表示任何独立性。针对这些一般性MAG，我们通过一种称为“幂DAG”的新型图论工具，优化了约减有序局部马尔可夫性质\citep{richardlocalmarkov}，由此得到的imset能够诱导正确模型，并且在温和条件下可在多项式时间内构建。