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 present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, using the parameterizing set representation of \citet{hu2020faster}. By construction, our imset also represents the Markov equivalence class of the MAG. We show that for many such graphs our proposed imset is \emph{perfectly Markovian} with respect to the graph 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. Unfortunately, for some models the representation does not work, and in certain cases does not represent any independences at all. We prove that it does work for \emph{simple} MAGs where there are only heads of size less than three, as well as for a large class of purely bidirected models. 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. Further we refine the ordered local Markov property, which relates to finding the best imsets representing general MAGs.

翻译：\ citet{ studenny2006 probablitic} 的设置是代表有条件独立模型的一种代数法。 当应用到这些模型时, 它们有许多有吸引力的属性, 并且它们对于使用定向环形图( DAG) 模型特别好。 特别是, DAG 的“ 标准” 设置是与其产生的独立的一对一对应的, 因而是其 Markov 等值类的标签。 我们提议扩展 标准, 以代表最大祖传图形( MAG) 模型 的标准 。 使用\ citet{ hu2020faster} 的参数化集表示这些模型。 通过构建, 我们的顶值还代表了 MAG 的 等值 。 对于许多这样的图形, 我们提议的设置的“ 标准” 是 与它所引发的独立的一对一对一对一, 因此, 我们只能用一个标准来测量一个定义模型的 ; 这给通常的 BIC 评分。 不幸的是, 对于一些模型来说, 代表不起作用, 在某些模型中, 我们的比 MAG 更能代表任何的大小。