Equivalence classes of DAGs (represented by CPDAGs) may be too large to
provide useful causal information. Here, we address incorporating tiered
background knowledge yielding restricted equivalence classes represented by
'tiered MPDAGs'. Tiered knowledge leads to considerable gains in
informativeness and computational efficiency: We show that construction of
tiered MPDAGs only requires application of Meek's 1st rule, and that tiered
MPDAGs (unlike general MPDAGs) are chain graphs with chordal components. This
entails simplifications e.g. of determining valid adjustment sets for causal
effect estimation. Further, we characterise when one tiered ordering is more
informative than another, providing insights into useful aspects of background
knowledge.

使用分层背景知识建立的 MPDAG 限制了 DAG 的等价类，简化了因果效应估计中的计算，同时提高了信息量。

我们随着时间变得更明智吗？关于分层背景知识因果等价性的研究

Do we become wiser with time? On causal equivalence with tiered  background knowledge

We present a graphical criterion for covariate adjustment that is sound and
complete for four different classes of causal graphical models: directed
acyclic graphs (DAGs), maximum ancestral graphs (MAGs), completed partially
directed acyclic graphs (CPDAGs), and partial ancestral graphs (PAGs). Our
criterion unifies covariate adjustment for a large set of graph classes.
Moreover, we define an explicit set that satisfies our criterion, if there is
any set that satisfies our criterion. We also give efficient algorithms for
constructing all sets that fulfill our criterion, implemented in the R package
dagitty. Finally, we discuss the relationship between our criterion and other
criteria for adjustment, and we provide new soundness and completeness proofs
for the adjustment criterion for DAGs.

我们提出了一个适用于四类因果图模型：定向无环图 (DAGs)、最大先祖图 (MAGs)、完成部分定向无环图 (CPDAGs) 和部分先祖图 (PAGs) 的变量调整图形标准，该标准统一了一个大的图形类的变量调整。此外，我们定义了一个明确的集合，如果有满足我们的标准的集合，则满足我们的标准。我们还提供了高效算法来构建所有满足我们标准的集合，并在 R 包 dagitty 中实现。最后，我们讨论了我们的标准与其他调整标准之间的关系，并为 DAG 的调整标准提供了新的完整性和有效性证明。

在祖先图的马尔可夫等价类中完成调整集的图形刻画和构造

Complete Graphical Characterization and Construction of Adjustment Sets  in Markov Equivalence Classes of Ancestral Graphs

Causal inference relies on the structure of a graph, often a directed acyclic
graph (DAG). Different graphs may result in different causal inference
statements and different intervention distributions. To quantify such
differences, we propose a (pre-) distance between DAGs, the structural
intervention distance (SID). The SID is based on a graphical criterion only and
quantifies the closeness between two DAGs in terms of their corresponding
causal inference statements. It is therefore well-suited for evaluating graphs
that are used for computing interventions. Instead of DAGs it is also possible
to compare CPDAGs, completed partially directed acyclic graphs that represent
Markov equivalence classes. Since it differs significantly from the popular
Structural Hamming Distance (SHD), the SID constitutes a valuable additional
measure. We discuss properties of this distance and provide an efficient
implementation with software code available on the first author's homepage (an
R package is under construction).

本文提出了结构干预距离（SID）作为有向无环图（DAG）之间差别的一种度量方法，并与流行的结构汉明距离（SHD）相比较。该度量方法适用于计算干预的图形，并且可以比较部分完成的有向无环图（CPDAGs）的马尔可夫等价类。