Pairwise causal background knowledge about the existence or absence of causal edges and paths is frequently encountered in observational studies. Such constraints allow the shared directed and undirected edges in the constrained subclass of Markov equivalent DAGs to be represented as a causal maximally partially directed acyclic graph (MPDAG). In this paper, we first provide a sound and complete graphical characterization of causal MPDAGs and introduce a minimal representation of a causal MPDAG. Then, we give a unified representation for three types of pairwise causal background knowledge, including direct, ancestral and non-ancestral causal knowledge, by introducing a novel concept called direct causal clause (DCC). Using DCCs, we study the consistency and equivalence of pairwise causal background knowledge and show that any pairwise causal background knowledge set can be uniquely and equivalently decomposed into the causal MPDAG representing the refined Markov equivalence class and a minimal residual set of DCCs. Polynomial-time algorithms are also provided for checking consistency and equivalence, as well as for finding the decomposed MPDAG and the residual DCCs. Finally, with pairwise causal background knowledge, we prove a sufficient and necessary condition to identify causal effects and surprisingly find that the identifiability of causal effects only depends on the decomposed MPDAG. We also develop a local IDA-type algorithm to estimate the possible values of an unidentifiable effect. Simulations suggest that pairwise causal background knowledge can significantly improve the identifiability of causal effects.

翻译：在观察性研究中，经常遇到关于因果边和因果路径存在与否的成对因果背景知识。此类约束使得受约束的马尔可夫等价有向无环图（DAG）子类中的共享有向边和无向边，能够表示为一个因果最大部分有向无环图（MPDAG）。本文首先给出了因果MPDAG的一个可靠且完备的图形化表征，并引入了因果MPDAG的最小表示。接着，通过引入一个称为直接因果子句（DCC）的新概念，我们为三种类型的成对因果背景知识（包括直接、祖先和非祖先因果知识）提供了一个统一的表示。利用DCC，我们研究了成对因果背景知识的一致性与等价性，并证明任何成对因果背景知识集都可以唯一且等价地分解为代表精炼马尔可夫等价类的因果MPDAG和一个最小残差DCC集。本文还提供了用于检查一致性与等价性、以及寻找分解后的MPDAG和残差DCC的多项式时间算法。最后，基于成对因果背景知识，我们证明了一个识别因果效应的充分必要条件，并意外地发现因果效应的可识别性仅取决于分解后的MPDAG。我们还开发了一种局部IDA型算法来估计不可识别效应的可能值。模拟结果表明，成对因果背景知识能显著提高因果效应的可识别性。