In the context of inferring a Bayesian network structure (directed acyclic
graph, DAG for short), we devise a non-reversible continuous time Markov chain,
the "Causal Zig-Zag sampler", that targets a probability distribution over
classes of observationally equivalent (Markov equivalent) DAGs. The classes are
represented as completed partially directed acyclic graphs (CPDAGs). The
non-reversible Markov chain relies on the operators used in Chickering's Greedy
Equivalence Search (GES) and is endowed with a momentum variable, which
improves mixing significantly as we show empirically. The possible target
distributions include posterior distributions based on a prior over DAGs and a
Markov equivalent likelihood. We offer an efficient implementation wherein we
develop new algorithms for listing, counting, uniformly sampling, and applying
possible moves of the GES operators, all of which significantly improve upon
the state-of-the-art.

在推断贝叶斯网络结构（有向无环图，DAG）的背景下，我们设计了一种非可逆连续时间马尔可夫链，称为 “因果 Zig-Zag 采样器”，该采样器针对一类观测等效（Markov 等价）DAG 的概率分布。这些类别以完成的部分有向无环图（CPDAG）表示。非可逆马尔可夫链依赖于 Chickering 的贪婪等价搜索（GES）中使用的操作符，并以动量变量进行了改进，从实证结果上显示其混合效果显著。可能的目标分布包括基于 DAG 先验和 Markov 等价似然的后验分布。我们提供了一种高效的实现，其中我们开发了新的算法来列出、计数、均匀采样和应用 GES 操作符的可能移动，所有这些都显著改进了现有技术水平。

动量驱动的因果结构学习：在有向无环图的 Markov 等价类上采样分布

Causal structure learning with momentum: Sampling distributions over  Markov Equivalence Classes of DAGs

Causal background knowledge about the existence or the absence of causal
edges and paths is frequently encountered in observational studies. The shared
directed edges and links of a subclass of Markov equivalent DAGs refined due to
background knowledge can be represented by a causal maximally partially
directed acyclic graph (MPDAG). In this paper, we first provide a sound and
complete graphical characterization of causal MPDAGs and give a minimal
representation of a causal MPDAG. Then, we introduce a novel representation
called direct causal clause (DCC) to represent all types of causal background
knowledge in a unified form. Using DCCs, we study the consistency and
equivalency of causal background knowledge and show that any causal background
knowledge set can be equivalently decomposed into a causal MPDAG plus a minimal
residual set of DCCs. Polynomial-time algorithms are also provided for checking
the consistency, equivalency, and finding the decomposed MPDAG and residual
DCCs. Finally, with 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 causal background knowledge can
significantly improve the identifiability of causal effects.

本文提出了一种新方法 —— 称作直接因果子句（DCC）来表述所有类型的因果背景知识，分析因果背景知识的一致性、等价性，任何因果背景知识集合都可分解成一个因果 MPDAG 和一个最小剩余 DCC 集合，并提供了用于检查一致性、等价性和查找分解的多项式算法。最后，作者们还发现，因果效应的可鉴定性仅取决于分解后的 MPDAG。