Causal models in statistics are often described using acyclic directed mixed graphs (ADMGs), which contain directed and bidirected edges and no directed cycles. This article surveys various interpretations of ADMGs, discusses their relations in different sub-classes of ADMGs, and argues that one of them -- the noise expansion (NE) model -- should be used as the default interpretation. Our endorsement of the NE model is based on two observations. First, in a subclass of ADMGs called unconfounded graphs (which retain most of the good properties of directed acyclic graphs and bidirected graphs), the NE model is equivalent to many other interpretations including the global Markov and nested Markov models. Second, the NE model for an arbitrary ADMG is exactly the union of that for all unconfounded expansions of that graph. This property is referred to as completeness, as it shows that the model does not commit to any specific latent variable explanation. In proving that the NE model is nested Markov, we also develop an ADMG-based theory for causality. Finally, we compare the NE model with the closely related but different interpretation of ADMGs as directed acyclic graphs (DAGs) with latent variables that is commonly used in the literature. We argue that the "latent DAG" interpretation is mathematically unnecessary, makes obscure ontological assumptions, and discourages practitioners from deliberating over important structural assumptions.

翻译：统计学中的因果模型常使用无环有向混合图（ADMG）进行描述，这类图包含有向边与双向边且不存在有向环。本文系统综述了ADMG的多种解释方式，探讨了其在ADMG不同子类中的关联，并论证其中一种解释——噪声扩展（NE）模型——应作为默认解释。我们支持NE模型基于两点观察：首先，在称为无混淆图（保留有向无环图与双向图大部分良好性质）的ADMG子类中，NE模型等价于包括全局马尔可夫模型与嵌套马尔可夫模型在内的多种其他解释；其次，任意ADMG的NE模型恰好是该图所有无混淆扩展对应NE模型的并集。此性质称为完备性，表明该模型无需依赖任何特定的隐变量解释。在证明NE模型属于嵌套马尔可夫模型的过程中，我们还建立了基于ADMG的因果理论框架。最后，我们将NE模型与文献中常用的、将ADMG解释为含隐变量的有向无环图（DAG）这一密切相关但不同的解释进行比较，指出"隐变量DAG"解释在数学上非必需，其隐含的本体论假设较为晦涩，且可能阻碍研究者对重要结构假设的深入考量。