Acyclic directed mixed graphs, also known as semi-Markov models represent the
conditional independence structure induced on an observed margin by a DAG model
with latent variables. In this paper we present a factorization criterion for
these models that is equivalent to the global Markov property given by (the
natural extension of) d-separation.

本研究提出了半马尔可夫模型的因子化准则，该准则等同于（自然扩展的）d - 分离给出的全局马尔可夫性质，以表示由具有潜在变量的 DAG 模型诱导的观察边际的条件独立结构。

无环有向混合图的分解准则

A factorization criterion for acyclic directed mixed graphs

Acyclic directed mixed graphs (ADMGs) are graphs that contain directed
($\rightarrow$) and bidirected ($\leftrightarrow$) edges, subject to the
constraint that there are no cycles of directed edges. Such graphs may be used
to represent the conditional independence structure induced by a DAG model
containing hidden variables on its observed margin. The Markovian model
associated with an ADMG is simply the set of distributions obeying the global
Markov property, given via a simple path criterion (m-separation). We first
present a factorization criterion characterizing the Markovian model that
generalizes the well-known recursive factorization for DAGs. For the case of
finite discrete random variables, we also provide a parameterization of the
model in terms of simple conditional probabilities, and characterize its
variation dependence. We show that the induced models are smooth. Consequently,
Markovian ADMG models for discrete variables are curved exponential families of
distributions.

研究了无向边和有向边混合，并且无向边形成无环图的 acyclic directed mixed graphs (ADMGs) 及其概率模型，提出了一种基于简单路径判据的全局马尔科夫性标准 ——m-separation，并得到了该模型的因子分解准则及离散随机变量的参数化表示，验证了该模型的光滑性，从而证明了马尔科夫性 ADMG 模型是曲率指数族。