Directed acyclic graph (DAG) models, also called Bayesian networks, impose
conditional independence constraints on a multivariate probability
distribution, and are widely used in probabilistic reasoning, machine learning
and causal inference. If latent variables are included in such a model, then
the set of possible marginal distributions over the remaining (observed)
variables is generally complex, and not represented by any DAG. Larger classes
of mixed graphical models, which use multiple edge types, have been introduced
to overcome this; however, these classes do not represent all the models which
can arise as margins of DAGs. In this paper we show that this is because
ordinary mixed graphs are fundamentally insufficiently rich to capture the
variety of marginal models.
We introduce a new class of hyper-graphs, called mDAGs, and a latent
projection operation to obtain an mDAG from the margin of a DAG. We show that
each distinct marginal of a DAG model is represented by at least one mDAG, and
provide graphical results towards characterizing when two such marginal models
are the same. Finally we show that mDAGs correctly capture the marginal
structure of causally-interpreted DAGs under interventions on the observed
variables.

介绍了一种新的超级图类 mDAGs，通过潜在投影操作从 DAG 的边缘获取 mDAG，每个独特的 DAG 模型边缘至少由一个 mDAG 表示，并提供了图形结果以表征两个边缘模型何时相同。最后，证明了 mDAGs 可以在观察变量干预下正确地捕捉 DAG 的边缘结构。

贝叶斯网络边缘图

Graphs for margins of Bayesian networks

We describe computationally efficient methods for learning mixtures in which
each component is a directed acyclic graphical model (mixtures of DAGs or
MDAGs). We argue that simple search-and-score algorithms are infeasible for a
variety of problems, and introduce a feasible approach in which parameter and
structure search is interleaved and expected data is treated as real data. Our
approach can be viewed as a combination of (1) the Cheeseman--Stutz asymptotic
approximation for model posterior probability and (2) the
Expectation--Maximization algorithm. We evaluate our procedure for selecting
among MDAGs on synthetic and real examples.

本文通过 Cheeseman-Stutz 渐近逼近和 Expectation-Maximization 算法的结合，提出了一种有效的方法来学习混合有向无环图模型（MDAGs），并探讨了参数和模型结构的搜索问题。