A Bayesian Network is a directed acyclic graph (DAG) on a set of $n$ random variables (the vertices); a Bayesian Network Distribution (BND) is a probability distribution on the random variables that is Markovian on the graph. A finite $k$-mixture of such models is graphically represented by a larger graph which has an additional ``hidden'' (or ``latent'') random variable $U$, ranging in $\{1,\ldots,k\}$, and a directed edge from $U$ to every other vertex. Models of this type are fundamental to causal inference, where $U$ models an unobserved confounding effect of multiple populations, obscuring the causal relationships in the observable DAG. By solving the mixture problem and recovering the joint probability distribution with $U$, traditionally unidentifiable causal relationships become identifiable. Using a reduction to the more well-studied ``product'' case on empty graphs, we give the first algorithm to learn mixtures of non-empty DAGs.

翻译：贝叶斯网络是定义在 $n$ 个随机变量（顶点）上的有向无环图（DAG）；贝叶斯网络分布（BND）是该随机变量集合上满足图马尔可夫性的概率分布。此类模型的有限 $k$-混合在图形上表现为一个更大的图，其中包含一个取值范围为 $\{1,\ldots,k\}$ 的附加"隐藏"（或"潜"）随机变量 $U$，且存在从 $U$ 指向其他每个顶点的有向边。这类模型是因果推断的基础，其中 $U$ 模拟了多个总体中未被观测的混杂效应，从而掩盖了可观测 DAG 中的因果关系。通过求解混合问题并恢复包含 $U$ 的联合概率分布，传统上不可识别的因果关系变得可识别。我们利用向空图上更成熟的"乘积"情形的归约，给出了首个学习非空 DAG 混合模型的算法。