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 on $U$, 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：贝叶斯网络是一组$n$个随机变量（顶点）上的有向无环图；贝叶斯网络分布是指在该图上满足马尔可夫性质的随机变量概率分布。此类模型的有限$k$混合可通过一个更大的图进行图形化表示，该图包含一个额外的“隐藏”（或“潜”）随机变量$U$（取值范围为$\{1,\ldots,k\}$），并存在从$U$到其他每个顶点的有向边。这类模型是因果推理的基础，其中$U$对多个总体的未观测混杂效应进行建模，从而掩盖了可观测有向无环图中的因果关系。通过求解混合问题并恢复$U$的联合概率分布，传统上不可识别的因果关系变得可识别。借助对空图上更成熟的“乘积”案例的归约，我们首次提出了学习非空有向无环图混合模型的算法。