Causal interactions among a group of variables are often modeled by a single causal graph. In some domains, however, these interactions are best described by multiple co-existing causal graphs, e.g., in dynamical systems or genomics. This paper addresses the hitherto unknown role of interventions in learning causal interactions among variables governed by a mixture of causal systems, each modeled by one directed acyclic graph (DAG). Causal discovery from mixtures is fundamentally more challenging than single-DAG causal discovery. Two major difficulties stem from (i) inherent uncertainty about the skeletons of the component DAGs that constitute the mixture and (ii) possibly cyclic relationships across these component DAGs. This paper addresses these challenges and aims to identify edges that exist in at least one component DAG of the mixture, referred to as true edges. First, it establishes matching necessary and sufficient conditions on the size of interventions required to identify the true edges. Next, guided by the necessity results, an adaptive algorithm is designed that learns all true edges using ${\cal O}(n^2)$ interventions, where $n$ is the number of nodes. Remarkably, the size of the interventions is optimal if the underlying mixture model does not contain cycles across its components. More generally, the gap between the intervention size used by the algorithm and the optimal size is quantified. It is shown to be bounded by the cyclic complexity number of the mixture model, defined as the size of the minimal intervention that can break the cycles in the mixture, which is upper bounded by the number of cycles among the ancestors of a node.

翻译：变量间的因果交互通常由单一因果图建模。然而，在某些领域中，这些交互最好由多个共存的因果图描述，例如在动态系统或基因组学中。本文探讨了干预在由混合因果系统（每个系统由一个独立的有向无环图建模）所控制的变量间因果交互学习中的、迄今未知的作用。从混合模型中学习因果发现本质上比单一有向无环图的因果发现更具挑战性。两大主要困难源于：（i）构成混合模型的各组分有向无环图骨架的固有不确定性，以及（ii）这些组分有向无环图之间可能存在的循环关系。本文应对这些挑战，旨在识别在混合模型中至少一个组分有向无环图中存在的边，即真实边。首先，本文建立了识别真实边所需干预规模相匹配的必要与充分条件。随后，基于必要性结果的指导，设计了一种自适应算法，该算法使用 ${\cal O}(n^2)$ 次干预学习所有真实边，其中 $n$ 为节点数。值得注意的是，若底层混合模型在其组分间不包含循环，则所用干预规模是最优的。更一般地，本文量化了算法所用干预规模与最优规模之间的差距。该差距被证明受限于混合模型的循环复杂度数，其定义为能够打破混合模型中循环的最小干预规模，该数值上界为节点祖先间循环的数量。