# Interventional Causal Discovery in a Mixture of DAGs

CoRR（2024）

Abstract

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 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.

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