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.

通过干预来学习混合因果模型中变量之间的因果关系是一项具有挑战性的任务，本文提出了匹配性的必要和充分条件以及一种自适应算法，用于学习混合因果模型中的所有真实边，具有最佳干预效果并在混合模型不包含循环关系时尺寸最小。

混合有向无环图中的干预因果发现

Interventional Causal Discovery in a Mixture of DAGs

Inferring causal interactions from observed data is a challenging problem,
especially in the presence of measurement noise. To alleviate the problem of
spurious causality, Haufe et al. (2013) proposed to contrast measures of
information flow obtained on the original data against the same measures
obtained on time-reversed data. They show that this procedure, time-reversed
Granger causality (TRGC), robustly rejects causal interpretations on mixtures
of independent signals. While promising results have been achieved in
simulations, it was so far unknown whether time reversal leads to valid
measures of information flow in the presence of true interaction. Here we prove
that, for linear finite-order autoregressive processes with unidirectional
information flow, the application of time reversal for testing Granger
causality indeed leads to correct estimates of information flow and its
directionality. Using simulations, we further show that TRGC is able to infer
correct directionality with similar statistical power as the net Granger
causality between two variables, while being much more robust to the presence
of measurement noise.

通过时间反转 Granger 因果检验可以准确地估计信息流及其方向性，并且相比于正向的 Granger 因果性具有更好的噪声容忍度。