The ability to conduct interventions plays a pivotal role in learning causal relationships among variables, thus facilitating applications across diverse scientific disciplines such as genomics, economics, and machine learning. However, in many instances within these applications, the process of generating interventional data is subject to noise: rather than data being sampled directly from the intended interventional distribution, interventions often yield data sampled from a blend of both intended and unintended interventional distributions. We consider the fundamental challenge of disentangling mixed interventional and observational data within linear Structural Equation Models (SEMs) with Gaussian additive noise without the knowledge of the true causal graph. We demonstrate that conducting interventions, whether do or soft, yields distributions with sufficient diversity and properties conducive to efficiently recovering each component within the mixture. Furthermore, we establish that the sample complexity required to disentangle mixed data inversely correlates with the extent of change induced by an intervention in the equations governing the affected variable values. As a result, the causal graph can be identified up to its interventional Markov Equivalence Class, similar to scenarios where no noise influences the generation of interventional data. We further support our theoretical findings by conducting simulations wherein we perform causal discovery from such mixed data.

本研究解决了从混合的干预和观察数据中剥离因果关系的挑战，特别是在存在高斯加性噪声的线性结构方程模型中。本文提出了一种新颖的方法，通过干预数据的多样性，有效恢复混合数据中的各个组成部分，并表明所需样本复杂性与干预对变量值变化的程度成反比，从而帮助识别因果图的干预马尔可夫等价类。

学习未知因果干预的混合模型