In this work, we consider the identifiability assumption of Gaussian linear structural equation models (SEMs) in which each variable is determined by a linear function of its parents plus normally distributed error. It has been shown that linear Gaussian structural equation models are fully identifiable if all error variances are the same or known. Hence, this work proves the identifiability of Gaussian SEMs with both homogeneous and heterogeneous unknown error variances. Our new identifiability assumption exploits not only error variances, but edge weights; hence, it is strictly milder than prior work on the identifiability result. We further provide a structure learning algorithm that is statistically consistent and computationally feasible, based on our new assumption. The proposed algorithm assumes that all relevant variables are observed, while it does not assume causal minimality and faithfulness. We verify our theoretical findings through simulations and real multivariate data, and compare our algorithm to state-of-the-art PC, GES and GDS algorithms.

中文翻译：

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具有均质和异质误差方差的高斯线性结构方程模型的可识别性

在这项工作中，我们考虑了高斯线性结构方程模型（SEM）的可识别性假设，其中每个变量均由其父项的线性函数加上正态分布误差确定。已经表明，如果所有误差方差相同或已知，则线性高斯结构方程模型是完全可识别的。因此，这项工作证明了具有均质和异质未知误差方差的高斯SEM的可识别性。我们新的可识别性假设不仅利用误差方差，还利用边缘权重。因此，它比可识别性结果以前的工作要严格得多。基于我们的新假设，我们进一步提供了一种统计上一致且计算上可行的结构学习算法。提出的算法假设所有相关变量均已观察到，虽然它不假定因果关系最小化和忠诚。我们通过仿真和真实的多元数据验证了我们的理论发现，并将我们的算法与最新的PC，GES和GDS算法进行了比较。