Half-trek criterion for identifiability of latent variable models
Half-trek criterion for identifiability of latent variable models
复制标题
DOI：
10.1214/22-aos2221
复制
发表时间：
2022-01
期刊：
影响因子：
--
通讯作者：
R. Barber;M. Drton;Nils Sturma;Luca Weihs
中科院分区：
文献类型：
--
作者：
R. Barber;M. Drton;Nils Sturma;Luca Weihs
我们考虑线性结构方程模型与潜在的变量，并开发了一个标准，以证明是否直接的因果关系之间的可观察变量是可识别的基础上观察到的协方差矩阵。线性结构方程模型假设观测变量和潜变量都求解一个具有随机噪声项的线性方程组。每个模型对应于一个有向图，其边表示作为方程系统中的系数出现的直接效应。先前的研究已经开发了各种方法来确定潜在投影框架中的直接效应的可识别性，其中潜在变量的混杂效应由噪声项之间的相关性表示。这种方法是有效的，当混杂是稀疏的，只影响观察变量的一小部分。相比之下，我们在本文中开发的新的潜在因素half-trek标准（LF-HTC）在原始的未投影的潜变量模型上运行，并且能够证明设置中的可识别性，其中一些潜变量也可能对许多甚至所有的可观测量产生密集的影响。我们的LF-HTC是一个有效的合理的可识别性的充分标准，在此条件下，直接影响可以唯一地恢复为所观察的随机变量的联合协方差矩阵的有理函数。当限制LF-HTC中的搜索步骤以考虑有界大小的潜变量的子集时，可以在时间上验证该准则在图的大小上是多项式的。
We consider linear structural equation models with latent variables and develop a criterion to certify whether the direct causal effects between the observable variables are identifiable based on the observed covariance matrix. Linear structural equation models assume that both observed and latent variables solve a linear equation system featuring stochastic noise terms. Each model corresponds to a directed graph whose edges represent the direct effects that appear as coefficients in the equation system. Prior research has developed a variety of methods to decide identifiability of direct effects in a latent projection framework, in which the confounding effects of the latent variables are represented by correlation among noise terms. This approach is effective when the confounding is sparse and effects only small subsets of the observed variables. In contrast, the new latent-factor half-trek criterion (LF-HTC) we develop in this paper operates on the original unprojected latent variable model and is able to certify identifiability in settings, where some latent variables may also have dense effects on many or even all of the observables. Our LF-HTC is an effective sufficient criterion for rational identifiability, under which the direct effects can be uniquely recovered as rational functions of the joint covariance matrix of the observed random variables. When restricting the search steps in LF-HTC to consider subsets of latent variables of bounded size, the criterion can be verified in time that is polynomial in the size of the graph.