Linear structural causal models (SCMs) are used to express and analyse the relationships between random variables. Direct causal effects are represented as directed edges and confounding factors as bidirected edges. Identifying the causal parameters from correlations between the nodes is an open problem in artificial intelligence. In this paper, we study SCMs whose directed component forms a tree. Van der Zander et al. (AISTATS'22, PLMR 151, pp. 6770--6792, 2022) give a PSPACE-algorithm for the identification problem in this case, which is a significant improvement over the general Gr\"obner basis approach, which has doubly-exponential time complexity in the number of structural parameters. In this work, we present a randomized polynomial-time algorithm, which solves the identification problem for tree-shaped SCMs. For every structural parameter, our algorithms decides whether it is generically identifiable, generically 2-identifiable, or generically unidentifiable. (No other cases can occur.) In the first two cases, it provides one or two fractional affine square root terms of polynomials (FASTPs) for the corresponding parameter, respectively.

翻译：线性结构因果模型（SCM）用于表达和分析随机变量之间的关系。直接因果效应用有向边表示，而混杂因素则用双向边表示。从节点间的相关性识别因果参数是人工智能领域的一个开放问题。本文研究有向分量构成树形结构的SCM。Van der Zander 等人（AISTATS'22, PLMR 151, pp. 6770--6792, 2022）针对此类情况给出了一个PSPACE算法用于识别问题，相较于通用Gröbner基方法（其时间复杂度随结构参数数量呈双指数增长），该方法实现了显著改进。在本工作中，我们提出了一种随机多项式时间算法，解决了树形SCM的识别问题。对于每个结构参数，我们的算法能判定其是否为一般可识别、一般2-可识别或一般不可识别（不存在其他情况）。在前两种情况下，算法分别为相应参数提供一个或两个分式仿射平方根项多项式（FASTPs）。