Stability of Linear Structural Equation Models of Causal Inference

We consider numerical stability of the parameter recovery problem in Linear Structural Equation Model (LSEM) of causal inference. Numerical stability is essential for the recovered parameters to be reliable. A long line of work starting from Wright (1920) has focused on understanding which sub-classes of LSEM allow for efficient parameter recovery. Despite decades of study, this question is not yet fully resolved. The goal of the present paper is complementary to this line of work: we want to understand the stability of the recovery problem in the cases when efficient recovery is possible. Numerical stability of Pearl's notion of causality was first studied in Schulman and Srivastava (2016) using the concept of condition number where they provide ill-conditioned examples. In this work, we provide a condition number analysis for the LSEM. First we prove that under a sufficient condition, for a certain sub-class of LSEM that are bow free (Brito and Pearl (2002)), parameter recovery is numerically stable. We further prove that randomly chosen input parameters for this family satisfy the condition with a substantial probability. Hence for this family, on a large subset of parameter space, recovery is stable. Next we construct an example of LSEM on four vertices with unbounded condition number. We then corroborate our theoretical findings via simulations as well as real-world experiments for a sociology application. Finally, we provide a general heuristic for estimating the condition number of any LSEM instance.