Robust Identifiability in Linear Structural Equation Models of Causal Inference

Karthik Abinav Sankararaman, Anand Louis, Navin Goyal

Abstract: We consider the problem of robust parameter estimation from observational data in the context of linear structural equation models (LSEMs). Under various conditions on LSEMs and the model parameters the prior work provides efficient algorithms to recover the parameters. However, these results are often about generic identifiability. In practice, generic identifiability is not sufficient and we need robust identifiability: small changes in the observational data should not affect the parameters by a huge amount. Robust identifiability has received far less attention and remains poorly understood. Sankararaman et al. (2019) recently provided a set of sufficient conditions on parameters under which robust identifiability is feasible. However, a limitation of their work is that their results only apply to a small sub-class of LSEMs, called ``bow-free paths.'' In this work, we show that for \emph{any} ``bow-free model'', in all but $\frac{1}{\poly(n)}$-measure of instances \emph{robust identifiability} holds. Moreover, whenever an instance is robustly identifiable, the algorithm proposed in Foygel et al., (2012) can be used to recover the parameters in a robust fashion. In contrast, for generic identifiability Foygel et al., (2012) proved that with measure $1$, instances are generically identifiable. Thus, we show that robust identifiability is a \emph{strictly} harder problem than generic identifiability. Finally, we validate our results on both simulated and real-world datasets.