  In this work, we consider the problem of robust parameter estimation from
observational data in the context of linear structural equation models (LSEMs).
LSEMs are a popular and well-studied class of models for inferring causality in
the natural and social sciences. One of the main problems related to LSEMs is
to recover the model parameters from the observational data. 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 significantly extend their work along
multiple dimensions. First, for a large and well-studied class of LSEMs, namely
``bow free'' models, we provide a sufficient condition on model parameters
under which robust identifiability holds, thereby removing the restriction of
paths required by prior work. We then show that this sufficient condition holds
with high probability which implies that for a large set of parameters robust
identifiability holds and that for such parameters, existing algorithms already
achieve robust identifiability. Finally, we validate our results on both
simulated and real-world datasets.
