  Recent advances in probabilistic generative modeling have motivated learning
Structural Causal Models (SCM) from observational datasets using deep
conditional generative models, also known as Deep Structural Causal Models
(DSCM). If successful, DSCMs can be utilized for causal estimation tasks, e.g.,
for answering counterfactual queries. In this work, we warn practitioners about
non-identifiability of counterfactual inference from observational data, even
in the absence of unobserved confounding and assuming known causal structure.
We prove counterfactual identifiability of monotonic generation mechanisms with
single dimensional exogenous variables. For general generation mechanisms with
multi-dimensional exogenous variables, we provide an impossibility result for
counterfactual identifiability, motivating the need for parametric assumptions.
As a practical approach, we propose a method for estimating worst-case errors
of learned DSCMs' counterfactual predictions. The size of this error can be an
essential metric for deciding whether or not DSCMs are a viable approach for
counterfactual inference in a specific problem setting. In evaluation, our
method confirms negligible counterfactual errors for an identifiable SCM from
prior work, and also provides informative error bounds on counterfactual errors
for a non-identifiable synthetic SCM.
