  Recent progress in Neural Causal Models (NCMs) showcased how identification
and partial identification of causal effects can be automatically carried out
via training of neural generative models that respect the constraints encoded
in a given causal graph [Xia et al. 2022, Balazadeh et al. 2022]. However,
formal consistency of these methods has only been proven for the case of
discrete variables or only for linear causal models. In this work, we prove
consistency of partial identification via NCMs in a general setting with both
continuous and categorical variables. Further, our results highlight the impact
of the design of the underlying neural network architecture in terms of depth
and connectivity as well as the importance of applying Lipschitz regularization
in the training phase. In particular, we provide a counterexample showing that
without Lipschitz regularization the NCM may not be asymptotically consistent.
Our results are enabled by new results on the approximability of structural
causal models via neural generative models, together with an analysis of the
sample complexity of the resulting architectures and how that translates into
an error in the constrained optimization problem that defines the partial
identification bounds.
