  Supervised learning approaches for causal discovery from observational data
often achieve competitive performance despite seemingly avoiding explicit
assumptions that traditional methods make for identifiability. In this work, we
investigate CSIvA (Ke et al., 2023), a transformer-based model promising to
train on synthetic data and transfer to real data. First, we bridge the gap
with existing identifiability theory and show that constraints on the training
data distribution implicitly define a prior on the test observations.
Consistent with classical approaches, good performance is achieved when we have
a good prior on the test data, and the underlying model is identifiable. At the
same time, we find new trade-offs. Training on datasets generated from
different classes of causal models, unambiguously identifiable in isolation,
improves the test generalization. Performance is still guaranteed, as the
ambiguous cases resulting from the mixture of identifiable causal models are
unlikely to occur (which we formally prove). Overall, our study finds that
amortized causal discovery still needs to obey identifiability theory, but it
also differs from classical methods in how the assumptions are formulated,
trading more reliance on assumptions on the noise type for fewer hypotheses on
the mechanisms.
