Causal discovery from data: the problem with “unfaithful” structural models

While I have shown a hypothetical example of a simple causal relationship that definitely cannot be learnt from data, here I present a more serious example that demonstrates a very common real world problem where pure data driven learning will fail. We know taking the contraceptive pill reduce the probability of pregnancy and pregnancy can cause Thrombosis. But taking the pill can cause thrombosis. Whereas in previous examples a confounding variable introduces a spurious relationship wrongly assumed to be causal, the danger here is the opposite. We might conclude from data alone that there is no relationship between taking the pill and thrombosis because of the confounding effect of pregnancy