Generative models can be trained to emulate complex empirical data, but are they useful to make predictions in the context of previously unobserved environments? An intuitive idea to promote such extrapolation capabilities is to have the architecture of such model reflect a causal graph of the true data generating process, to intervene on each node of this graph independently of the others. However, the nodes of this graph are usually unobserved, leading to a lack of identifiability of the causal structure. We develop a theoretical framework to address this challenging situation by defining a weaker form of identifiability, based on the principle of independence of mechanisms.