  Discarding undesirable measurement results in Bell experiments opens the
detection loophole that prevents a conclusive demonstration of nonlocality. As
closing the detection loophole represents a major technical challenge for many
practical Bell experiments, it is customary to assume the so-called fair
sampling assumption (FSA) that, in its original form, states that the
collectively postselected statistics are a fair sample of the ideal statistics.
Here, we analyze the FSA from the viewpoint of causal inference: We derive a
causal structure that must be present in any causal model that faithfully
encapsulates the FSA. This provides an easy, intuitive, and unifying approach
that includes different accepted forms of the FSA and underlines what is really
assumed when using the FSA. We then show that the FSA can not only be applied
in scenarios with non-ideal detectors or transmission losses, but also in ideal
experiments where only parts of the correlations are postselected, e.g., when
the particles' destinations are in a superposition state. Finally, we
demonstrate that the FSA is also applicable in multipartite scenarios that test
for (genuine) multipartite nonlocality.
