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Vaniver comments on Causal Diagrams and Causal Models - Less Wrong
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Fascinating; thanks for the papers! Those look like they describe continuous and discrete distributions; does my statement hold for binary variables?
Aren't binary variables a discrete distribution?
Yes, but they contain less information. Check out figure 2 of the Peters paper (which describes discrete distributions). If you have an additive noise model, so Y is X plus noise, then by looking at the joint pdf you can distinguish between X causing Y and Y causing X by the corners. This doesn't seem possible if X and Y can only have 2 values (since you get a square, not a trapezoid).