I read an argument about variables $A,B,C,D$ that are not mutually independent. It supposed the existence of $P(A,B,C,D)$ where $A \perp B \mid \{C,D\}$ and $C\perp D \mid \{A,B\}$ (with $\perp$ denoting independence). But I'm not sure if there is such a distribution or how I would go about finding one. I tried coming up with some examples, but even in the simple binary case there are 16 values I need to decide on, and it's a bit overwhelming.

How would I know if there's a distribution like this? How could I come up with one?

  • $\begingroup$ If (A,C)⊥(B,D) this should work... $\endgroup$ – Xi'an Feb 3 '19 at 10:21
  • $\begingroup$ @Xi'an That's a good point; is there any example that's not as trivial? $\endgroup$ – Hatshepsut Feb 3 '19 at 20:17

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