# Conditional independence of four variables

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?

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