I was trying to form an example where I had 3 r.v.s such that the distribution describing them had more conditional independencies or independencies than the directed graphical model corresponding to it.
I was told that if we let $X_1, X_2$ be two independent fair coin flips (WLOG 1 indicating heads, 0 indicating tails) and then define $X_3 = X_1 \oplus X_2$ i.e. be the XOR of the result of the coins. Consider its directed graphical model:
I believe that the directed graphical model has less (conditional and marginal) independencies than its distribution. Does someone understand why is this case? Can someone provide me with a rigorous explanation of why this example works? If this is not the case, can someone provide me with a different example? Its ok if it has more than 3 r.vs. If it only involves fair coins that would be even better!
In general, how do you form this kind of example? What is the intuition behind it? Its not 100% intuitive for me why this should work or why its obvious or how do you come up with it.