Indeed, If X is independent of both Y and X xor Y, that violates the compositional semigraphiod axioms (assuming X is nondeterministic.) Although it could still happen e.g. in the uniform distribution on X x Y. In the example in the post, I mean for X to be independent of X xor Y and for X to not be independent of Y.
Or wait, I’m dumb, that can definitely happen if X and Y are coin flips. But I feel like this doesn’t add up with the other stuff, will need to read more carefully.
I’m a bit confused about how X can be independent of both Y and of (X xor Y). What would a probability distribution where this holds look like?
Indeed, If X is independent of both Y and X xor Y, that violates the compositional semigraphiod axioms (assuming X is nondeterministic.) Although it could still happen e.g. in the uniform distribution on X x Y. In the example in the post, I mean for X to be independent of X xor Y and for X to not be independent of Y.
I think one thing that confuses me is, wouldn’t Y also be before X then?
Nope, we have X⊥X⊕Y, but not Y⊥X⊕Y. That breaks the symmetry.
Ah of course! So many symbols to keep track of 😅
Or wait, I’m dumb, that can definitely happen if X and Y are coin flips. But I feel like this doesn’t add up with the other stuff, will need to read more carefully.