Thanks for the feedback. There’s a condition which I assumed when writing this which I have realized is much stronger than I originally thought, and I think I should’ve devoted more time to thinking about its implications.
When I mentioned “no information being lost”, what I meant is that in the interaction A→B, each value b∈B (where B is the domain of PB) corresponds to only one value of a∈A. In terms of FFS, this means that each variable must be the maximally fine partition of the base set which is possible with that variable’s set of factors.
Under these conditions, I am pretty sure that A⊥C⟹A⊥C|B
Thanks for the feedback. There’s a condition which I assumed when writing this which I have realized is much stronger than I originally thought, and I think I should’ve devoted more time to thinking about its implications.
When I mentioned “no information being lost”, what I meant is that in the interaction A→B, each value b∈B (where B is the domain of PB) corresponds to only one value of a∈A. In terms of FFS, this means that each variable must be the maximally fine partition of the base set which is possible with that variable’s set of factors.
Under these conditions, I am pretty sure that A⊥C⟹A⊥C|B