I was thinking about causality in terms of forced directional arrows in Bayes nets, rather than in terms of d-separation. I don’t think your example as written is helpful because Bayes nets rely on the independence of variables to do causal inference: is equivalent to .
It’s more important to think about cases like where causality can be inferred. If we change this to by adding noise then we still get a distribution satisfying (as and are still independent).
Even if we did have other nodes forcing (such as a node which is parent to , and another node which is parent to ), then I still don’t think adding noise lets us swap the orders round.
On the other hand, there are certainly issues in Bayes nets of more elements, particularly the “diamond-shaped” net with arrows . Here adding noise does prevent effective temporal inference, since, if and are no longer d-separated by , we cannot prove from correlations alone that no information goes between them through .
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