Bayesian Nets. No active path from X to Y, versus No inactive paths from X to Y

Question

I am learning d-seperation in Bayes nets for my A.I. class.

What this involves is considering all paths from some node X to Y (representing random variables) and seeing whether such paths are active or inactive. By definition, a path is active if there is exist a tuple along the path that is inactive (essentially blocking the propagation of information between X and Y).

I know that if ALL paths are inactive, we can conclude X and Y are independent.

Now there are two other cases. 1) All paths are active 2) Some paths active, some inactive

My question is, does case (1) allow us to make some stronger conclusion about the independence of X and Y, versus case (2). Something like this

1) X and Y are NOT independent 2) X and Y are not necessarily independent.

Answer 1

No, you cannot make a stronger conclusion. That is because Bayes nets are "I-maps" (in Pearl's terminology): d-separation implies independence but independence does not imply d-separation. See section 3.1.3 of Probabilistic Reasoning in Intelligent Systems.