I known that for give $x,y$, if we have
Then we call $x,y$ are independent.
For marginal independence, I found the definition here.
Random variable $x$ is marginal independent of random variable $y$ if: $$p(x|y)=p(x)$$
However, I cannot see the difference between them.
The question come to my mind when I want to figure out a question about Bayesian network. In 3-way Bayesian network, there are three nodes A,B,C in a common parent struct.
A | | B C
A is the parent of B and C. It says B and C are conditional independent given A, and I can see it because:
But, when the value of A is unknown, why $B,C$ are not independent?