Confusion about the concept of Bayesian Networks

Question

I read a lot about Bayesian networks, including focused literature. However, what I have not yet understood properly is this:

What is the use case of Bayesian networks that contain of more than 2 levels? Suppose I have a graph A -> X; B -> X. I am interested in X. Then I can set up a model with conditional probabilities P(X|A,B). That's interesting. Now suppose there is also C -> A. If I want to find P(X|A,B), I never need to know C. So when would that relationship matter?

In a more general sense: The Markov condition says that only the parents are relevant for the probability of a node. If I want to train a classifier to learn some node's conditional probability, wouldn't it be sufficient to just track its parents? What to do with the rest of the network? Thus, wouldn't I always end up with a 2-layered net, one layer for the inputs and one for the targeted value that is to be trained?

Answer 1

Of course when the values of A and B are known, the value of C doesn't give information anymore about A. The value of C becomes useful in the case value of A is unknown. That is the value of C tells some info about the value of A and further, tells a little bit about X. It can be said there's an active trail from C to X through A. If A is known the trail becomes inactive. And another note is that trails in Bayesian networks behave very differently from those in Markov networks.