Parental Markov Condition Example I'm currently reading a text on Bayesian networks and the text is giving some very crude interpretations of what appear to be some of the most important foundations of the subject.
It states the following:

Theorem 1.2.7 (Parental Markov Condition):
  A necessary and sufficient condition for a probability distribution P to be Markov relative a [directional acyclic graph] G is that every variable be independent of all it's nondescendants (in G), conditional on it's parents.

I understand the premise of what is happening here. Given some point X in the graph, if you condition over all parent nodes, it should be independent of all non-descendants. I guess the confusion for me arrises around how this makes the process Markov relative. Maybe that is the weak part of my understanding. Could someone please provide a example of graph which fails this condition and hence fails being Markov relative to this graph?
I've done some study on Markov Chains and I feel like maybe their definitions differ in some way that's been lost on me.
Thanks
 A: I'm not really an expert on the domains so I may say some stupid things. Sorry in advance.
Up to my understanding the difference with Markov chains is that in Markov chains you have a sequence of random variable $(X_i)$ where the $X_i$ can take as value a state of your Markov chain $X_i$ representing the probability distribution to be in a state at step $i$.
But here the random variables are associated to a certain state. Each random variable represent the probability that the variable take a certain  value and the graph represent the link between this variable.
For example, suppose that the sky is either sunny or cloudy, that the weather is either dry or rainy at that I'm either dry or wet.
Let $S$ be the random variable associated to the sky $W$ to the weather and $M$ to me. And let P be the probability distribution such that $P(s,r,\_)=0$ $P(\_,d,w)=0$ $P(s,d,d)=1/2$ $P(c,d,d)=1/4$a and $P(c,r,w)=1/4$. (i.e there is probability 1/2 to be cloudy and if it's cloudy there is 1/2 chance that it's rainy and if it's rainy I'm wet).
The graph $G:$ $S\to W\to M$ represent that the weather depend only on the sky and the fact that I'm wet depend only if it's rainy or not  And thus P is Markov relative to G.
But if you consider the graph $G':$ $M\to S \to W$ P is not Markov relative to G since the probability that I'm wet is not independent of $W$.
I hope it's clearer now ...
