# Markov condition on collider

I am studying Bayesian Networks using the Neapolitan book (Learning Bayesian Networks). In section 1.3.2 it is stated the following:

Definition 1.9 Suppose we have a joint probability distribution $$P$$ of the random variables in some set $$V$$ and a DAG $$G = (V, E)$$. We say that $$(G, P )$$ satisfies the Markov condition if for each variable $$X \in V$$, $${X}$$ is conditionally independent of the set of all its non-descendants given the set of all its parents. Using the notation established in Section 1.1.4, this means if we denote the sets of parents and non-descendants of $$X$$ by $$PA_X$$ and $$ND_X$$ respectively, then $$I_P ({X}, ND_X |PA_X )$$.

Then he proceeds to give some examples and he gives four simple cases:

1. $$V \gets C \to S$$
2. $$V \to C \to S$$
3. $$V \gets C \gets S$$
4. $$V \to C \gets S$$

i.e. the final item is the collider. When talking about the examples, the author says:

$$(G, P)$$ satisfies the Markov condition if $$G$$ is the DAG in Figure 1.3 (1), (2), or (3). However, $$(G, P)$$ does not satisfy the Markov condition if $$G$$ is the DAG in Figure 1.3 (4) because $$I_P ({V}, {S})$$ is not the case."

And here I could not understand. He is stating that in the collider, the parents are not independent. However, to my knowledge, the parents in the collider are indeed independent and are d-connected conditional to the common descendent. It is clear that $$p(v,c,s) = p(c|v,s)p(v)p(s)$$. This implies that $$V$$ and $$S$$ are in fact independent. So, my question is: why the graph in (4) does not obey the Markov condition as said by the author?

• Does this answer your question? Graphical dependence in the DAG X->Z<-Y Apr 9, 2021 at 0:13
• Partially. I am updating my question to be more specific. Apr 9, 2021 at 2:01
• It sounds like a mixup about the difference between 'independent' and 'conditionally independent'. Apr 9, 2021 at 13:35