Thread: I and D-Maps of independence relations

1. I and D-Maps of independence relations

By defintion the (closed) graph G= (V, V x V) over a set of statistical variabeles is an I-map for any independence relation over V.
I find this counter intuitive, take for example the Independence relation over V defined by the statements $\displaystyle I(\{V_1\}, \{V_2, V_3\}, \{V_4\})$ and $\displaystyle I(\{V_2\}, \{V_1, V_4\}, \{V_3\})$. This tells me V1 and V4 are independent given V2 & V3 and that V2 is independent of V3 given V1 and V4. This would correspond to the following separation statements: $\displaystyle <\{V_1\} | \{V_2, V_3\} | \{V_4\}>_G$ and $\displaystyle <\{V_2\} | \{V_1, V_4\} | \{V_3\}>_G$.

But in the completed graph there's an edge from V1 to V4 and from V2 to V3. And so basically there exist 4 I-Maps for this indepence relation, the combinations 'between' the closed graph (V, VxV) and the graph in which there is no edge from V1 to V4 and V2 to V3.
In my opinion this does not correspond to the separation/independence statements. So clearly my intuition is lacking. Could anyone give some more clarification on this?

And as a sidemark, is anyone aware of some nice (online) tool to draw graphs?