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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 $V = \{V_1, V_2, V_3, V_4 \}$, a set of statistical variables. But in the completed graph there's an edge from V1 to V4 and from V2 to V3. I find this counter intuitive, take for example the Independence relation over V defined by the statements $I(\{V_1\},\{V_2,V_3\},\{V_4\})$ and $I(\{V_2\},\{V_1,V_4\},\{V_3\})$ . 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: $<\{V_1\}|\{V_2,V_3\}|\{V_4\}>$ and $<\{V_2\}|\{V_1,V_4\}|\{V_3\}>$ .

So basically there exist 4 I-Maps for this indepence relation, the combinations 'between' the closed graph (V, VxV) and the graphs 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? Some good explanations on d-separation, I-maps and D-Maps would be greatly appreciated as well.

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What are $V_1, V_2, V_3$, and $V_4$? –  Qiaochu Yuan Oct 2 '12 at 6:21
    
Statistical variables –  fuaaark Oct 2 '12 at 6:28

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