- E. Xing, S. Ma, M. Reddy. 2015. 3: Representation of Undirected Gm. Abstract: A DAG G is a perfect-map (P-map) for a distribution P if I(G) = I(P ). A DAG G is a minimal I-map for P if it is an I-map for P and the removal of any single edge in G would make it no longer an I-map of P. A distribution can have several minimal I-maps. However, not every distribution has a perfect map as a DAG. We can use proof by contradiction to show this. Say we have a distribution over four random variables A, B, C, and D, where A ⊥⊥ C|{B,D} and B ⊥⊥ D|{A,C}. By drawing out all possible DAGs using these four random variables, we can see that none of them can represent both of these conditional independencies. However, the undirected graph below can:
