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Unformatted text preview: Probablistic Graphical Models, Spring 2007 Homework 2 solutions December 6, 2007 1 Markov Networks Solution due to Yongjin Park Strong Union: X Y  Z = X Y  Z,W (1) Transitivity: ( X A  Z ) ( A Y  Z ) = ( X Y  Z ) (2) 1. See Figure 1. Figure 1: BN G of which I ( G ) does not satisfy these two properties. 2. The global Markov assumption defines X Y  Z if and only if sep H ( X ; Y  Z ), or there is no other path between X and Y that does not pass through Z . Due to the monotonicity of separation, sep H ( X ; Y  Z ) = se H ( X ; Y  Z ) , (3) where Z Z and { X,Y } / Z . Then it follows that any set of nodes Z s.t. Z Z , X Y  Z = X Y  Z . Since { Z } { Z,W } , the strong union property holds in I ( H ). As for the transitivity, we can prove by showing the contrapositive proposition is correct: A, X Y  Z = X A  Z A Y  Z (4) There can be three cases of A without violating X Y  Z , A belongs to the only path between X and Y where sep H ( X ; Y  Z ). If A is between X and Z then sep H ( X ; A  Z ) but sep H ( A ; Y  Z ); thus A Y  Z . The mirror case is also true. A is connected to either X or Y , but not simultaneously. If a path between X and A , but not between Y and A , sep H ( A ; X  Z ) but sep H ( A ; Y  ). The mirror case is also true. Thus, either X A  Z or A Y  Z . A is not connected with either X or Y at all. Trivially, X A Y A . 1 3. Show I l ( H ) = I p ( H ). Equivalently we can prove by showing that if X Y  N H ( X ), then X Y X { X,Y } , for all disjoint sets X,Y . By the definition of Imap, X Y  N H ( X ) sep H ( X ; Y  N H ( X )) . (5) Again exploiting the monotonicity of d separation (Eq. 3), if N H ( X ) Z , Z dseparates X and Y , i.e. sep H ( X ; Y  Z ). Inductively, adding each node A [ X { X,Y } N H ( X )] to Z , i.e. Z Z { A } , where Z is initially N H ( X ), sep H ( X ; Y  N H ( X )) = = sep H ( X ; Y  Z ) = sep H ( X ; Y  Z ) = = sep H ( X ; Y X { X,Y } ) . (6) In sum, X Y  N H ( X ) = X Y X { X,Y } , and I l ( H ) = I p ( H )....
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This note was uploaded on 05/25/2008 for the course MACHINE LE 10708 taught by Professor Carlosgustin during the Fall '07 term at Carnegie Mellon.
 Fall '07
 CarlosGustin

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