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Unformatted text preview: 1 Introduction to Probabilistic Graphical Models Lecturer: Eran Segal Problem set 2  Solutions 1. General notes: • Please note that ∑ y P ( X  Y = y ) 6 = P ( X ). Moreover, it can be easily verified that ∑ y P ( X  Y = y ) might not even represent a valid distribution. • Conditional probability P ( X,Y  Z ) is not defined for the case P ( Z ) = 0, but the conditional independence ( X ⊥ Y  Z ) is defined to hold for P ( Z ) = 0. In questions (a) and (b) one need to justify divisions by, or conditioning on, such zero probability cases (which in those cases is pretty much straight forward from definitions). In question (c), one utilizes the positivity of the distribution to justify such manipulations. • Recall that ( X ⊥ Y  Z ) holds if and only if P ( X,Y  Z ) = P ( X  Z ) P ( Y  Z ), but also if and only if P ( X  Z ) = P ( X  Y,Z ) (by definition). (a) Answer: Assume: ( X ⊥ Y,W  Z ) (1) = ⇒ P ( X,Y,W  Z ) = P ( X  Z ) P ( Y,W  Z ) Case analysis: i. Case: P ( Z,W ) = 0 = ⇒ trivial from definition of conditional independence. ii. Case: P ( Z ) = 0 = ⇒ P ( Z,W ) = 0 = ⇒ by case i . iii. Case: P ( Z,W ) ,P ( Z ) 6 = 0 = ⇒ P ( W  Z ) 6 = 0 . P ( X  Z,W ) = P ( X,Z,W ) P ( Z,W ) = P ( X,W  Z ) P ( W  Z ) = ∑ y P ( X,y,W  Z ) P ( W  Z ) By (1): = ∑ y P ( X  Z ) P ( y,W  Z ) P ( W  Z ) = P ( X  Z ) P ( W  Z ) P ( W  Z ) = P ( X  Z ) . (This is in fact the decomposition rule). Hence, P ( X  Z,W ) P ( Y  Z,W ) = P ( X  Z ) P ( Y,W  Z ) P ( W  Z ) By (1): = P ( X,Y,W  Z ) P ( W  Z ) = P ( X,Y  W,Z ) = ⇒ ( X ⊥ Y  Z,W ) 2 (b) Answer: Assume: ( X ⊥ W  Z,Y ) (2) ( X ⊥ Y  Z ) (3) Case analysis: i. Case: P ( Z ) = 0 , or P ( Y,Z ) = 0 , or P ( X,Y,Z ) = 0 = ⇒ simple. ii. Otherwise: P ( X,Y,W  Z ) = P ( X,Y  Z ) P ( W  X,Y,Z ) By (3): = P ( X  Z ) P ( Y  Z ) P ( W  X,Y,Z ) By (2): = P ( X  Z ) P ( Y  Z ) P ( W  Y,Z ) = P ( X  Z ) P ( Y,W  Z ) = ⇒ ( X ⊥ Y,W  Z ) (c) Answer: Assume: ( X...
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This note was uploaded on 02/05/2010 for the course CS CS229 taught by Professor Eransegal during the Fall '07 term at École Normale Supérieure.
 Fall '07
 EranSegal

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