2 x y z and x y w z imply x w z 1 3 x y w z and y w z

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2. ( X Y | Z ) and ( X, Y W | Z ) imply ( X W | Z ). 1
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3. ( X Y, W | Z ) and ( Y W | Z ) imply ( X, W Y | Z ). 1.4 [3 pts] Provide an example of a distribution P ( X 1 , X 2 , X 3 ) where for each i ̸ = j , we have that ( X i X j ) I ( P ), but we also have that ( X 1 , X 2 X 3) / I ( P ). 1.5 [8 pts] Figure 1: Graphical Model for Prob. 1.5 Let X , Y , Z be binary random variables with joint distribution given by the graphical model shown above ( v -structure). We define the following shorthands: a = P ( X = t ); b = P ( X = t | Z = t ); c = P ( X = t, | Z = t, Y = t ) 1. For all the following cases, provide examples of conditional probability tables (CPTs) (and compute the quantities, a , b , c ), which make the statements true: (a) a > c (b) a < c < b (c) b < a < c 2. Think of X , Y as causes and Z as a common effect, and for all the above cases summarize (in a sentence or two) why the statements are true for your examples. (Hint: Think about positive and negative correlations along edges) 2 Graph independencies [12pt] 2.1 [4 pts] Let X = { X 1 , ..., X n } be a random vector with distribution given by the graphical model in Figure 2. Consider variable X 1 . What is the minimal subset of the variables, A ⊆ X -{ X 1 } , such that X 1 is independent of the rest of the variables, X - A ∪ { X 1 } , givan A ? Justify your answer.
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