# Figure 1 the new marginalized bayes net 42 the

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Figure 1: The new marginalized Bayes net 4.2 The procedure is as follows: 1) Remove node X i 2) For each parent of X i , create an edge to each child of X i 3) Number each child of X i from k = 1 . . . n where n is the number of children of X i . For each k , create an edge from child X k to children X k + 1 . . . X n . Also, create an edge from each parent of X k to the children X k +1 . . . X n . (If we don’t add these extra edges we create an independency not in the original graph). We can see why this works by actually performing the marginization. Consider 3-node subgraphs: 7
X W Y where we want to remove W: w P ( X, W, Y ) = w P ( X ) P ( W | X ) P ( Y | W ) = P ( X ) w P ( W | X ) P ( Y | W, X ) = P ( X ) w P ( Y, W | X ) = P ( X ) P ( Y | X ) which implies that X and Y are not necessarily independent, meaning we need an edge between X and Y . For X W Y where we want to remove W: w P ( X, W, Y ) = w P ( W ) P ( X | W ) P ( Y | W ) = w P ( W, X ) P ( Y | W, X ) = w P ( Y, W, X ) = P ( X ) P ( Y | X ) Again, this implies that X and Y are not necessarily independent, meaning we need an edge between X and Y . For X W Y where we want to remove W: w P ( X, W, Y ) = w P ( X ) P ( Y ) P ( W | X, Y ) = P ( X ) P ( Y ) w P ( W | X, Y ) = P ( X ) P ( Y )(1) = P ( X ) P ( Y ) X and Y are marginally independent before marginalizing W , and remain so after marginal- ization. One other thing to consider is the case N X W Y . If we condition on X , then N is not necessarily independent of Y . But after we marginalize W we have N X Y . In this case, if we condition on X then N is independent of Y . Therefore, we need an edge between N and Y otherwise we introduce an extra independence not in the original graph. 8
5 Bayesian Network Inference 5.1 Given that r 2 and r 3 have been observed the nodes that are d-connected to (influenced by) r 1 are the following: u 1 .Age, u 1 .Gender, b 1 .Genre, b 2 .Genre, u 2 .Age, u 2 .Gender, r 4 , r 5 . Aside : In the na¨ ıve Bayes model one rating is independent of another rating given Θ NB . In the elaborate model two ratings can be dependent given Θ EL . In this question, knowing r 1 influences r 4 and r 5 , even if Θ EL is given to you. This is the basic motivation for template models in relational learning. 5.2 The answers to the three na¨ ıve Bayes queries are 0 . 7716, 0 . 3719, and 0 . 5815. The answers to the three elaborate model queries are 0 . 2778, 0 . 1800, and 0 . 6654.

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