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ps1add - Queries Outside a Clique February 8 2008 Consider...

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Queries Outside a Clique February 8, 2008 Consider a query P ( Y | e ) where the variables Y are not present together in a single clique. One naive approach is to construct a clique tree where we force one of the cliques to contain Y . However, this approach forces us to tailor our clique tree to different queries, negating many of its advantages. An alternative approach is to perform variable elimination over a calibrated clique tree. Example 0.1: Consider a clique tree obtained from a chain structured network A—B—C—D, with an appropriate set of factors F . Our clique tree in this case would have three cliques C 1 = { A, B } , C 2 = { B, C } , and C 3 = { C, D } . Assume that we have calibrated the clique tree, so that the clique potentials are equal to the marginals of P F as in Corollary 11.2.7. Assume that we now want to compute the probability P F ( B, D ). If the entire clique tree is calibrated, so is any (connected) subtree T . Letting T consist of the two cliques C 2 and C 3 , it follows from Theorem 11.2.11 that: P F ( B, C, D ) = π T .
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