P gf f t p f f t 426 3420 3725 920 41 3 all

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P ( G,F F = T ) P ( F F = T ) = 0 . 426 × 0 . 342+0 . 3725 × 0 . 278 0 . 62 = 0 . 145692+0 . 103555 0 . 62 0 . 402 (41) 3. All descendants of G are irrelevant for this query, because their joint conditional probability sums to one. P ( G = T ) = T F,F F P ( G = T | TF, FF ) P ( FF | TF ) P ( TF ) = T F P ( TF ) F F P ( G | TF, FF ) P ( FF | TF ) = 0 . 1 × (0 . 9 × 0 . 8 + 0 . 7 × 0 . 2) + 0 . 9 × (0 . 5 × 0 . 6 + 0 . 1 × 0 . 4) = 0 . 1 × (0 . 72 + 0 . 14) + 0 . 9 × (0 . 3 + 0 . 04) = 0 . 086 + 0 . 306 = 0 . 392 (42) 8
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4. P ( M = T | K = T ) = P G P ( M = T,K = T | G ) P ( G ) P G P ( K = T | G ) P ( G ) = P G P ( M = T | G ) P ( K = T | G ) P ( G ) P G P ( K = T | G ) P ( G ) = 0 . 6 × 0 . 7 × 0 . 392+0 . 5 × 0 . 6 × 0 . 608 0 . 7 × 0 . 392+0 . 6 × 0 . 608 = 0 . 16464+0 . 1824 0 . 2744+0 . 3648 = 0 . 34704 0 . 6392 0 . 5429 (43) 7 7.1 Given W = k i =1 ( { Y i }∪ Pa Y i ), we let Y = k i =1 Y i and Z = W - Y . We construct the network B prime over W by throwing away those edges of the original network B which end in Z . Thus, all nodes in Z have no parent. We also carry over the CPTs for each Y i to B prime (since all the parents of Y i in B also exist in B prime ). The nodes in Z are set to have discrete uniform probabilities. Sort the nodes in Y topologically, so that there are no descendants of Y i in { Y 1 , . . . , Y i - 1 } . P B prime ( Y | Z ) = P B prime ( Y, Z ) P B prime ( Z ) = producttext k i =1 P B prime ( Y i | Y 1 , . . . , Y i - 1 , Z ) producttext w Z P B prime ( w ) producttext w Z P B prime ( w ) = k productdisplay i =1 P B ( Y i | Pa Y i ) = τ ( W ) 7.2 From the procedural definition of variable elimination, it follows that all the intermediate factors are of form summationdisplay X 1 ,...,X n k productdisplay i =1 P ( Y i | Pa ( Y i )) (44) We can now use the result from section 7.1 that every product of a set of BN basic CPTs is a conditional probability in some network. Now if we have a factor that represents the conditional probability P ( A | B ) and sum out a variable X A (i.e. marginalizing it), we get a new conditional probability P ( A \ X | B ) as a result. Also X cannot be summed out if X B because it would mean that some CPTs involving X are not included in the current factor (every variable has a corresponding CPT P ( X | Pa ( X )), maybe with Pa ( X ) ≡∅ ; and from the construction in Problem 7.1, inclusion of this CPT means X A ) which is not possible from the procedural definition of variable elimination. Therefore, every intermediate factor of variable elimination corresponds to a conditional probability in some network. 9
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