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ps4_sol - 1 Introduction to Probabilistic Graphical Models...

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1 Introduction to Probabilistic Graphical Models December 26, 2006 Problem set 4 - Solutions Lecturer: Eran Segal 1. (a) This can be done by summing out the variables that we wish to remove and incorpo- rating their effects into the leak noise parameter. The important thing to see here is that the summing out can be done efficiently. P ( F i = f 0 i , D 1 , ..., D ) = X D +1 ,...,D k P ( F i = f 0 i , D 1 . . . D k ) = X D +1 ,...,D k P ( F i = f 0 i | D 1 ...D k ) · P ( D 1 . . . D k ) = X D +1 ,...,D k P ( F i = f 0 i | D 1 ...D k ) · k Y i =1 P ( D i ) = X D +1 ,...,D k (1 - λ 0 ) k Y j =1 (1 - λ j ) d j · k Y i =1 P ( D i ) = (1 - λ 0 ) Y j =1 (1 - λ j ) d j P ( D j ) X D +1 ,...,D k k Y i = +1 (1 - λ i ) d i · P ( D i ) For each D i where i > ‘ the term in parentheses can be simplified as follows: (1 - λ i ) P ( D i = 1) + P ( D i = 0) = 1 - λ i P ( D i = 1) And so the entire term in parentheses can be written as: k Y i = +1 (1 - λ i P ( D i = 1)) We will denote this term as A . Notice that A can be computed efficiently in the price of O ( k ) instead of a trivial summing out in F i ’s CPD, in the price of O (2 k ).
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