ps4_sol - 1 Introduction to Probabilistic Graphical Models...

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Unformatted text preview: 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 i , D 1 , ..., D ‘ ) = X D ‘ +1 ,...,D k P ( F i = f i , D 1 . . . D k ) = X D ‘ +1 ,...,D k P ( F i = f i | D 1 ...D k ) · P ( D 1 . . . D k ) = X D ‘ +1 ,...,D k P ( F i = f i | D 1 ...D k ) · k Y i =1 P ( D i ) = X D ‘ +1 ,...,D k (1- λ ) k Y j =1 (1- λ j ) d j · k Y i =1 P ( D i ) = (1- λ ) ‘ 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 ....
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This note was uploaded on 02/05/2010 for the course CS CS229 taught by Professor Eransegal during the Fall '07 term at École Normale Supérieure.

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ps4_sol - 1 Introduction to Probabilistic Graphical Models...

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