hw5sol - 10-708 Homework 5 Solutions Problem 1: Gaussian...

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10-708 Homework 5 Solutions Problem 1: Gaussian Graphical Models [25 pts] Gaussian graphical model over three variables x 1 , x 2 , x 3 with (1) P x 1 , x 2 , x 3 1    2  3 2 1 2 exp 1  2 x  T  1 x   where x x 1 , x 2 , x 3 and  1 , 2 , 3 are column vectors, and is a 3 3 covariance matrix. Let (2) 1 2 1 0 1 2 1 0 1 2 , 2 .75 .5 .25 .5 1.0 .5 .25 .5 .75 (a) Suppose  1 . Are there any marginal independencies among the components of x? There is one instance of marginal independence: x 1 x 3 , since 13 0. (b) Suppose  2 . Are there any marginal independencies among the components of x? No. There are no zero elements in 2 . (c) Take the following parameterization: (3) X 1 N 0, 1 X 2 X 1 N x 1 , 1 X 3 X 2 N x 2 , 1 hw5sol.nb12/6/06 Printed by Mathematica for Students
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Joint distribution: (4) p x 1 , x 2 , x 3 p x 1 p x 2 x 1 p x 3 x 2 1    2  exp 1  2 x 1 2 1    2  exp 1  2 x 2 x 1 2 1    2  exp 1  2 x 3 x 2 2 1   2  3 2 exp 1  2 x 1 2 1  2 x 2 x 1 2 1  2 x 3 x 2 2 1   2  3 2 exp 1  2 2 x 1 2 2 x 2 2 x 3 2 2 x 1 x 2 2 x 2 x 3  1   2  3 2 exp 1  2 2 x 1 2 2 x 2 2 x 3 2 2 x 1 x 2 2 x 2 x 3  1   2  3 2 exp 1  2 x 1 x 2 x 3 T 2
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This note was uploaded on 05/25/2008 for the course MACHINE LE 10708 taught by Professor Carlosgustin during the Fall '07 term at Carnegie Mellon.

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hw5sol - 10-708 Homework 5 Solutions Problem 1: Gaussian...

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