Exercise 19.2 - CI properties of Gaussian graphical models

Answers

For question (a), we have:

Σ = ( 0.75 0.5 0.25 0.5 1.0 0.5 0.25 0.5 0.75 ) ,

and:

Λ = Σ − 1 = ( 2 − 1 0 − 1 2 − 1 0 − 1 2 ) .

Thus we have independency: X 1 ⊥ X 2 | X 3 . This introduces a MRF like:

For question (b): The inverse of Σ contains no zero element, hence no conditional independency inhabits in this model. Therefore there has to be edge between any two nodes.

This model cancels the marginal independency X 1 ⊥ X 3 . But it is possible to model this property by Bayesian network with two directed edges X 1 → X 2 and X 3 → X 2 . The UGM is obtained by moralizing X 1 and X 3 from the directed version.

For question (c), we only have to consider the terms inside the exponential:

− 1 2 { x 1 2 + ( x 2 − x 1 ) 2 + ( x 3 − x 2 2 ) } ,

from which it is easy to see the precision matrix and covariance matrix are:

Λ = ( 2 − 1 0 − 1 2 − 1 0 − 1 1 ) , Σ = ( 1 1 1 1 2 2 1 2 3 ) .

For question (d), the only conditional independency is X 1 ⊥ X 3 | X 2 , which is in accordance with the following model:

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2021-03-24 13:42
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