Exercise 19.3 - Independencies in Gaussian graphical models

For question (a), the model implies that:

• Given $X_2$, $X_1$ is indenpendent from $X_3$.

Hence the inverse of its covariance matrix should be zero at $(1,3)$-th entry. Thus $A$ and $D$ are candidates.

For question (b), the candidates are $C$ and $D$.

For question (c), the PGM tells that:

• $X_1$ is indenpendent from $X_3$.

Hence its covariance should be zero at its $(1,3)$-th entry. Therefore $C$ and $D$ could be the covariance matrix.

For question (d), the candidates are $A$ and $B$.

For question (e), recall (4.68). So $A$ is true while $B$ is not. It is safe to derive the marginal distribution from the joint Gaussian by partitioning the covariance, but not its inverse.