Read a Gaussian graphical model

Topic: Graphical models · more questions on this topic
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A mean-zero Gaussian vector has positive-definite precision matrix

K=\begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}.

You may use \rho_{ij\cdot\mathrm{rest}}=-K_{ij}/\sqrt{K_{ii}K_{jj}}.

  1. Draw the undirected graph and state the conditional independence it encodes.

  2. Calculate the nonzero partial correlations and explain why the entries of K are not themselves partial correlations.

  3. Are X_1 and X_3 marginally independent? Justify your answer. The covariance entry \Sigma_{13}=1/4 may be used.

  1. The graph is the chain 1-2-3, with edges 12 and 23 and no edge 13. It implies X_1\perp X_3\mid X_2 for the specified nonsingular Gaussian model.

  2. \rho_{12\cdot 3}=\rho_{23\cdot 1}=1/2 and \rho_{13\cdot 2}=0. The conversion uses a minus sign and normalises by the precision diagonals, so the raw precision entries are not the partial correlations.

  3. No. In fact

\Sigma=K^{-1}=\frac14\begin{pmatrix} 3 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 3 \end{pmatrix}.

The nonzero covariance 1/4 implies marginal dependence for this Gaussian vector. The marginal correlation is 1/3. Conditional separation does not remove the marginal association carried through X_2.