Conditional association and sparsity

Topic: Graphical models · more questions on this topic
‹ Read a Gaussian graphical model

A Gaussian vector has precision matrix

K=\begin{pmatrix} 1 & -0.2 & 0 \\ -0.2 & 1 & 0.3 \\ 0 & 0.3 & 1 \end{pmatrix}.

1. Draw its graph, state the conditional independence, calculate the two nonzero partial correlations, and explain why a missing edge need not mean zero marginal correlation.

2. Explain the purpose of the penalty in the graphical lasso, how a tuning parameter may be chosen, and why selected edges should not automatically be described as causal effects.

Worked answer (click to show)

1. The edges are 12 and 23, and X_1\perp X_3\mid X_2. The partial correlations are +0.2 and -0.3, using the negative standardised precision entries. The path through X_2 can give a marginal association: in this example (K^{-1})_{13}=-0.06/0.87\neq 0.

2. A typical objective is \log\det K-\operatorname{tr}(SK)-\lambda\sum_{i\neq j}|K_{ij}| over K\succ 0. The penalty regularises and can create exact off-diagonal zeros, trading fit for a sparser graph. Choose \lambda by a stated criterion such as the extended information criterion or a held-out likelihood, with the selection performed inside the intended assessment procedure. An edge records conditional association under the fitted Gaussian model, not the direction or the effect of an intervention. Omitted variables, sampling error, and misspecification remain possible.