Directed Gaussian Graphical Models

Definition
The Gaussian graphical model given by the DAG \(G=(V,E)\) is
\(\begin{align*} \mathcal N(G) &=\bigl\{N_m(\mu,\Sigma)\colon \mu\in\mathbb R^m,\ \Sigma\in\mathrm{PD}_m,\ N_m(\mu,\Sigma)\text{ satisfies the global Markov property for }G\bigr\}\\ &=\bigl\{N_m(\mu,\Sigma)\colon \mu\in\mathbb R^m,\ \Sigma\in\mathrm{PD}_m,\ \text{density factorizes according to }G\bigr\}. \end{align*}\) Composition Property
For \(X\sim N_m(0,\Sigma)\), positivity implies the intersection axiom and moreover \(A\perp\!\!\perp B\mid C \;\Longleftrightarrow\; u\perp\!\!\perp v\mid C \quad\text{for all }u\in A,\ v\in B,\) since \(\Sigma_{A,B}=0\iff\Sigma_{u,v}=0\).

Parametrization via Edge Weights

Define
\(\R^E =\{\,B=(\omega_{v u})\in\mathbb R^{m\times m}\colon \omega_{v u}=0\text{ if }u\notin\pa(v)\}, \quad \mathrm{Diag}^+ =\{\,\Omega=\mathrm{diag}(\epsilon_{vv})\colon \epsilon_{vv}>0\}.\) Let \(I\) be the \(m\times m\) identity. The map \(\phi_G:\R^E\times \mathrm{Diag}^+ \;\to\;\mathrm{PD}_m, \qquad \phi_G(B,\Omega) =(I - B)^{-1}\,\Omega\,(I - B)^{-T}\) provides a parametrization of \(\mathcal N(G)\).

Theorem (Parametrization)
\(N_m(\mu,\Sigma)\in\mathcal N(G)\) if and only if \(\Sigma=\phi_G(B,\Omega)\) for some \(B\in\mathcal R_E,\ \Omega\in\mathrm{Diag}^+\).

Lemma
For any DAG \(G=(V,E)\) and \(B\in\mathcal R_E\), \(\det(I - B)=1\) and thus \(I - B\) is invertible.

Theorem
The parametrization map \(\phi_G\) is injective. If \(\Sigma=\phi_G(B,\Omega)\), then \(B_{v,\pa(v)}=\Sigma_{v,\pa(v)}\,\Sigma_{\pa(v),\pa(v)}^{-1}, \quad \epsilon_{vv} =\Sigma_{vv}-\Sigma_{v,\pa(v)}\,\Sigma_{\pa(v),\pa(v)}^{-1}\,\Sigma_{\pa(v),v}.\)

Corollary
\(\{\Sigma\in\mathrm{PD}_m:\,N(0,\Sigma)\in\mathcal N(G)\}\) has dimension \(|V|+|E|\).

Trek Rule

Lemma
For \(B\in\mathcal R_E\), the entries of the path‐matrix \((I - B)^{-1}\) are \([(I - B)^{-1}]_{v u} =\sum_{\pi\in\Pi(u,v)}w(\pi),\) where \(\Pi(u,v)\) is the set of directed paths from \(u\) to \(v\) and \(w(\pi)=\prod_{(x\to y)\in\pi}\omega_{y x}\).

Theorem (Trek Rule)
If \(\Sigma=(I - B)^{-1}\,\Omega\,(I - B)^{-T}\), then \(\Sigma_{u v} =\sum_{\phi\in\mathcal T(u,v)}w(\phi),\) where \(\mathcal T(u,v)\) are all treks from \(u\) to \(v\), and each trek \(\phi\) with top \(z\) has weight \(\;w(\phi)=\epsilon_{zz}\,w(\pi_L)\,w(\pi_R)\).

Linear Structural Equation Models

In the SEM for \(G\), each \(X_v =\omega_{0v} +\sum_{u\in\pa(v)}\omega_{v u}\,X_u +\zeta_v, \quad \zeta_v\overset{\mathrm{iid}}{\sim}N(0,\epsilon_{vv}).\) Vectorizing gives \((I - B)X=\omega_0+\zeta\), so \(X=(I - B)^{-1}\omega_0+(I - B)^{-1}\zeta, \quad \Var{X}=(I - B)^{-1}\,\Omega\,(I - B)^{-T},\) recovering the same family \(\mathcal N(G)\).