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Write $V=\{1,\dots ,p\}$. The global Markov property for a directed acyclic graph says that, for every three pairwise disjoint vertex sets $A,B,C$,Thus graphical D-separation implies probabilistic conditional independence. Causal minimality means that $P$ has this global Markov property for a directed acyclic graph, but has it for no proper subgraph of $\mathcal{G}$ obtained by deleting edges while keeping all vertices. Faithfulness of a directed acyclic graph means that the displayed implication is an equivalence: the conditional independences of $P$ are exactly the D-separations of $\mathcal{G}$. In particular it includes the global Markov property for a directed acyclic graph.

A causally minimal distribution can nevertheless have an extra independence. Here is a causally minimal nonfaithful Gaussian model. Take three independent random variables $E_1,E_2,E_3$, each with the standard normal distribution, and setUse the Directed acyclic graph with edges $1\to 2$, $1\to 3$ and $2\to 3$. The equations give a directed acyclic graph factorization, so the global Markov property for a directed acyclic graph holds. Alternatively this complete acyclic graph imposes no nontrivial D-separations. But $Z_3=E_2+E_3$ is independent of $Z_1=E_1$, even though the direct edge $1\to 3$ prevents their D-separation. Hence faithfulness of a directed acyclic graph fails.

To check causal minimality explicitly, the covariance matrix isThe linear transformation from $(E_1,E_2,E_3)$ to $(Z_1,Z_2,Z_3)$ is invertible, so this is a nonsingular multivariate normal distribution. Deleting $1\to 2$ would require $Z_1\perp Z_2$, whereas $\mathrm{Cov}(Z_1,Z_2)=1$. Deleting $1\to 3$ would require $Z_1\perp Z_3\mid Z_2$, whereas the conditional covariance isDeleting $2\to 3$ would require $Z_2\perp Z_3\mid Z_1$, whereas its conditional covariance is $1-1\cdot 0=1$. By Gaussian conditional independence, each nonzero conditional covariance rules out the required independence. Every proper subgraph is contained in a graph obtained by a single edge deletion, and further deletion can only add Markov restrictions. Thus no proper subgraph is Markov for $P$, proving causal minimality.

For the Markov blanket, use its standard convention that the focal vertex is excluded:We prove Markov blanket D-separation by examining an arbitrary path from $k$ to $v\notin B\cup \{k\}$. If its first edge points into $k$, its next vertex is a parent of $k$, hence a conditioned noncollider in $B$, and the path is blocked. Otherwise the path starts $k\to h$ at a child $h\in B$. If the next edge points out of $h$, then $h$ is again a conditioned noncollider. If it points into $h$, the path begins $k\to h\leftarrow u$. Although conditioning on the collider $h$ opens that local segment, $u$ is another parent of $h$, hence belongs to $B$. The outgoing arrow $u\to h$ makes $u$ a noncollider on the path, so conditioning on $u$ blocks it. Since $v$ is outside $B$, none of these blocking vertices can be the endpoint.

Every path is therefore blocked by $B$. Applying the global Markov property for a directed acyclic graph givesThis proves the requested screening property without any positivity assumption on $P$.

Finally assume faithfulness of a directed acyclic graph and let $A$ have the stated screening property. Then $k$ and every vertex outside $A\cup \{k\}$ must be D-separated by $A$. If a parent or child $h$ of $k$ were missing from $A$, the one-edge path between them would be active, a contradiction. Thus every parent and child is in $A$. If another parent $u$ of a child $h$ were missing from $A$, the path $k\to h\leftarrow u$ would be active: its only interior vertex is the collider $h$, already in $A$. This is again a contradiction. ConsequentlyUnder faithfulness the graphical blanket is contained in every screening set. This is the minimal Markov blanket under faithfulness; the preceding argument also shows that the blanket itself is a screening set.

The moral graph of a Directed acyclic graph is formed by joining any two parents of a common child and then removing the directions from all edges. For D-separation, a path in the underlying undirected graph is active given $S$ when every noncollider on it is outside $S$, and every collider has itself or a descendant in $S$. Disjoint vertex sets $A$ and $B$ are D-separated by $S$ if no such active path connects them.

The adjacency characterization isA one-edge path cannot be blocked by conditioning on other vertices. For a nonadjacent pair, choose the later vertex $v$ in a topological ordering. The earlier vertex $u$ is a nondescendant and is not a parent of $v$, so the parents of $v$ separate the pair by the local Markov property of a directed acyclic graph.

Use faithfulness of a directed acyclic graph to mean that graphical and probabilistic independences agree, for all disjoint sets:This includes the global Markov property of a directed acyclic graph as well as the absence of extra independences. The conditional independence graph of $P$ is the undirected graph in which $u-v$ is absent exactly when $Z_u\perp Z_v\mid Z_{V\setminus \{u,v\}}$.

To identify this graph, condition on every vertex other than $u,v$. Any noncollider in the interior of a path is then conditioned on, so an active path must have only colliders internally. Two adjacent interior vertices cannot both be colliders: the arrow joining them has an arrowhead at only one of its ends. Thus an active path can have no interior vertex, giving a direct edge, or just one, giving $u\to w\leftarrow v$. In the latter case the conditioned common child $w$ activates the path. These possibilities are exactly the edges added or retained by moralization. Faithfulness therefore provesThis is why the moral graph equals the conditional independence graph under faithfulness.

For Gaussian graph estimation, collect the data into an $n\times p$ matrix $D$ and let $S_n=n^{-1}{\sum }_i{x}_i{x}_i^T$. The precision matrix $\Theta ={\Sigma }^{-1}$ has ${\Theta }_{jk}=0$ precisely for absent graph edges, by Gaussian conditional independence. In neighbourhood selection, fit a Lasso regression of each column on the others:Select the nonzero coefficients as neighbours of $j$, then make the graph undirected using either the OR rule, retaining an edge selected in at least one regression, or the AND rule, requiring both. These are instances of Nodewise Lasso.

The Graphical Lasso estimates the whole precision matrix jointly, for example byRead off its nonzero off-diagonal entries as graph edges. This expression uses an off-diagonal penalty; an all-entry penalty is another convention.

To reduce computation in the PC algorithm, start its skeleton search from an estimate of the conditional independence graph rather than the complete graph. The exact graph contains the true DAG skeleton, so oracle tests can remove its extra edges while retaining the true adjacencies. Fewer initial edges and smaller neighbourhoods reduce both candidate pairs and candidate conditioning sets.

For correctness of the orientation stage, preserve valid separating-set records. With an exact conditional independence graph, an initially absent pair $u,v$ has the full complement $V\setminus \{u,v\}$ as a separating set. Record that set, and record the usual discovered sets for edges removed later. Leaving an initially absent pair with an empty default could falsely mark an unshielded noncollider as a collider. This describes PC initialization from a conditional independence graph; with an estimated initial graph, screening errors can propagate into the output.

The vector $(y_1,y_2,y_3)$ is Jointly Gaussian. Assuming $R>0$, Gaussian conditional independence givesThus the required condition is $R{R}_{13}=R_{12}{R}_{23}$. Under it, conditioning on $y_1$ supplies no further information after $y_2$, and the Gaussian process regression posterior isBoth the conditional expectation and conditional variance depend only on $y_2,t_2,t_3$; neither contains $y_1$ or $t_1$.