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The completed partially directed acyclic graph of a Markov equivalence of directed acyclic graphs class has their common skeleton of a directed graph. An edge is directed exactly when its orientation agrees in every member of the class, and is otherwise undirected. It can be constructed by enumerating all acyclic orientations with the prescribed unshielded colliders and retaining only the common directions; practical PC algorithms use orientation propagation instead of enumeration.

The global Markov property for a directed acyclic graph requires, for all pairwise disjoint vertex sets $A,B,C$, that D-separation of $A$ and $B$ by $C$ in $G$ implies $Z_A\perp Z_B\mid Z_C$. Two Markov equivalent directed acyclic graphs encode exactly the same such D-separations. The structural characterization is that they have the same skeleton of a directed graph and the same unshielded colliders. A distribution has faithfulness of a directed acyclic graph when its conditional independences are exactly those encoded by D-separation; this includes the global Markov implication as well as its converse.

The population PC algorithm uses an exact conditional independence oracle. Start with the complete undirected graph, and conditioning-set size $\ell =0$. At each size, examine ordered adjacent pairs $(u,v)$. Test subsets $C\subseteq \mathrm{Adj}(u)\setminus \{v\}$ of size $\ell$ until independence $Z_u\perp Z_v\mid Z_C$ is found or all such subsets have been examined. On finding it, delete $u-v$ and record a symmetric separating set $\mathrm{Sep}(u,v)=\mathrm{Sep}(v,u)=C$. Ordered pairs ensure that subsets from either endpoint's neighbours are considered. Increase $\ell$ until no adjacent ordered pair has enough remaining neighbours to supply a set of that size. Rechecking eligibility after deletion, or freezing the neighbours at each level and then deleting, gives the same population skeleton under the hypothesis below.

Next, for each unshielded triple $u-v-w$ with $u,w$ nonadjacent, orient $u\to v\leftarrow w$ precisely when $v\notin \mathrm{Sep}(u,w)$. Propagate directions that are forced without introducing a directed cycle or an additional unshielded collider. In particular, $a\to b-c$ with $a,c$ nonadjacent forces $b\to c$; an undirected edge $a-b$ with a directed path from $a$ to $b$ forces $a\to b$; and $a-b,a-c,a-d,c\to b,d\to b$ with $c,d$ nonadjacent forces $a\to b$. The first avoids a new unshielded collider, the second a directed cycle. For the third, $b\to a$ would force $c\to a$ and $d\to a$ to avoid cycles, creating the prohibited unshielded collider $c\to a\leftarrow d$.

For a fully specified exact completion, list all acyclic orientations of the recovered skeleton which extend the collider arrows and have exactly the recorded unshielded colliders. Direct an edge if and only if all listed graphs orient it the same way; otherwise leave it undirected. This finite enumeration implements the definition of the completed partially directed acyclic graph; orientation propagation is the usual efficient way to obtain the same completion. The equivalence-class proof below uses the exact completion description and does not need an unproved completeness claim about a chosen subset of propagation rules.

To prove correctness, assume faithfulness of a directed acyclic graph to $G^0$. A true edge $u\to v$ cannot be D-separated by any set excluding its endpoints: its length-one path is always active. Thus a true edge is never deleted. Conversely, take nonadjacent $u,v$ and choose their order so that $u$ is earlier than $v$ in a topological ordering of $G^0$. Then $u$ is a nondescendant and nonparent of $v$. Its parent set $\mathrm{Pa}(v)$ gives D-separation of $u$ and $v$. To see this local graphical fact, a path beginning $v\leftarrow q$ is blocked by the conditioned parent $q$, a noncollider. A path beginning with an arrow out of $v$ must encounter a first collider before reaching the nondescendant $u$. That collider is a descendant of $v$, and neither it nor any of its descendants can be a parent of $v$ in an acyclic graph. Hence conditioning on $\mathrm{Pa}(v)$ does not open it.

All parent edges survive. If $u-v$ has not already been deleted, $\mathrm{Pa}(v)$ remains among $v$'s neighbours excluding $u$. It is therefore tested by size $|\mathrm{Pa}(v)|$, and faithfulness forces deletion. The size loop cannot stop prematurely while this set remains eligible. This proves exact recovery of the skeleton of a directed graph.

For an unshielded triple $u-v-w$, if it is a noncollider in $G^0$, its length-two path remains active unless $v$ is conditioned on. Thus every separator of $u,w$ contains $v$. If it is a collider, conditioning on $v$ opens the length-two path, so no separator can contain $v$. Faithfulness turns these graphical statements into the recorded separator test. Therefore the collider phase recovers precisely all unshielded colliders. By the skeleton and collider characterization of Markov equivalence, the acyclic extensions with exactly these colliders are exactly the Markov equivalence of directed acyclic graphs class of $G^0$. In particular that set is nonempty, and the completion returns its CPDAG. This proves population PC correctness under faithfulness.

For the four-variable example, the only removable pairs are $2,4$, with separator $\varnothing$, and $1,3$, with separator $\{2,4\}$. The remaining skeleton has edges $1-2,1-4,2-3,3-4$. The unshielded triples $2-1-4$ and $2-3-4$ both have center outside the separator of $2,4$, so both are unshielded colliders. Every remaining edge is thereby oriented:The other two unshielded triples are noncolliders, consistent with the separator $\{2,4\}$ of $1,3$. There is no undirected edge left, so the equivalence class consists of this unique Directed acyclic graph. This identification is about the given faithful observational distribution; general PC algorithm outputs need not identify a unique causal direction.

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.

With an exact conditional independence oracle and faithfulness of a directed acyclic graph, the skeleton phase of the PC algorithm deletes precisely the nonedges. A true edge cannot be D-separated by a set excluding its endpoints. For nonadjacent vertices, choose the later vertex $v$ in a topological ordering; its parents separate it from the earlier nonparent. True parent edges survive throughout, so this separator remains available among the algorithm's candidate neighbour sets. For an unshielded triple $u-v-w$, any recorded separator of $u,w$ excludes $v$ exactly when the triple is an unshielded collider. The skeleton and collider characterization of Markov equivalence therefore identifies the correct equivalence class. Directions compelled throughout that class give its completed partially directed acyclic graph.

Two Directed acyclic graphs are Markov equivalent directed acyclic graphs if and only if they have the same skeleton of a directed graph and the same unshielded colliders. Thus these two structures determine the observational equivalence class. This structural theorem is what converts the skeleton and collider phases of the PC algorithm into identification of an equivalence class.

The skeleton of a directed graph is the undirected graph obtained by replacing every directed edge with an undirected edge and forgetting its orientation. In causal structure learning, the PC algorithm first recovers this adjacency structure before learning orientations.