[proofplan] The proof uses faithfulness to convert a graphical d-connection into a probabilistic dependence. Fix an edge $u\to v$ and condition on all other parents of $v$. The single-edge path from $u$ to $v$ is d-connected under this conditioning set, because neither endpoint is conditioned on and the path has no interior vertices. Faithfulness therefore rules out the corresponding conditional independence, which is exactly the parent-child dependence required by causal minimality. [/proofplan] [step:Fix an edge and name the relevant conditioning set] Let $u\to v$ be an arbitrary directed edge of $G$. Define the conditioning set \begin{align*} C:=\operatorname{pa}_G(v)\setminus\{u\}. \end{align*} Since $G$ is a directed acyclic graph and $u\to v$ is an edge, $v\notin \operatorname{pa}_G(v)$ and $u\notin C$. Thus the singleton sets $\{u\}$ and $\{v\}$ and the set $C$ are pairwise disjoint. [/step] [step:Show that the edge path from $u$ to $v$ remains d-connected given the other parents] Consider the path $\pi$ in $G$ consisting of the single directed edge $u\to v$. This path has endpoints $u$ and $v$ and has no interior vertices. Therefore there is no non-collider interior vertex of $\pi$ that could lie in $C$, and there is no collider interior vertex of $\pi$ whose descendant status must be checked. Since neither endpoint belongs to $C$, the path $\pi$ is active given $C$. Hence $\{u\}$ and $\{v\}$ are not d-separated by $C$ in $G$. [guided] We want to understand whether conditioning on the other parents of $v$ can block the direct edge $u\to v$. Let \begin{align*} C:=\operatorname{pa}_G(v)\setminus\{u\}. \end{align*} The path we use is the one-edge path \begin{align*} u\to v. \end{align*} A path is blocked by a conditioning set only through its interior vertices: non-colliders are blocked when they are conditioned on, while colliders are opened only when they or their descendants are conditioned on. This particular path has no interior vertices. Consequently there is no non-collider to block and no collider to analyze. The endpoints also do not create a blockage in the definition of d-separation. We have $u\notin C$ by construction, and $v\notin C$ because a vertex cannot be its own parent in a directed acyclic graph. Thus the single-edge path from $u$ to $v$ is active given $C$. Therefore $\{u\}$ and $\{v\}$ are not d-separated by $C$ in $G$. [/guided] [/step] [step:Use faithfulness to convert d-connection into conditional dependence] By the preceding step, $\{u\}$ and $\{v\}$ are not d-separated by $C$ in $G$. Since $\mathbb P_X$ is faithful to $G$, conditional independence under $\mathbb P_X$ is equivalent to d-separation in $G$. Therefore the failure of d-separation implies \begin{align*} X_u \not\!\perp\!\!\!\perp X_v \mid X_C. \end{align*} Substituting the definition of $C$ gives \begin{align*} X_u \not\!\perp\!\!\!\perp X_v \mid X_{\operatorname{pa}_G(v)\setminus\{u\}}. \end{align*} [/step] [step:Conclude causal minimality for every edge] The edge $u\to v$ was arbitrary. Hence for every parent $u$ of every vertex $v$, the [random variable](/page/Random%20Variable) $X_v$ is conditionally dependent on $X_u$ given the remaining parents of $v$. This is precisely causal minimality for $(G,\mathbb P_X)$. Therefore faithfulness of $\mathbb P_X$ to $G$ implies causal minimality. [/step]