PC distinguishes a chain from an unshielded collider

Let $W_1,\dots ,W_m$ be a finite independent family of scalar normal variables with means ${\mu }_r$ and variances ${\sigma }_r^2$. For any real coefficients $a_1,\dots ,a_m,b$, [F1]–[F2] give the characteristic function of $b+{\sum }_r{a}_r{W}_r$ as $\exp \text{\hspace{0.17em}⁣}(it(b+{\sum }_r{a}_r{\mu }_r)-\frac{t^2}{2}{\sum }_r{a}_r^2{\sigma }_r^2)$; scalar uniqueness identifies this as the normal law with that mean and variance. Every scalar projection of a finite affine image of $W=(W_1,\dots ,W_m)$ is such a sum, so [F3] proves that every such image is multivariate normal, including when a projection has variance zero.

The chain model's density is $\varphi (x)\varphi (y-x)\varphi (z-y)$, where $\varphi$ is the standard-normal density. Its factors are exactly the parent probability kernels $K_X(dx)=N(0,1)(dx)$, $K_Y(x,dy)=N(x,1)(dy)$, and $K_Z(y,dz)=N(y,1)(dz)$, so the full-joint law factorizes in the topological order $X,Y,Z$. All three vertices are observed and [F8] gives the Markovian SCM condition. Its moments are $\mathrm{Var}(X)=1$, $\mathrm{Var}(Y)=2$, $\mathrm{Var}(Z)=3$, and distinct covariances $\mathrm{Cov}(X,Y)=1$, $\mathrm{Cov}(Y,Z)=2$, $\mathrm{Cov}(X,Z)=1$. The vector is multivariate normal by step 1.1, and its covariance matrix is positive definite because $(X,Y,Z)$ is an invertible triangular linear transform of $(U_X,U_Y,U_Z)$. The conditional covariance formula gives $\mathrm{Cov}(X,Z\mid Y)=1-\frac{1\cdot 2}{2}=0,\mathrm{Cov}(X,Y\mid Z)=1-\frac{1\cdot 2}{3}=\frac{1}{3},\mathrm{Cov}(Y,Z\mid X)=2-\frac{1\cdot 1}{1}=1.$ Together with the nonzero marginal covariances, [F4] shows that the only conditional independence between distinct variables is $X\perp Z\mid Y$. By [F10], the only d-separated vertex pair is also $X,Z$ given $Y$: the adjacent pairs have an active one-edge path, and the $X$–$Z$ path is blocked only by conditioning on $Y$. Any conditional independence between vertex sets would imply the corresponding coordinate independences, so the set-level independence and d-separation models also agree. This verifies faithfulness in [F7].

The collider model's density is $\varphi (x)\varphi (z)\varphi (y-x-z)$. Its factors are exactly the parent probability kernels $K_X(dx)=N(0,1)(dx)$, $K_Z(dz)=N(0,1)(dz)$, and $K_Y(x,z,dy)=N(x+z,1)(dy)$, so the full-joint law factorizes in the topological order $X,Z,Y$. All vertices are observed and [F8] gives the Markovian SCM condition. Its variances are $\mathrm{Var}(X)=\mathrm{Var}(Z)=1$, $\mathrm{Var}(Y)=3$, its covariances are $\mathrm{Cov}(X,Y)=\mathrm{Cov}(Z,Y)=1$, and $\mathrm{Cov}(X,Z)=0$. The vector is multivariate normal by step 1.1, and this covariance matrix is positive definite by the invertible triangular map from the exogenous variables. The conditional covariances are $\mathrm{Cov}(X,Z\mid Y)=0-\frac{1\cdot 1}{3}=-\frac{1}{3},\mathrm{Cov}(X,Y\mid Z)=1,\mathrm{Cov}(Y,Z\mid X)=1.$ Therefore [F4] shows that $X\perp Z$ marginally, but there is no conditional independence after conditioning on the third variable and no independence for an adjacent pair. By [F10], this is exactly the collider DAG's d-separation list: the collider path is blocked unconditioned and opened by conditioning on $Y$, while each adjacent pair has an active one-edge path. Any set conditional independence would imply coordinate independences, so [F7] holds at the set level as well.

Thus PC does not delete any pair at level $0$. At level $1$, $X-Z$ is deleted and its separator is recorded as $\{Y\}$; $X-Y$ and $Y-Z$ remain because step 2.1 shows that each is dependent both marginally and conditional on the third variable. The sole unshielded triple has its middle vertex in the separator, so [F5] or [F6] leaves both edges undirected. No Meek rule applies without a directed edge, and the output is the undirected chain CPDAG. The run terminates under [F9], since step 2.1 verifies the chain model's discovery-regime hypotheses.

PC deletes $X-Z$ at level $0$ and records the empty separator. It retains $X-Y$ and $Y-Z$ by their nonzero marginal covariances. Since $Y$ is absent from the separator, [F5] orients $X\to Y\leftarrow Z$; the output is the collider CPDAG. The run terminates under [F9], since step 2.2 verifies the collider model's discovery-regime hypotheses. These calculations verify both claimed outputs without relying on the diagrams.