Violation of Faithfulness in Structural Causal Models

Question

While reading "Elements of Causal Inference" by Peters et al. I stumbled over something I'm not sure about regarding "Example 6.34 (Violations of Faithfulness)". A link to the book can be found here.

There, a linear Gaussian SCM is defined as

$X:=N_X$, $Y:=aN_X+N_Y$, $Z:=bY+cX+N_Z$ where $N_X \sim \mathcal{N}(0, \sigma_X^2)$, $N_Y \sim \mathcal{N}(0, \sigma_Y^2)$ and $N_Z \sim \mathcal{N}(0, \sigma_Z^2)$.

The corresponding DAG for the SCM is given by $\mathcal{G}_1$. Now the book states that if the SCM is parameterized with $a\cdot b -c = 0$, $\mathcal{G}_1$ is not faithful to the distribution $P$ induced by said SCM. That's still clear to me.

But now the book states that it is easy to construct an SCM inducing same distribution, but the DAG depicted in $\mathcal{G}_2$. I'm having trouble to construct this SCM. With the constraint $a\cdot b -c = 0$, the distribution were looking for is given by

$X=N_X$, $Y=aN_X + N_Y$ and $Z=bN_Y + N_Z$.

I tried to use the properties of Gaussian distributions to define the SCM for $\mathcal{G}_2$ via

$X' = N_X$, $Z'=bN_Y + N_Z$ and $Y'= \tilde{a}X' + \tilde{b}Z'$ where $\tilde{a} = a$ and $\tilde{b} = \sqrt{\frac{\sigma_Y^2}{b\sigma_Y^2 + \sigma_Z^2}} $.

Now with this, the parameters of the Gaussians should be equal in both SCMs. Even the (in)dependencies work out as far as I can tell. Yet, I doubt they have the same joint distributions as knowing the value of $Z'$ and $X'$ completelly determines the value of $Y'$ which is not the case for the first SCM.

Where am I going wrong?

Answer 1

Your mistake is in the definition of $Y'$. Actually, $Y'= \tilde{a}X' + \tilde{b}Z' + N_{Y'}$, with $N_{Y'} \sim \mathcal{N}(0, \sigma_{Y'}^2)$.

In order to determine $\tilde{a}, \tilde{b}$ and $\sigma_{Y'}^2$, notice that:

1. the joint distribution of $X',Y',Z'$ is Gaussian (since the independent $X'$ and $Z'$ are normally distributed and also the conditional $Y'|X',Z'$ is Gaussian with mean a linear function of $X',Z'$)
2. the joint distribution of $X,Y,Z$ is Gaussian (using the same type of arguments)
3. the two distributions must have the same parameters

In particular:

1. $Cov(X,Y)=Cov(X',Y') \Rightarrow ... \Rightarrow \tilde{a}=a$
2. $Var(Y)=Var(Y') \Rightarrow ... \Rightarrow \tilde{b}^2(b^2\sigma_Y^2+\sigma_{Z}^2) +\sigma_{Y'}^2 = \sigma_Y^2$
3. $Cov(Y,Z)=Cov(Y',Z') \Rightarrow ... \Rightarrow \tilde{b}(b^2\sigma_Y^2+\sigma_Z^2) =b\sigma_Y^2$

After some calculations we get $\tilde{b}=\frac{b\sigma_Y^2}{b^2\sigma_Y^2 + \sigma_Z^2}$ and $\sigma_{Y'}^2=\frac{\sigma_Y^2\sigma_Z^2}{b^2\sigma_Y^2 + \sigma_Z^2}$.