What's the MAG of this underlying DAG?

Question

I am studying causal discovery, with an interest on constraint-based algorithms like FCI (Fast Causal Inference).

I want to know what's the Maximal Ancestral Graph (MAG) of this underlying DAG (example taken from Fig-1 in Jiji Zhang et,al)

In this DAG, $\textbf{O}=\{A,Ef,R\}$ is the set of observed variables, $\textbf{L}=\{H\}$ is the set of unobserved variables, and $\textbf{S}=\{Sel\}$ is the set of unobserved variables that are mistakenly conditioned on (i.e. the selection variables).

I use the rules given by Sect 2.3 in Jiji Zhang et,al and compute the MAG over $\textbf{O}$ of this underlying DAG as:

$A-E$, $E\rightarrow R$, $A \rightarrow R$

My computing process is:

• $A$ is adjacent with $R$ because there is an inducing path $\langle A,E,H,R\rangle$ w.r.t. $\textbf{L},\textbf{S}$ between $A$ and $R$.
• $A \rightarrow R$ because $A$ is an ancestor of $\textbf{S}$, but $R$ is not an ancestor of $A$ nor $\textbf{S}$.
• $A-E$ because $A$ is an ancestor of $E$ (and $\textbf{S}$), $E$ is also an ancestor of $\textbf{S}$.
• $E\rightarrow R$ because there is an inducing path $\langle E,H,R\rangle$ between $E$ and $R$, and $E$ is an ancestor of $\textbf{S}$, but $R$ is not an ancestor of $E$ nor $\textbf{S}$.

Since the MAG of this underlying DAG is not given in Jiji Zhang's paper, I am worrying whether my result is correct. Especially about the orientation between $A$ and $E$ (it looks very strange to me because $A$ is actually a cause of $E$ in the underlying DAG).

Am I giving the right results? If not, can anyone point out which step or rule is mistakenly used?

Answer 1

The MAG looks like this:

The steps:

The edge between $Ef$ and $R$: There is an inducing path between $Ef$ and $R$ relative to $\langle\{H\},\emptyset\rangle$, so there must be an edge. The edge is bidirected because of rule (2)(c) in the paper: $Ef\notin\bf{An}_\mathcal{G}(\{R\}\cup\emptyset)$ and $R\notin\bf{An}_\mathcal{G}(\{Ef\}\cup\emptyset)$. Thus $$ Ef\leftrightarrow R. $$

The edge between $A$ and $Ef$: There is an inducing path between $A$ and $Ef$ relative to $\langle\emptyset, \emptyset\rangle$, so there must be an edge. The edge is directed from $A$ to $Ef$ because of rule (2)(a) in the paper: $A\in\bf{An}_\mathcal{G}(\{Ef\}\cup\emptyset)$ and $Ef\notin\bf{An}_\mathcal{G}(\{A\}\cup\emptyset)$. Thus: $$ A\to Ef. $$

The edge between $A$ and $R$: There is an inducing path between $A$ and $R$ relative to $\langle\{H\}, \{Sel\}\rangle$, so there must be an edge. The edge is directed from $A$ to $R$ because of rule (2)(a) in the paper: $A\in\bf{An}_\mathcal{G}(\{R\}\cup\{Sel\})$ and $R\notin\bf{An}_\mathcal{G}(\{A\}\cup\{Sel\})$. Thus: $$ A\to R. $$

Answer 2

After checking Zhang's paper, I believe my original solution is correct.

The MAG of the causal DAG is indeed $A - E$, $A \to R$, $E \to R$.

This is because when deciding adjacency and edges' orientation, the set L and S must be respectively the set of latent variables (in our case $\{H\}$) and the set of selection variables (in our case $\{S\}$).

This MAG actually correspond with Zhang's explanation in Sect. 2.3, for example:

• $A - E$ because $A$ is the cause of some selection variable $Sel$, $E$ is the cause of some selection variable $Sel$.
• $A \to E$ is wrong because $A \to E$ requires $E$ is not the cause of $A$ and not the cause of any selection variable.

Indeed, the MAG of the DAG looks quite strange. This is also illustrated by Zhang in his paper, "The positive causal information is admittedly less informative than one would wish, when the possibility of selection bias is allowed."