# DAG – Understanding the MAG of an Underlying DAG

artificial intelligencecausalitydag

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?

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: $$A\to R.$$