# Why are these 2 MAGs Markov Equivalent? DAGs, MAGs, and PAGs

On page 1443 of the linked paper, the authors present the following causal DAG (Directed Acyclic Graph) with a latent variable (Profession).

On the following page, they present the 2 MAGs below with the one on the left (a) described as "the causal MAG that corresponds to the causal DAG". The one on the right (b) is presented as a Markov-equivalent MAG to the causal MAG.

Finally, on page 1445, the authors present the PAG that corresponds to the causal MAG. Please see below.

Can someone help me think through why the 2 MAGs (Maximal Ancestral Graph) are Markov Equivalent (imply the same constraints by the m-separation criterion)? I would also like to convince myself that the MAG on the left is indeed the causal MAG for the causal DAG? How is a MAG is related to a DAG and a PAG to a MAG? As I understood it: a PAG is Markov-equivalent class of MAGs, but a MAG is not a Markov-equivalent class of DAGs (instead a PCDAG is). Is that correct? I ask these questions because I am not yet fully comfortable with all these terms: PAGs, MAG, m-separation, active paths, etc.

Reference: Causal Reasoning with Ancestral Graphs

1. The two MAGs are indeed Markov-equivalent because they share the same $$m$$-separation statements. They have identical adjacencies among the set $$\{I,PHS, L, G\}$$. Besides, $$m$$-separation statements in MAG$$_a$$ include $$I\perp PHS$$, with the conditioning set being empty. This is also an $$m$$-separation statement in MAG$$_b$$, because the second condition for an active path is violated, as the collider $$S$$ is not an ancestor of the empty set. In the end, all $$m$$-separation statements in MAG$$_a$$ and MAG$$_b$$ are equivalent.

2. Yes, MAG$$_a$$ is the projection of the causal DAG (including latent $$P$$) into the observed variables $$\{I,PHS, S, L, G\}$$.

3. DAG vs CPDAG vs MAG vs PAG:

• A causal DAG represents all directed causal relationships among a set of variables, including latent ones, and only includes directed arrows.

• The Markov-equivalent class of a DAG, representing all DAGs with the same $$d$$-separation statements is called either a PDAG of a CPDAG. Its purpose is to represent marginal independences that can be learned from statistical independences in data.

• A MAG is a graph that includes bidirected arrows and meets two conditions: i) no (almost) directed cycles, and ii) no inducing paths between non-neighbors. A DAG with latent variables can always be projected into a MAG that includes only the observed variables. Its purpose is to represent marginal independences that remain when latent variables are "ignored".

• A PAG is a graphical representation of the Markov-equivalent class of a MAG, representing a set of several MAGs with the same $$m$$-separation statements. Its purpose is to represent marginal independences that can be learned from statistical independences in data after ignoring latent variables.

In other words, a DAG represents the direct relationships between variables. However, it is not possible to learn the underlying DAG from data alone because other DAGs may induce the same independences. Instead, what can actually be learned from data is the CPDAG. If latent variables are present in your DAG, the marginal independences can be represented in a MAG. Yet, just like with DAGs, the MAG cannot be identified directly from data. Therefore, the best you can do with data is to identify the PAG.

• +1: Can you please expand on #2? What does it mean for MAGa to be the projection of the causal DAG (including latent P) into the observed variables? Commented May 24 at 19:05
• @ColorStatistics Sure. Last paragraph of page 1442 in the paper presents a procedure that takes as an input a DAG and a grouping of its variables into $O$ (observed) and $L$ (latent); and it outputs a MAG having only variables in $O$. This is what I call the "projection" of the DAG into $O$, even though the paper doesn't use that name. It is just the "MAG constructed from the DAG". It bears resemblance to a latent projection which is a similar procedure that transform DAGs into graphs with bidirected edges (ADMG). Check this: stat.ucla.edu/~zhou/courses/Stats212_Latent.pdf Commented May 24 at 20:54