# sigma-separation question in cyclic causal graph - understanding sigma-separation

## Main Question

In https://arxiv.org/pdf/1807.03024.pdf, a generalization of d-separation in DAGs is introduced, called $$\sigma$$-separation for cyclic graphs.

1. I am wondering how $$v_1 \perp v_6$$ using the $$\sigma$$-separation criterion?
2. Based on that answer, why is $$v_1 \not\perp v_6 | v_3, v_5$$

## Background

The following graph is given as Figure 3:

and they then provide the following table comparing the d-separation statements and the $$\sigma$$-separation statements:

### Background details on a path to be $$\sigma$$-connected (i.e. similar to d-connected)

$$\newcommand{\pperp}{\perp\kern-5pt\perp} \newcommand{\npperp}{\not\perp\kern-8pt\perp}$$ First, please note that there are different definitions of a path out there and in the definition of $$Z$$-$$\sigma$$-open paths, they use the following definition for a path (unfortunately, they don't give this definition in the paper, at least I didn't find it):

A path as a sequence of consecutive edges in the graph, without any restrictions on the types or orientations of the edges involved.

In particular, this means that a path can also go through the same node or edge multiple times.

You cannot find a path from $$v_1$$ to $$v_6$$ that does not contain a collider: A path starting with $$v_1\to v_2\to v_5$$ contains $$v_2$$ as a collider, so we have to go via $$v_1\to v_2\to v_3\to v_4$$. But however we continue this path to $$v_6$$, there will always be a collider in this path. Thus, all paths from $$v_1$$ to $$v_6$$ contain colliders, thus $$v_1 \pperp v_6$$.

Using the notation $$Z=\{v_3, v_5\}$$, we need to find a path from $$v_1$$ to $$v_6$$ that is $$Z$$-$$\sigma$$-open. Once we have that, we have shown that $$v_1\npperp v_6 | Z$$.

Now, the path $$p = v_1\to v_2\to v_3\to v_4\to v_5\leftarrow v_4\leftrightarrow v_6$$ is a $$Z$$-$$\sigma$$-open path, because:

• $$p$$ is $$Z$$-$$\sigma$$-open at $$v_1$$ because $$v_1\not\in Z$$ (point 1. in the definition),
• $$p$$ is $$Z$$-$$\sigma$$-open at $$v_2$$ because of (c) in the definition,
• $$p$$ is $$Z$$-$$\sigma$$-open at $$v_3$$ because of (c) in the definition (this is different from d-separation!),
• $$p$$ is $$Z$$-$$\sigma$$-open at $$v_4$$ because of (c) in the definition,
• $$p$$ is $$Z$$-$$\sigma$$-open at $$v_5$$ because of (a) in the definition,
• $$p$$ is $$Z$$-$$\sigma$$-open at $$v_4$$ because of (b) in the definition, and
• $$p$$ is $$Z$$-$$\sigma$$-open at $$v_6$$ because $$v_6\not\in Z$$ (point 1. in the definition).

In a nutshell:
We have a $$\sigma$$-open path through the conditioned node $$v_3$$ because at $$v_3$$ the path "stays inside a loop" and the path is open at the conditioned node $$v_5$$ simply because it is a collider.

• wow awesome! thanks so much for the in-depth explanation Commented Aug 28, 2022 at 22:24
• Are you aware of any implementations/algorithms that can check if some X is sigma-separated from Y given Z? Commented Aug 31, 2022 at 21:43