# Why a path in a causal graph can have edges not all with the same direction?

In a fork, A <- C -> B, A and B are independent given C. We can say that A and B are d-separated or the path between them is blocked by C, given C. So information does not flow (why?). In a chain, A -> C -> B, A and B are independent given C. Again, we can say that A and B are d-separated or the path between A and B is blocked by C, given C. However, in a collider, the situation is somehow the opposite. In a collider, A -> C <- B, A and B are independent NOT given C, but if C (or any descendant of C) is given they are dependent.

What exactly is a path in a casual model? A path can apparently have edges not all of the same direction, but why? I read that a path somehow tells us if information flows. Which info? Why does a path would allow information to flow (or not)? In which circumstances does info flow or not along a specific path (i.e. a chain, fork and collider)? In which sense information flows or not flows along these paths?

The challenge in DAGs is that a single probability distribution can support multiple possible DAGS.

A path in a causal model represents a relation to the "do" operator. If you "do" A, what is the potential outcome of B or C? If the probability distribution for either node shifts as a result of "doing" A, then there is a causal path between them. This is all covered in "Causality" by Pearl.

The notion that a causal path shows a "flow of information" is vague and non-specific, and when you consider the statistical definition of information as a theoretical quantity, it's wrong. It's unfortunate that causality has a lot of overlapping nomenclature with probability, but probability and statistics only serve to quantify causal relationships, not identify them. When you draw the DAG, you impose a structure to the probability distribution that allows you to estimate potential outcomes. If you change that structure, the results and inference are completely different. As an aside, this is a common issue in structural equation modeling.

• @nbro marginal dependence means $P(A|B) \ne P(A)$ as opposed to conditionally independent $P(A|B,C) = P(A|C)$. Commented Jan 4, 2019 at 16:24
• @nbro thanks you're right. I'll revise my answer. On another note, I added to my answer to address the second half of your question... I don't see how it relates to the confusion about confounders, colliders, and mediators, though. Commented Jan 4, 2019 at 17:09
• I don't think paths have to do with the do operator. For example, in the collider network, P(B | do(A)) = P(B). There is no path given do(A). And yet there is a path between A and B if C is unobserved. I also disagree that paths showing flow of information is "vague"; it's just a visual language for describing dependence. Commented Nov 17, 2022 at 18:58
• @NeilG By referring to a DAG, I mean that the edges represent causation, so A->B means P(B|do(A)) != P(B). Same notation in Pearl's Causality. Depicting dependence would preclude using arrows rather than undirected edges, because Cov(A,B) = Cov(B,A). Anyway, I can't visualize the collider example you are describing. Commented Dec 20, 2023 at 16:19

A path between variables just means that they are dependent. We call it a path because it is a path—that obeys the d-separation rules—in a causal graph.

have edges not all of the same direction, but why?

You answered this question in your question. It's because of the d-separation rules.

In which circumstances does info flow or not along a specific path (i.e. a chain, fork and collider)?

You answered this in your question. You can review the d-seperation rules if you want a formal definition.

In a fork, A <- C -> B, A and B are independent given C. We can say that A and B are d-separated or the path between them is blocked by C, given C. So information does not flow (why?).

How can information flow from A to B if C were observed? What more does A tell you about B? If C's unobserved, information would flow because A tells you about C, which tells you about B.

However, in a collider, the situation is somehow the opposite.

You can reason about this through common sense. If A and B cause C, then, not knowing C, A tells you nothing about B. Given C, then potentially A or B may be responsible for C, and knowing that, for example, A couldn't have caused C, implies that B did. This is explaining away.

Let's assume that effects have many causes. For example, we can have $$A$$ and $$B$$ causing $$C$$, and $$B$$ and $$D$$ causing $$E$$. The causal diagram is below.

If we somehow keep constant (adjust, control, regress, match, select....) the common causes of $$C$$ and $$E$$, which is $$B$$, the changes they will have will come from their other causes ($$A$$ for $$C$$ and $$D$$ for $$E$$), which are independent, that is, $$A \mathrel{\unicode{x2AEB}} D$$. That's why you can measure some dependence (due to the common causes), but it will disappear once you adjust for the common causes. This is an example of confounding ($$A \leftarrow B \rightarrow C$$), and we would have the same reasoning for a chain ($$A \rightarrow B \rightarrow C$$). If the relationship between $$A$$ and $$C$$ are somehow interrupted, they become independent. $$A$$ causes $$B$$ that causes $$C$$. If $$B$$ is a system and we turn it off, $$A$$ won't have an effect on $$C$$, for the only way to cause $$C$$ was through $$B$$.

Things get trickier when we have a collider ($$A \rightarrow B \leftarrow C$$). If the only relationship between $$A$$ and $$C$$ is through their common effect $$B$$, they're independent. But if you adjust for their common child, you get some information (correlation, statistical dependence). One nice example is throwing dice. If I throw two dice, one after the other, it's reasonable to assume each throw is independent of the other. If I ask you to guess what number I got in the first throw, you won't have anything to help you answer this. If I ask about the second throw, the same. If I tell you the result in my first throw, this doesn't help you guess the second throw. So indeed they're independent. However... If I tell you the first throw and the sum of both results (which is caused by both throws, their sums), you can guess the second throw.

What exactly is a path in a casual model? A path can apparently have edges not all of the same direction, but why? I read that a path somehow tells us if information flows. Which info? Why does a path would allow information to flow (or not)? In which circumstances does info flow or not along a specific path (i.e. a chain, fork and collider)? In which sense information flows or not flows along these paths?

In this context we're discussing, a path is a set of edges between two nodes that does not contain colliders ($$\rightarrow \leftarrow$$). It may look counterintuitive to call $$A \leftarrow B \leftarrow C \leftarrow D$$ a path between $$A$$ and $$D$$ or from $$A$$ to $$D$$, with edges oriented backward, but it is a path just like $$A \leftarrow B \leftarrow C \rightarrow D$$.

What is this info? It's a measure of statistical dependence. In some circumstances, this is spurious statistical dependence (in the sense that it is not direct). If I want to measure the dependence between $$A$$ and $$C$$ but they are only connected through a confounder $$B$$ ($$A \leftarrow B \rightarrow C$$), not adjusting for $$B$$ will still give me information, but a biased one. It can help with prediction, but intervening on $$A$$ won't have an effect on $$C$$ because they're not directly related.

There are sets of rules that help such analyses, but there are also analytical demonstrations. I tried to explain to you in an intuitive way, with daily examples.