# 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.

• I'm not sure I understand that A and B are marginally dependent not given C. What do you mean by "marginally dependent" not given C? – nbro Jan 4 '19 at 16:16
• @nbro marginal dependence means $P(A|B) \ne P(A)$ as opposed to conditionally independent $P(A|B,C) = P(A|C)$. – AdamO Jan 4 '19 at 16:24
• But isn't A and B independent in the collider case? I don't get why you say they are dependent. I read that those causes are independent (to start with). They only become dependent given C (i.e. the collider), which I explained in the answers I gave to my own question here: stats.stackexchange.com/q/385519/82135. – nbro Jan 4 '19 at 17:02
• @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. – AdamO Jan 4 '19 at 17:09