d-separation is a fundemntal concept in the SCM, graphical flavour of causal inference, as well as in Bayesian nets more generally. It gives a systematic way to predict whether covariates represented by a causal graph will show dependence or independence in any dataset that is generated according to a particualr causal graph. The "d" part of the name thus reference to "dependence". To understand the concept better, we first need to have a quick introduction to the relationships between variables that make up a graph.
The fundamental building blocks of a causal graph, decribing every possible relationship between any three related variables, are the following:
-
Confounders: These are variables which are common causes of two others, i.e. for some random variables X, Y, Z, X <- Z -> Y. Confounders imply both a) dependence between X and Y, X¬⫫Y, and b)conditional independence given Z, X⫫Y|Z. Alternatively, some sources refer to this conditional indepepndence of X and Y (the children) given the common cuase Z (the parent) as Z "screening off" X and Y from each other.
Confounders are the main threats behind Confounding Bias, and a common example can be if we try to explain the relationship between ice cream consumption (X) and number of death cases caused by drowning in pools (Y). While X and Y can show positive correlation, it is likely that the two have a lurking common cause behind both, hot weather (Z). As such, it is likely that we have the following structure
Ice_creatm_consumption (X) <- hot_weather (Z) -> death_cases_in_pools (Y).
(Confounders are related to the Reichenbach's common cause principle, see principle 1.1 in Elements of Causal Inference)
-
Mediators: variables in the causal graph that can thought of as the mechanism describing a relationship between two, more remotely associated variables, i.e.
X -> Z -> Y (treatment -> mediator -> outcome).
For example, a mediator could represent the process by which taking aspirin alleviates headaches. While taking an aspirin pill is know to reduce headaches, it is not the pill itself, but the multiple chemical reactions of the contents of the pill that reduce the headache, i.e.
medication (X) -> chemical_recation (Z) -> Y.
While Z again renders X and Y dependent, X¬⫫Y, conditioned on Z, i.e. the mechanism, we once again lose the association between X and Y, i.e. X⫫Y|Z.
-
Colliders: when two seemingly unrelated phenomena, represented by random variables, are related by a relationship to a third one, we see colliders. This phenomenon si also referred to as v-structure or immorality¹, with a slight traditional family moralistic undertone, derived from the fact we are looking at a child node from two parent nodes which are not themselves adjacent). In other words, we look at a case where both some X and Y point at a third variable Z,
X -> Z <- Y.
For example, a study might look at the occurrence of two diseases, say cancer (X) and tuberculosis (Y). While the two might not be correlated in the general population, if we relate them by performing the study at a particular hospital (Z), we could be linking the two -- making them seem negatively associated when we condition on presence in hospital Z. Following this logic, we see that in this case, unlike with confounders or mediators, cancer (X) and tuberculosis (Y) might be marginally independent, X⫫Y, and yet become dependent when conditioning on hospital presence (Z), X¬⫫Y|Z. This example is also described as Berkson's paradox.
When trying to systemtically reason through bigger networks and graphs, where nodes are often conncected via more than one path, we can use the insights about dependence and independence above in a more formalized manner. That is where Pearl's d-separation comes in.
A path p is blocked by a set of nodes Z, if and only if
- p contains a chain of nodes A -> B -> C (i.e. B is a mediator) of a fork A <- B -> C (i.e. B is a confounder), such that the middle node B is in Z (i.e., B is conditioned on), or
- p contains a collider A -> B <- C (i.e. B is a collider) such that the collision node B is not in Z, and no descendant of B is in Z. If Z blocks every path between two nodes X and Y, then X and Y are d-separated, conditional on Z, and thus are independent conditional on Z.
With this definition, we can start to reason through cases where we would like to d-separate variables, and ones where we would like to maintain variables d-connected. For example, when studying a causal effect of interest, we would want to block paths with confounders, that could trick us into thinking some variable X causes another Y, where really their relationship is explained by a third common cause (just like ice cream consumption doesn't cause pool drownings in the example above). On the other hand, we would also not like to block mediation paths (e.g. when trying to assess the efficacy of a medication for reducing headache we would like to keep the mediating mechnism active, and not condition on it, such that the effect of medication taking on headache appearance will seem to disappear); and we would similalry not like to open currently blocked paths with colliders in them, as when studying the relation between two diseases, we would not like to condition on a random variables, such as hospitalization, that would make the two seem related when in the general population they are not.
d-sepeation underlies many topics in the graphical interpretation of causal inference, and features, for example, in the Backdoor criterion and Frontdoor criterion for causal effect recovery/estimation.
- Pearl's d-separation without tears
- Richard Scheines's d-separation tutorial
- Pieter Abbeel's d-separation video for Bayesian nets
- d-seperation, chpater 2.4 in causal inference in statistics: A Primer
- For a slightly different flavour, see Elements of Causal Inference, chapter 6, definition 6.1