d-separation · Tae Hyun Kim (Lowell)

Definition

d-separation (directional separation) is a graphical criterion in a DAG for determining whether two sets of variables are conditionally independent given a third set.

Formal Definition:

In a DAG $G$ , if $Z$ d-separates $X$ and $Y$ :

$X \perp_G Y \mid Z$

every path between $X$ and $Y$ is blocked by $Z$ .

Path Blocking Rules

Three Elementary Structures

Structure	Name	Blocked by $Z$ ?
$A \rightarrow B \rightarrow C$	Chain	blocked if $B \in Z$
$A \leftarrow B \rightarrow C$	Fork	blocked if $B \in Z$
$A \rightarrow B \leftarrow C$	Collider	blocked if $B \notin Z$ and $De(B) \cap Z = \emptyset$

Blocking Rule Summary

A path $\pi$ is blocked by $Z$ if:

A non-collider (chain or fork) intermediate node is in $Z$
A collider intermediate node and its descendants are not in $Z$

Intuitive Understanding

Core idea:

d-separation = “blocking the flow of information”

Fork (Common Cause):
    A ← B → C

    - Given B: A, C independent (confounding removed)
    - Not given B: A, C associated (via B)

Chain (Mediation):
    A → B → C

    - Given B: A, C independent (mediator blocked)
    - Not given B: A, C associated (causal path)

Collider (Common Effect):
    A → B ← C

    - Not given B: A, C independent (default blocked)
    - Given B: A, C associated! (collider bias)

d-separation Algorithm

Input

DAG $G$
Sets $X$ , $Y$ , $Z$

Procedure

For each path π between X and Y:
    blocked = False
    For each node B on path π (excluding endpoints):
        If B is a non-collider on π:
            If B ∈ Z:
                blocked = True
                break
        Else (B is a collider on π):
            If B ∉ Z and De(B) ∩ Z = ∅:
                blocked = True
                break

    If not blocked:
        return "X and Y are NOT d-separated by Z"

return "X and Y are d-separated by Z"

Faithfulness Assumption

Markov Property

$P$ is Markov relative to DAG $G$ : $X \perp_G Y \mid Z \implies X \perp_P Y \mid Z$

d-separation in the graph → probabilistic independence

Faithfulness

$P$ is faithful to DAG $G$ : $X \perp_P Y \mid Z \implies X \perp_G Y \mid Z$

probabilistic independence → d-separation in the graph

Combined (Perfect Map): $X \perp_P Y \mid Z \iff X \perp_G Y \mid Z$

Examples

Example 1: Simple Chain

X → Y → Z

$X \perp_G Z \mid Y$ ? Yes (Y blocks the chain)
$X \perp_G Z$ ? No (path open)

Example 2: Fork (Confounding)

    C
   ↙ ↘
  X   Y

$X \perp_G Y \mid C$ ? Yes (C blocks the fork)
$X \perp_G Y$ ? No (confounding path open)

Example 3: Collider

X → C ← Y

$X \perp_G Y$ ? Yes (collider blocks by default)
$X \perp_G Y \mid C$ ? No (conditioning opens collider)

Example 4: Complex Graph

    A
   ↙ ↘
  B   C
   ↘ ↗
    D

$A \perp_G D$ ? Check all paths:
- $A \rightarrow B \rightarrow D$ : Open
- Not d-separated
$A \perp_G D \mid B$ ?
- $A \rightarrow B \rightarrow D$ : Blocked (B ∈ Z)
- $A \rightarrow C \rightarrow D$ : Open
- Not d-separated
$A \perp_G D \mid \{B, C\}$ ?
- $A \rightarrow B \rightarrow D$ : Blocked
- $A \rightarrow C \rightarrow D$ : Blocked
- d-separated!