6.3.2. D-Separation

D-Separation in Bayes Nets

Conditional independence relationships in Bayesian Networks can be directly read from the graph through D-separation, a graphical criterion that determines whether two nodes are independent given a set of observed nodes. Consider the three canonical three-node patterns:

• Causal Chain ($X\to Y\to Z$): it follows that $X/\perp Z$, but $X\perp Z\mid Y$.
• Common Cause ($X\leftarrow Y\to Z$): it follows that $X/\perp Z$, but $X\perp Z\mid Y$.
• Common Effect (Collider) ($X\to Y\leftarrow Z$): it follows that $X\perp Z$, but $X/\perp Z\mid Y$, or any other descendant of $Y$.

A path between two variables is said to be active if all its triples (segments of three nodes):

• …form a chain or common cause, and the middle node is unobserved.
• …form a common effect, and the middle node or one of its descendants is observed.

If all paths between $X$ and $Y$ are inactive given a set of observed nodes $Z$, then $X\perp Y\mid Z$. Otherwise, $X$ and $Y$ are D-connected and potentially dependent.

In practice, the D-separation algorithm proceeds as follows:

1. Shade all observed nodes $Z$.
2. Enumerate all undirected paths between $X$ and $Y$.
3. For each path, decompose it into triples and determine which are active.
4. If every path is inactive, $X\perp Y\mid Z$.