# d-separation

Given a directed graph $$G = (\mathbf{V},\mathbf{E})$$, a path $$P$$ between $$V_1,V_2 \in \mathbf{V}$$ is blocked by a set $$\mathbf{B}\subset \mathbf{V}$$ if:

• $$\exists$$ non-collider $$B$$ in the path $$P$$ such that $$B \in \mathbf{B}$$.
• $$\exists$$ collider $$B$$ in the path $$P$$ such that $$B \notin \mathbf{B}$$.

Sets of variables $$\mathbf{X},\mathbf{Y} \subset \mathbf{V}$$ are d-separated by $$\mathbf{B}$$, sometimes denote $$\mathbb{I}_G(\mathbf{X},\mathbf{Y}|\mathbf{B})$$ if all paths between $$\mathbf{X},\mathbf{Y}$$ are blocked by $$\mathbf{B}$$.

The notion of d-separation is important since they allow us to identify all conditional independencies that hold in Bayesian Networks (BNs), namely given a BN $$(G,\mathbb{P})$$, the distribution $$\mathbb{P}$$ entails all and only all the conditional independencies that are identified by d-separation in $$G$$ (Theorem 2.1 (Neapolitan and others 2004)).

## Thoughts

Created: 2022-04-04 Mon 23:39

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