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d-separation II [3 P]

Let $ X$ be a random variable corresponding to some node in a Bayesian Network. Denote by $ N_{-X}$ the set of all variables in the system (i.e. variables that correspond to some node in the Network) except for $ X$ .

Find the smallest set of nodes $ D(X) \subseteq N_{-X}$ so that $ P(X\vert N_{-X}) = P(X\vert D(X))$ , i.e. variable $ X$ is conditionally independent of all other variables in $ N_{-X}$ given $ D(X)$ . Use the fact that $ X$ is conditionally independent of all its non-descendants given its parents (this is true by construction of the Bayesian Network). Specify precisely what types of nodes (their relation to $ X$ ) have to be included in $ D(X)$ and explain why.



Haeusler Stefan 2010-01-26