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Markov Blanket [4 P]

Let $ X$ be a random variable corresponding to some node in a Bayesian Network. Recall that a Markov Blanket of $ X$ is a set of variables consisting of $ X$'s parents, children and children's parents in the Network. Thus

$\displaystyle MB(X) = Parents(X) \cup Children(X) \cup [ Parents(Children(X)) - \{ X \} ]$    

Denote by $ V_{-X}$ the set of all variables in the system (i.e. variables that correspond to some node in the Network) except for $ X$.

a)
[2 P]

Show that $ P(X\vert V_{-X}) = P(X\vert MB(X))$, i.e. a variable is conditionally independent of all other variables given its Markov Blanket. 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). Be sure to specify what a "non-descendant" node is.

b)
[2 P]

Show (e.g. by counter example) that $ Parents(Children(X))$ cannot be left out of the definition of a Markov Blanket. So you need to show that there is a network in which some $ X$ is not conditionally independent of all other nodes given $ Parents(X) \cup Children(X)$.


next up previous
Next: d-separation [4* P] Up: MLA_Exercises_2007 Previous: Bayes' Decision Theory [5+2*
Haeusler Stefan 2007-12-03