(4) (5 points) Consider the following Bayesian network of 6 variables. Set $X = B$ and $Y = E$. Specify a set $Z$ such that: $X \perp\kern-5pt\perp Y | Z$ (in other words, $X$ is independent of $Y$ given $Z$). (i) $Z = \{A\}$ (ii) $Z = \{C\}$ (iii) $Z = \{F\}$
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In other words, knowing Z makes X and Y independent. We are given a Bayesian network and the variables X=B and Y=E. Show more…
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Let's look at the Bayesian networks below. We use the notation X ∑ Y to denote that the variable X is independent of Y, and (X₁ ∑ X₂|Y) to denote X₁ is independent of X₂ conditioned on Y. For each of these three network, write the factored joint distribution. For example, for a BN with structure X → Y, the factored joint distribution is P(X,Y) = P(X)P(Y|X). Using the joint distribution you wrote down for Fig(i), write down a formula for P(B, E). Now prove that B ∑ E in Fig(i). Similarly, prove that B ∑ M|A in the Bayesian network of Fig(ii), and M ∑ J|A in the Bayesian network of Fig(iii).
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