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Consider the Bayesian Network Structure Below, decide whether the statements are true or false.

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a) If every variable in the network has a Boolean state, then the Bayesian network can be represented with 18 numbers (probabilities).

b) $G \perp \!\!\! \perp A$ (G is independent of A)

c) $E \perp \!\!\! \perp H | \{D,G\}$ (E and H are conditionally independent given D and G).

d) $E\perp \!\!\! \perp H | \{C,D,F\}$ (E and H are conditionally independent given C,D and F).

For a) I believe the formula for the number of probabilities needed is $n \times 2^k$ where $n$ is the number of variables in the network and $k$ is the maximum number of parents any given node has. This gives $8 \times 2^2 = 32$ probabilities needed so the statement is false according to my calculations.

b) For the rest of the problems I applied the d-separation algorithm as explained here http://web.mit.edu/jmn/www/6.034/d-separation.pdf Using this approach I found that G and A are indeed independent.

c) True (using d-separation).

d) Also true (d-separation).

Could anyone please verify/correct my answers? I feel like at least one of the problems $b,c$ or $d$ should be false.

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I disgree with a) and d)

a) I don't understand your formula.. The network has 9 nodes with binary state hence it has $2^9$ configurations.

d) Following the moralization step you have a connection between $E$ and $G$ which links $E$ and $H$.

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  • $\begingroup$ And then for c) you get independence since you don't have the link between $E$ and $G$ because the node $F$ is not in the ancestral graph? In other words, we should only draw the line between pairs of variables with a common child in the moralization step if the child is on the ancestral graph? $\endgroup$
    – Pame
    Jan 27, 2021 at 21:20
  • $\begingroup$ yes but a stronger point for c) is that $G$ is a given! $\endgroup$
    – TheCG
    Jan 28, 2021 at 6:55

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