Number of Variables Needed to Represent Bayesian Network and Independence

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

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.

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 $$H$$.
• 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?
• yes but a stronger point for c) is that $G$ is a given! Commented Jan 28, 2021 at 6:55