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I have the following network:
I am told that B is independant of D. Why is this the case? Shouldnt that they are both connected to C break that independence based on the V shape?
In this case, I am told that B is not indep of D. Why is this the case?
A graphical model is the graphical representation of conditional probabilities.
So, your question can be answered by checking what set of conditional probabilities the model assumes.
By definition, your model is equivalent to the following set of equations :
$$ \begin{aligned} P (A, B, C, D) &= P(A) P(B|A) P(C|B) P(C |D) \\[7pt] P (C | A) &= \sum_{b \in B} P (C | b) P(b | A)\\[7pt] P(B|D) &= P (B) \end{aligned} $$
thus by definition $P(B|D) = P (B)$.
(Further, you can show that $P(D | C ) \neq P(B |C)$)
Again by definition,
$$ \begin{aligned} P (A, B, C, D) &= P(D) P(C|D) P(B|C, D) P(A|B) \\[7pt] P (A | C) &= \sum_{b \in B} P (A | b) P(b | C) \\[7pt] P (B | D) &= \sum_{c \in C} P (B | D, c) \end{aligned} $$
note that $P (B | D) \neq P(B)$
