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I have the following network:

enter image description here

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?

enter image description here

In this case, I am told that B is not indep of D. Why is this the case?

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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.


Above figure

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 $P(D | C ) \neq P(B |C)$)


Below figure

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 $P (B | D) \neq P(B)$

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