# Can't understand D seperation [duplicate]

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.

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