Belief networks: independence examples

1 b) Conditioning on $C$ makes A and B (graphically) dependent $p(A,B|C)\ne p(A|C)p(B|C)$. Although the causes are a priori independent, knowing the effect $C$ can tell us something about how the causes colluded to bring about the effect observed.

2. Conditioning on $D$, a descendent of a collider $C$ makes $A$ and $B$  (graphically) dependent. $p(A,B|D)\ne p(A|D)p(B|D)$

3 a) $p(A,B,C)=$ $p(A|C)p(B|C)p(C)$. Here there is a 'cause' $C$ and independent 'effects' $A$ and $B$.

3 b) Marginalizing over $C$ makes $A$ and $B$ (graphically) dependent. $p(A,B)\ne p(A)p(B)$. Although we don't know the 'cause', the 'effects' will nevertheless be dependent. 

3 c) Conditioning on $C$ makes $A$ and $B$  independent. $p(A,B|C)=p(A|C)p(B|C)$. If you know the 'cause' $C$, you know everything about how each effect occurs, independent of the other effect. This is also true for reversing the arrow from $A$ to $C$—in this case, $A$ would 'cause' $C$ and then $C$ would 'cause' $B$. Conditioning on $C$ blocks the ability of $A$ to influence $B$.

Finally, these following graphs all express the same conditional independence assumptions.

David Barber

p42-43