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Let us consider variables $A,B,C,D$ such that

  • $A$ is conditionally independent of $B$ given $C$, i.e. $A \bot B|C$
  • $A$ is conditionally dependent on $B$ given $D$, i.e. $A \not\bot B|D$

Are $A$ and $B$ conditionally independent of each other given $C,D$? Are they not? Can we conclude anything about their conditional independence?

That is, can we say anything about the following claim? $A \bot B|C,D$

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A situation where $A \perp B \mid C, D$ (dependence arrows run top-to-bottom, i.e. $C \to A$):

D       C
      /   \
     A     B

A situation where $A \not\perp B \mid C, D$:

  C
 / \
A   B
 \ /
  D

So we can't say one way or the other.

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