I have a diagram of a Bayes network as shown below:
$$\begin{array}{c}A&&&&B&&&&C\\&\searrow&&\swarrow\\&&D&&&&E\\&&&\searrow&&\swarrow\\&&&&F\end{array}$$
The only independence assumptions between the random variables A to F are those enforced by the shape of this network.
The professor said that "conditioned on D, we have A is conditionally independent of B".
But from what I have read and learnt from $d$-separation, wouldn't that be incorrect? Since $d$-separation states that if we have a path $X\rightarrow Y\leftarrow Z$, $X$ is guaranteed to be independent of $Z$ if $Y$ and its descendents are not in the evidence set, i.e. they are not been conditioned on.
So $A$ is only independent of $B$ if $D$ and $F$ is not conditioned on?
Could the answer for this be confirmed please?
Yes, $D$ is a common effect of nodes $A$ and $B$. Its presence in the conditioning set will block the only path between them. They are not d-separated.
Still, this does not prohibit them from being conditionally independent.
D-separation entails conditional independence, but conditional independence does not need d-separation.
However, the graph alone is not enough to proclaim that the nodes are conditionally independent. Are there any attached tables the professor may be using to say so?