One of the fun thing about graphical models is that arguments can be done by looking at diagrams (kind of like a diagram chase in algebraic topology). One such trick is from R.D. Shachter’s paper in UAI called “Bayes-Ball: The Rational Pastime (for Determining Irrelevance and Requisite Information in Belief Networks and Influence Diagrams)” (see it here. for example). This is a handy method for figuring out conditional independence relations, and is a good short-cut for figuring out when certain conditional mutual information quantities are equal to 0. The diagram below shows the different rules for when the ball can pass through a node or when it bounces off. Gray means that the variable is observed (or is in the conditioning). I tend to forget the rules, so I made this little chart summary to help myself out.