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.
An Ergodic Walk 
“Pastime” is correct. Not “Pasttime”.
Whoops, my bad. Fixed now.
Thanks for the summary. Now, are the values conditionally independent if the path is blocked or if the path is not blocked? I can never remember.
If the “ball” can go through from A to B then A and B are dependent.
Hi Anand, Thanks for the paper link.
And if I’m right, even if the ball can go through, the nodes may still be independent. The only thing we can say with certainty is that if the ball *cannot* go through, the nodes are independent.
I should have been clearer — if the ball is blocked they are conditionally independent. That’s correct.