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.
October 20, 2010 at 1:43 pm
“Pastime” is correct. Not “Pasttime”.
October 20, 2010 at 1:48 pm
Whoops, my bad. Fixed now.
November 20, 2011 at 7:39 am
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.
November 21, 2011 at 12:06 pm
If the “ball” can go through from A to B then A and B are dependent.
September 16, 2012 at 3:56 pm
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.
September 17, 2012 at 8:05 am
I should have been clearer — if the ball is blocked they are conditionally independent. That’s correct.