In Bayesian network graphs, how to systematically search for pairs of conditional independent nodes, and the associated condition node(s)? Is there some simple rules or algorithms to follow?
There is the Bayes-Ball Algorithm, described in this paper: http://www.gatsby.ucl.ac.uk/~zoubin/course05/BayesBall.pdf
On page 22 of these slides there is a handy diagram that summarizes the algorithm: http://ai.stanford.edu/~paskin/gm-short-course/lec2.pdf
Yes, you can find conditionally independent nodes by evaluating the graphical structure of the Bayesian network. Check out the notion of D-separability which provides 3 rules:
1) X -> Y -> Z
2) X <- Y -> Z
For both (1) and (2), Z is conditionally independent of X given Y
3) X -> Y <- Z
Andrew Moore has some explanation of these rules on his website.