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?
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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.