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In general the dependence can be arbitrarily complex as the following identity illustrates,P x P x 1 P x 2 x 1 P x 3 x 1,x 2
...Therefore,to capture all dependence data we would need to condition each variable in turn on a steadily increasing set of other variables
...where m 1,m 2,...Pt x P x 1 P x 2 x 1 P x 3 x 2 P x 4 x 2 P x 5 x 2 P x 6 x 5 Notice how similar the A 2 assumption is to the independence assumption A 1,the only difference being that in A 2 each factor has a conditioning variable associated with it
...The permutation and the function j
...write the function Pt x the way I did with xi as the unconditioned variable,and hence the root of the tree,and all others consistently conditioned each on its parent node,in fact any one of the nodes of the tree could be singled out as the root as long as the conditioning is done consistently with respect to the new root node
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