Statistics 4240 - Data Mining

Hi David & all. I have a question about the "k-wise" dependence that the guest lecturer mentioned last class. I would like to understand more about how k-wise dependence actually arises -- i.e., what kind of processes give rise to k-way dependence (without evidencing lower-order dependencies)? The simplest example I can think of is the XOR function. Thus, in C = xor(A,B), where A and B are independent binary variables, we have a 3-wise dependency without any pairwise dependencies. In other words, all pairs {A,B}, {A,C}, {B,C} are independent, but the triple {A,B,C} has dependency; that is, p(A,B,C) does not equal p(A)p(B)p(C). What is the special property (if there is one) of the XOR function that creates this situation? What other distributions have this property? Are they common or very rare? And how does one synthesize a distribution with this property? Does k-wise dependence (without lower-order dependencies) represent some form of singularity in the probability space? If anyone has any insights (or ideas for further reading), I would be eager to hear them. Thanks very much. -- Dave F.