A Appendix A.1 Sufficiency of J

We describe the proofs for the sufficiency results from Section 5 here. Note that the first term in Equation 7 is zero by the conditional independence assumptions in Figure 8. Equating the expansions, we can see that to satisfy our assumption that The first term in Equation 9 is zero by the conditional independence assumptions in Figure 8. First note that the statement is not trivially true. Comparing this with the statement we'd like to prove, we can see that the key idea is to show that the MI equivalence implies that p ( Y | Z, X = x)= p ( Y | Z) . The didactic examples are computed as follows.