Supplementary Materials A Theoretical proofs

We first prove the direction Z T SI (Z; T) = 0, which is equivalent to prove I ( Z; T) = 0 SI (Z; T) = 0 . We now consider the direction SI (Z; T) = 0 Z T . Now consider the moment generating functions. T)) > 0 for the chosen h, g, θ, ϕ (since h, g are not constant functions). A.2 Proof of Theorem 2 Theorem 2. Provided that each h We prove the contrapositive, i.e. rather than show LHS = RHS, we show that RHS = LHS .