Supplementary Material for VDE and GCFN A Theoretical Details and Proofs Notation We use the expectation operator in different contexts in the proof

We use the expectation operator in different contexts in the proof. Here, we show the full derivation of the lower bound for negative mutual-information. We derive the lower bound for the general case where there are both observed and unobserved confounders. The VDE optimization involves the expectations of distributions with parameters with respect to a distribution that also has parameters. In our experiments, we let the control function be a categorical variable.