This in turn means that J var = H(Z) − H(q, p) is a lower bound on I(X, Z), hence the name.
the first term is (using (10) again) the second term is (as a sum over batches) In contrast, the numerator term estimated as an average over minibatches is and the two terms of its gradient with respect to q (corresponding to (12) and 13  the difference between (12) and 14is where the difference between (13) and (15) is Adding these differences gives the second result.