Minimizing the difference in the marginal distributions can be regarded as a coarse approximation to the minimization of the distance.
we have to concede that the above argument is fairly informal, as the generalization bounds do not directly apply to our case: (1) our feature representation is learned from the same data as the classifier, (2) we cannot guarantee that the existence of a domainindependent scoring function is preserved under the learned transformation x→z and (3) in our setting we have access not only to samples from P (z|x, θ) but also to the distribution itself.