I have came across this statement several time now
Maximizing likelihood is equivalent to minimizing KL-Divergence
(Sources: Kullback–Leibler divergence and Maximum likelihood as minimizing the dissimilarity between the empirical distriution and the model distribution)
I would like to know in applications such as VAE, why use KL- divergence then over MLE? In which applications would you choose one over the other? And any specific reason for it given both are equivalent?