  In this paper we propose two novel bounds for the log-likelihood based on
Kullback-Leibler and the R\'{e}nyi divergences, which can be used for
variational inference and in particular for the training of Variational
AutoEncoders. Our proposal is motivated by the difficulties encountered in
training VAEs on continuous datasets with high contrast images, such as those
with handwritten digits and characters, where numerical issues often appear
unless noise is added, either to the dataset during training or to the
generative model given by the decoder. The new bounds we propose, which are
obtained from the maximization of the likelihood of an interval for the
observations, allow numerically stable training procedures without the
necessity of adding any extra source of noise to the data.
