  The variational autoencoder (VAE) is a powerful generative model that can
estimate the probability of a data point by using latent variables. In the VAE,
the posterior of the latent variable given the data point is regularized by the
prior of the latent variable using Kullback Leibler (KL) divergence. Although
the standard Gaussian distribution is usually used for the prior, this simple
prior incurs over-regularization. As a sophisticated prior, the aggregated
posterior has been introduced, which is the expectation of the posterior over
the data distribution. This prior is optimal for the VAE in terms of maximizing
the training objective function. However, KL divergence with the aggregated
posterior cannot be calculated in a closed form, which prevents us from using
this optimal prior. With the proposed method, we introduce the density ratio
trick to estimate this KL divergence without modeling the aggregated posterior
explicitly. Since the density ratio trick does not work well in high
dimensions, we rewrite this KL divergence that contains the high-dimensional
density ratio into the sum of the analytically calculable term and the
low-dimensional density ratio term, to which the density ratio trick is
applied. Experiments on various datasets show that the VAE with this implicit
optimal prior achieves high density estimation performance.
