Variational autoencoders
A variational autoencoder (VAE) is a generative model proposed by Kingma and Wellin (in their work Kingma D. P., Wellin M., Auto-Encoding Variational Bayes, arXiv:1312.6114 [stat.ML]) that partially resembles a standard autoencoder, but it has some fundamental internal differences. The goal, in fact, is not finding an encoded representation of a dataset, but determining the parameters of a generative process that is able to yield all possible outputs given an input data-generating process.
Let's take the example of a model based on a learnable parameter vector 
and a set of latent variables 
that have a probability density function 
. Our goal can, therefore, be defined as the research of the 
parameters that maximize the likelihood of the marginalized distribution 
(obtained through the integration of the joint probability 
):
If this problem could be easily solved in closed form, a large set of samples drawn from the 
data-generating process...
