  Variational autoencoders (VAEs) are one class of generative probabilistic
latent-variable models designed for inference based on known data. We develop
three variations on VAEs by introducing a second parameterized encoder/decoder
pair and, for one variation, an additional fixed encoder. The parameters of the
encoders/decoders are to be learned with a neural network. The fixed encoder is
obtained by probabilistic-PCA. The variations are compared to the Evidence
Lower Bound (ELBO) approximation to the original VAE. One variation leads to an
Evidence Upper Bound (EUBO) that can be used in conjunction with the original
ELBO to interrogate the convergence of the VAE.
