So suppose we have a neural network that aims to map values from one distribution to another. That is to say the inputs do not belong to the same distribution as the targets.
It then follows that, technically the usual formulation of the ELBO would not be applicable even if we incorporate the targets into the reconstruction term.
But is there a formulation of the ELBO that would allow us to do so in a more Bayesian manner, i.e., create predictions of the targets through a predictive distribution by sampling from the posterior then passing that through the decoder to predict the target?
