But aside from that, there is an information-theoretic view on why you might prefer VAEs over AEs. In short, having p(z|x) not be point-mass (aka an ordinary AE) allows you to bound the information flow through the bottleneck. KL loss on p(z|x) forces the network to be honest about how much information it is cramming into z for the purposes of reconstruction.
To unpack that a bit: in theory, even a single real-valued latent variable z could store an arbitrary amount of information (if the encoder and decoder conspired cleverly enough). But if you make z stochastic, or in other words if your encoder's job is to calculate the parameters of a distribution from which you sample z, you're essentially introducing a noisy channel in the middle of your network, and you can then bound how much information is flowing across that channel. But to do that you still need to use KL divergence loss to encourage p(z|x) to approximate your chosen latent distribution, otherwise your encoder and decoder might cheat, e.g. by using near-point-mass z as a way to turn back into ordinary AEs again.
Or in deep learning speak, it's a form of regularization with a particularly rich and interpretable statistical motivation.
What I don't really intuit is: is it just basically doing regularization, or is the interpretation in terms of learning to infer the posterior meaningful?
> in mean-field variational inference, we have parameters for each datapoint ... In the variational autoencoder setting, we do amortized inference where there is a set of global parameters ...
Mean-field implies the variational posterior is modelled as factorising over the different latent variables involved. Some latent variables can be local (unique to a data point) and some can be global (shared across data points).
