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?
Putting a sparsity loss on z in a regular AE will encourage the code to have smaller magnitudes, and with relu those units will tend to saturate to zero, yes.
But the original point was that even a single continuous unit can be used to transmit an arbitrary amount of information. Not so much that this happens in practice, because the encoder and decoder would need access to something like modulo to do the most obvious kinds of cheating, but just that from an information theory point of view you can't really talk about how much information a continuous variable transmits unless you are transmitting it over a noisy channel and can measure entropies of distributions (and indeed you can formally derive how a given KL loss bounds the information transmitted by z).
> 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?
Both, which I think is really nice. You can look at it either way.
The Bayesian interpretation is powerful because you now have a principled way to calculate p(x), which you didn't have before. And you can introduce multiple latent variables in your network (as long as no layers take inputs from both ordinary and sampling layers) and so you have some flexibility to do limited forms of graphical modelling that supports efficient forward inference and GPU acceleration. And the inference machinery can be trained via cheap backpropagation instead of expensive sampling.
> 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).