you may bug me on this. i work too closely with alp :)

edward does not implement completely implement advi yet. the piece that is missing is the automated transformation of constrained latent variable spaces to the real coordinate space. however, edward offers much more flexibility in specifying the dependency structure of the posterior approximation. diagonal is, just like in stan, the fastest and easiest. however introducing structure (e.g. assigning a dense normal to some of the latent variables while assigning a diagonal to others) is much easier in edward.