I'm working through variational Bayesian methods at the moment, and I think I have a grasp of the bigger picture. Where I sometimes have trouble is with the exact details of how it can be implemented. Right now, this centrs on the idea of a mean field variational family. Specifically, Blei et al. say the following:
In this review we focus on the mean-field variational family, where the latent variables are mutually independent and each governed by a distinct factor in the variational density. A generic member of the mean-field variational family is
$$q(z) = \prod_ {j=1}^m q_j (z_j )$$
I'm afraid that I can't see how a distribution can be expressed as a product in this way without being reduced to a constant. Clearly, I'm missing something fundamental, but I seem to be going around in circles trying to google the answer.
Can anyone supply some intuition?