What is the meaning of "mean-field"? In lots of Bayesian papers, people use variational approximation. In lots of them they call it  "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this context? 
 A: I believe the mean-field approximation used in mean-field variational Bayes is the assumption that the posterior approximation factorizes over the parameters
$$q(\mathbf{\theta}) = q_1(\theta_1) q_2(\theta_2) \dots q_n(\theta_n)$$
A: Mean-field approximation is a way to simplify the variational Bayes procedure. MFA makes it possible to use coordinate ascent to find the approximating function. See https://github.com/idnavid/misc/blob/master/variationalbayes_doc1.ipynb
A: I found some intuitions that might answer this; based on the definition of "mean-field" at Wikipedia, 
mean field theory (MFT also known as self-consistent field theory) studies the behaviour of large and complex stochastic models by studying a simpler model. Such models consider a large large number of small interacting individuals who interact with each other. The effect of all the other individuals on any given individual is approximated by a single averaged effect, thus reducing a many-body problem to a one-body problem.
So basically approximating the inference and learning problem, using independence assumptions and decomposition into several products, brings the notion of "mean-field" approximation. 
