What is the meaning of "mean-field"?

Solution 1:

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.

Solution 2:

I believe the mean-field approximation used in mean-field variational Bayes is the assumption that the posterior approximation factorizes over the parameters

$$q(\mathbf{\theta}) = q_1(\theta_1) q_2(\theta_2) \dots q_n(\theta_n)$$

Solution 3:

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