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?
$\endgroup$3 Answers
$\begingroup$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.
$\endgroup$ $\begingroup$I believe the mean-field approximation used in mean-field variational Bayes is the assumption that the posterior approximation factorizes over the parameters
$$q(\mathbf{\theta}) = q_1(\theta_1) q_2(\theta_2) \dots q_n(\theta_n)$$
$\endgroup$ 3 $\begingroup$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
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