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In variational inference, the mean-field family of probability distributions is the set of distributions that factors over its terms (i.e. each component is independent of all others). This allows us to use "mean field approximations" of a probability distribution which may not have independent components.

Why is it called this? I.e. how is this related to the meaning of mean-field in physics/statistical mechanics? A system of $N$ interacting particles can be described using a mean field approximation if we replace pairwise interactions with an averaged interaction. In other words the force on a particle $x$ is no longer described by $\sum_{y} K(x,y)$ (sum of pairwise interactions with other particles $y$) but instead by $\int K(x,y) d\rho(y)$ where $\rho$ is the probability distribution of particles.

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