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# Mean field inference in a general probabilistic setting

*Proceedings of the Seventh International Workshop on Artificial Intelligence and Statistics*, PMLR R2, 1999.

#### Abstract

We present a systematic, model-independent formulation of mean field theory (MFT) as an inference method in probabilistic models. "Model-independent" means that we do not assume a particular type of dependency among the variables of a domain but instead work in a general probabilistic setting. In a Bayesian network, for example, you may use arbitrary tables to specify conditional dependencies and thus run MFT in any Bayesian network. Furthermore, the general mean field equations derived here shed a light on the essence of MFT. MFT can be interpreted as a

*local*iteration scheme which relaxes in a consistent state (a solution of the mean field equations). Iterating the mean field equations means propagating information through the network. In general, however, there are multiple solutions to the mean field equations. We show that improved approximations can be obtained by forming a weighted mixture of the multiple mean field solutions. Simple approximate expressions for the mixture weights are given. The benefits of taking into account multiple solutions are demonstrated by using MFT for inference in a small Bayesian network representing a medical domain. Thereby it turns out that every solution of the mean field equations can be interpreted as a ’disease scenario’.