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Mean field inference in a general probabilistic setting

We present a systematic, model-independent formulation of mean eld 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 eld 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 eld equations). Iterating the mean eld equations means propagating information through the network. In general, however , there are multiple solutions to the mean eld equations. We show that improved approximations can be obtained by forming a weighted mixture of the multiple mean eld solutions. Simple approximate expressions for the mixture weights are given. The bene-ts 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 eld equations can be interpreted as a `disease scenario'.