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Approximate interference technique in Bayesian networks

Variational message passing (VMP) is an approximate inference technique for continuous or discrete valued Bayesian networks, with conjugate exponential parents, developed by John Winn. VMP was developed as a means of generalizing the approximate variational methods used by such techniques as latent Dirichlet allocation, and works by updating an approximate distribution at each node through messages in the node's Markov blanket.

Parents send their children the expectation of their sufficient statistic while children send their parents their natural parameter, which also requires messages to be sent from the co parents of the node.

Because all nodes in VMP come from exponential families and all parents of nodes are conjugate to their children nodes, the expectation of the sufficient statistic can be computed from the normalization factor.

The algorithm begins by computing the expected value of the sufficient statistics for that vector. Then, until the likelihood converges to a stable value (this is usually accomplished by setting a small threshold value and running the algorithm until it increases by less than that threshold value), do the following at each node:
Get all messages from parents. Get all messages from children (this might require the children to get messages from the co parents). Compute the expected value of the nodes sufficient statistics.

Because every child must be conjugate to its parent, this has limited the types of distributions that can be used in the model. For example, the parents of a Gaussian distribution must be a Gaussian distribution (corresponding to the Mean) and a gamma distribution (corresponding to the precision, or one over in more common parameterizations). Discrete variables can have Dirichlet parents, and Poisson and exponential nodes must have gamma parents. More recently, VMP has been extended to handle models that violate this conditional conjugacy constraint.