Belief propagation can describe a particular kind of Message passing process in Networks, and the related algorithm for performing Inference in Conditional random fields. It is approximate (except for tree graphical models)

• The forward-backward algorithm of CRFs is a special case, it is belief propagation for the linear chain case. Belief propagation yields exact inference if the CRF graph is a tree, by computing marginals. How to use
• On graphs with loops, belief propagation can be used to do approximate inference. But one has to be careful with divergences (c.f. infinities in Quantum field theory, and loops in Feynman diagrams).

Neural networks [3.10] : Conditional random fields - belief propagation. See Conditional random field, Factor graph

Example

9 - 1 - Belief Propagation- Probabilistic Graphical Models - Professor Daphne Koller

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Belief propagation algorithm

defines cluster graph

Belief propagation in statistical physics of constraint satisfaction problems. In the context of Statistical physics, it's called cavity method

In Coding theory, they call it sum-product algorithm

Belief propagation in factor graphs with factors involving two variables only

Cluster graphs must satisfy the family preservation and the running intersection properties

Bethe cluster graph is a commonly used type of cluster graph, which comes from Statistical physics

Convergence of belief propagation implies callibration (consistency among factors of marginal probabilities)

Many general purpose libraries.

See also related concepts/models in Dynamical systems on networks