This chapter discusses a general method for approximating marginals of large graphical models. This powerful technique has been discovered independently in various fields: statistical physics (under the name ‘Bethe Peierls approximation’), coding theory (‘sum-product’ and ‘min-sum’ algorithms), and artificial intelligence (‘belief propagation’). It is based on an exchange of messages between variables and factors, along the edges of the factor graph. These messages are interpreted as probability distributions for the variable in a graph where a cavity has been dug. The chapter also discusses the statistical analysis of these messages in large random graphical models: density evolution and the replica symmetric cavity method.
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