Tensor networks contraction and the belief propagation algorithm

Belief Propagation is a well-studied message-passing algorithm that runs over graphical models and can be used for approximate inference approximation of local marginals. The resulting approximations are equivalent to the Bethe-Peierls statistical mechanics. Here we show how this adapted world PEPS tensor networks as an contraction scheme. We further resultant ``mean field'' in Simple-Update algorithm, thereby showing latter essentially approximation. This shows one simplest algorithms schemes approximating marginals general, paves way using improvements BP algorithms.