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