  The loop series provides a formal way to write down corrections to the Bethe
entropy (and/or free energy) of graphical models. We provide methods to
rigorously control such expansions for low-density parity-check codes used over
a highly noisy binary symmetric channel. We prove that in the asymptotic limit
of large size, with high probability, the Bethe expression gives an exact
formula for the entropy (per bit) of the input word conditioned on the output
of the channel. Our methods also apply to more general models.
