  A new method for analyzing low density parity check (LDPC) codes and low
density generator matrix (LDGM) codes under bit maximum a posteriori
probability (MAP) decoding is introduced. The method is based on a rigorous
approach to spin glasses developed by Francesco Guerra. It allows to construct
lower bounds on the entropy of the transmitted message conditional to the
received one. Based on heuristic statistical mechanics calculations, we
conjecture such bounds to be tight. The result holds for standard irregular
ensembles when used over binary input output symmetric channels. The method is
first developed for Tanner graph ensembles with Poisson left degree
distribution. It is then generalized to `multi-Poisson' graphs, and, by a
completion procedure, to arbitrary degree distribution.
