  Random factor graphs provide a powerful framework for the study of inference
problems such as decoding problems or the stochastic block model.
Information-theoretically the key quantity of interest is the mutual
information between the observed factor graph and the underlying ground truth
around which the factor graph was created; in the stochastic block model, this
would be the planted partition. The mutual information gauges whether and how
well the ground truth can be inferred from the observable data. For a very
general model of random factor graphs we verify a formula for the mutual
information predicted by physics techniques. As an application we prove a
conjecture about low-density generator matrix codes from [Montanari: IEEE
Transactions on Information Theory 2005]. Further applications include phase
transitions of the stochastic block model and the mixed $k$-spin model from
physics.
