  We consider generic optimal Bayesian inference, namely, models of signal
reconstruction where the posterior distribution and all hyperparameters are
known. Under a standard assumption on the concentration of the free energy, we
show how replica symmetry in the strong sense of concentration of all
multioverlaps can be established as a consequence of the Franz-de Sanctis
identities; the identities themselves in the current setting are obtained via a
novel perturbation coming from exponentially distributed "side-observations" of
the signal. Concentration of multioverlaps means that asymptotically the
posterior distribution has a particularly simple structure encoded by a random
probability measure (or, in the case of binary signal, a non-random probability
measure). We believe that such strong control of the model should be key in the
study of inference problems with underlying sparse graphical structure (error
correcting codes, block models, etc) and, in particular, in the rigorous
derivation of replica symmetric formulas for the free energy and mutual
information in this context.
