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We consider generic optimal Bayesian inference, namely, models of signalreconstruction where the posterior distribution and all hyperparameters areknown. Under a standard assumption on the concentration of the free energy, weshow how replica symmetry in the strong sense of concentration of allmultioverlaps can be established as a consequence of the Franz-de Sanctisidentities; the identities themselves in the current setting are obtained via anovel perturbation coming from exponentially distributed "side-observations" ofthe signal. Concentration of multioverlaps means that asymptotically theposterior distribution has a particularly simple structure encoded by a randomprobability measure (or, in the case of binary signal, a non-random probabilitymeasure). We believe that such strong control of the model should be key in thestudy of inference problems with underlying sparse graphical structure (errorcorrecting codes, block models, etc) and, in particular, in the rigorousderivation of replica symmetric formulas for the free energy and mutualinformation in this context.

Strong replica symmetry in high-dimensional optimal Bayesian inference