高维最优贝叶斯推理中的强副本对称性
Strong replica symmetry in high-dimensional optimal Bayesian inference
摘要：
我们考虑通用的最优贝叶斯推理，即，信号重建模型的后验分布和所有的超参数是已知的。在一个标准的假设下的自由能的浓度，我们展示了如何复制对称性的浓度的所有multioverlaps的强烈意义上可以建立作为后果的Franz-de Sanctis身份，身份本身在当前的设置是通过一个新的扰动来自指数分布的“边观测”的信号。多重重叠的集中意味着后验分布渐近地具有由随机概率测度（或者，在二进制信号的情况下，非随机概率测度）编码的特别简单的结构。我们认为，这种强有力的控制模型应该是关键的推理问题的研究与底层稀疏的图形结构（纠错码，块模型等），特别是在严格推导副本对称公式的自由能和互信息在这种情况下。
英文摘要：
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.
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