论文标题
高维最佳贝叶斯推断中的强副本对称性
Strong replica symmetry in high-dimensional optimal Bayesian inference
论文作者
论文摘要
我们考虑一般的最佳贝叶斯推断，即信号重建模型，其中已知后验分布和所有超参数。在自由能浓度的标准假设下，我们展示了如何以强烈的浓度意义上的所有多verlap的浓度来建立复制对称性。当前设置中的身份本身是通过来自信号的指数分布的“侧面观察”的新型扰动获得的。多重视图的浓度意味着渐近的后验分布具有由随机概率度量（或在二进制信号的情况下）编码的特别简单的结构。我们认为，对模型的这种强大控制应该是研究基础稀疏图形结构（误差校正代码，块模型等）的推理问题的关键，尤其是在这种情况下的自由能和共同信息的副本对称公式的严格推导中。
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.
