高维后验推断的高斯比较定理
Gaussian Comparison Theorems for High-Dimensional Posterior Inference
- Department of Physics, Duke University（杜克大学物理系）
- Departments of Statistical Science and Electrical and Computer Engineering, Duke University（杜克大学统计科学与电气与计算机工程系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出非高斯高维模型后验推断的高斯比较理论，通过矩条件建立自由能及后验误差的夹逼界，并应用于稀疏伯努利超图推断。
AI中文摘要：
我们为非高斯高维模型中的后验推断发展了一套高斯比较理论。该框架允许模型设定错误，且不要求后验分布本身近似于高斯分布。直接在可分函数空间上处理似然过程，我们建立了由前两阶矩差异和局部三阶矩控制的显式非渐近自由能界。一个扰动-凸性论证将这些比较转化为后验均方误差和后验方差的夹逼界，包括在不可微相变处的一侧结论。在高维乘积模型中，这些条件简化为局部矩控制和典型的特征-半径缩放。在较弱的正则性要求下，所得普适性理论适用于广泛的似然模型类别，从指数族到更一般的正则局部对数似然。该框架在稀疏伯努利超图推断上进行了说明。
英文摘要：
We develop a Gaussian comparison theory for posterior inference in non-Gaussian high-dimensional models. The framework allows for model misspecification and does not require the posterior distribution itself to be approximately Gaussian. Working directly with likelihood processes on separable function spaces, we establish explicit nonasymptotic free energy bounds controlled by first two moment discrepancies and local third moments. A perturbation-and-convexity argument transfers these comparisons to sandwich bounds for posterior mean squared error and posterior variance, including one-sided conclusions at nondifferentiable phase transitions. In high-dimensional product models, these conditions reduce to local moment controls and a canonical feature-radius scaling. Under weak regularity requirements, the resulting universality theory applies to a broad class of likelihood models, ranging from exponential families to more general regular local log-likelihoods. The framework is illustrated on sparse Bernoulli hypergraph inference.
