标题： 高维贝叶斯线性回归中的中心极限定理与低信噪比 显示英文标题

标题： CLT in high-dimensional Bayesian linear regression with low SNR

摘要： 我们研究高维贝叶斯线性回归中线性统计量的中心极限定理，使用的是乘积先验。 与现有文献侧重于后验收缩不同，我们在一个不收缩的框架下工作，其中似然函数和先验都不占主导地位。 这是由具有有界信噪比的现代高维数据所驱动的。 这项工作迈出了理解此类情况下后验的一维投影以及后验均值的极限分布的第一步。 类似于收缩情形，得到的极限分布是正态分布，但它们很大程度上依赖于所选先验，并围绕后验的平均场近似展开。 我们研究了两个感兴趣的模型来说明这一现象——白噪声设计和（误指的）贝叶斯模型。 作为应用，我们在任何误指的先验下构造可信区间并计算其覆盖概率。 我们的证明依赖于随机场伊辛模型的贝里-埃斯滕类型界限的最新发展，以及一阶和二阶庞加莱不等式。 值得注意的是，我们的结果不需要对先验有任何稀疏性假设。 显示英文摘要

摘要： We study central limit theorems for linear statistics in high-dimensional Bayesian linear regression with product priors. Unlike the existing literature where the focus is on posterior contraction, we work under a non-contracting regime where neither the likelihood nor the prior dominates the other. This is motivated by modern high-dimensional datasets characterized by a bounded signal-to-noise ratio. This work takes a first step towards understanding limit distributions for one-dimensional projections of the posterior, as well as the posterior mean, in such regimes. Analogous to contractive settings, the resulting limiting distributions are Gaussian, but they heavily depend on the chosen prior and center around the Mean-Field approximation of the posterior. We study two concrete models of interest to illustrate this phenomenon -- the white noise design, and the (misspecified) Bayesian model. As an application, we construct credible intervals and compute their coverage probability under any misspecified prior. Our proofs rely on a combination of recent developments in Berry-Esseen type bounds for Random Field Ising models and both first and second order Poincar\'{e} inequalities. Notably, our results do not require any sparsity assumptions on the prior.