High dimensional statistics deals with the challenge of extracting structured information from complex model settings. Compared with the growing number of frequentist methodologies, there are rather few theoretically optimal Bayes methods that can deal with very general high dimensional models. In contrast, Bayes methods have been extensively studied in various nonparametric settings and rate optimal posterior contraction results have been established. This paper provides a unified approach to both Bayes high dimensional statistics and Bayes nonparametrics in a general framework of structured linear models. With the proposed two-step model selection prior, we prove a general theorem of posterior contraction under an abstract setting. The main theorem can be used to derive new results on optimal posterior contraction under many complex model settings including stochastic block model, graphon estimation and dictionary learning. It can also be used to re-derive optimal posterior contraction for problems such as sparse linear regression and nonparametric aggregation, which improve upon previous Bayes results for these problems. The key of the success lies in the proposed two-step prior distribution. The prior on the parameters is an elliptical Laplace distribution that is capable to model signals with large magnitude, and the prior on the models involves an important correction factor that compensates the effect of the normalizing constant of the elliptical Laplace distribution.

中文翻译：

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贝叶斯结构化线性模型的通用框架

高维统计处理从复杂模型设置中提取结构化信息的挑战。与越来越多的频率论方法相比，可以处理非常普遍的高维模型的理论上最优的贝叶斯方法相当少。相比之下，贝叶斯方法已在各种非参数设置中得到广泛研究，并且已经建立了速率最优后收缩结果。本文在结构化线性模型的一般框架中为贝叶斯高维统计和贝叶斯非参数提供了一种统一的方法。通过提出的两步模型选择先验，我们证明了抽象设置下后收缩的一般定理。主要定理可用于在许多复杂模型设置（包括随机块模型、图形估计和字典学习）下推导出最优后收缩的新结果。它还可以用于为稀疏线性回归和非参数聚合等问题重新推导最佳后验收缩，从而改进这些问题的先前贝叶斯结果。成功的关键在于提议的两步先验分布。参数的先验是椭圆拉普拉斯分布，能够对大信号进行建模，模型的先验涉及一个重要的校正因子，用于补偿椭圆拉普拉斯分布的归一化常数的影响。它还可以用于为稀疏线性回归和非参数聚合等问题重新推导最佳后验收缩，从而改进这些问题的先前贝叶斯结果。成功的关键在于提议的两步先验分布。参数的先验是椭圆拉普拉斯分布，能够对大信号进行建模，模型的先验涉及一个重要的校正因子，用于补偿椭圆拉普拉斯分布的归一化常数的影响。它还可以用于为稀疏线性回归和非参数聚合等问题重新推导最佳后验收缩，从而改进这些问题的先前贝叶斯结果。成功的关键在于提议的两步先验分布。参数的先验是椭圆拉普拉斯分布，能够对大信号进行建模，模型的先验涉及一个重要的校正因子，用于补偿椭圆拉普拉斯分布的归一化常数的影响。