标题： $L_2$-范数后验收缩在方差未知的高斯模型中 显示英文标题

标题： $L_2$-norm posterior contraction in Gaussian models with unknown variance

摘要： 基于测试的方法是建立后验收缩率的基本工具。 尽管由于存在理想的检验函数，Hellinger度量具有吸引力，但它在高斯模型中不能直接应用，因为将Hellinger度量转化为更直观的度量通常需要强有界性条件。 当方差已知时，可以通过使用似然比检验直接相对于$L_2$-度量构造检验函数来解决这个问题。 然而，当方差未知时，现有的结果有限，并且依赖于限制性假设。 为了克服这一限制，我们推导了一个针对未知方差设置相对于$L_2$-度量的检验函数，并基于基于测试的方法提供了后验收缩的充分条件。 我们将这个结果应用于高维回归和非参数回归的分析。 显示英文摘要

摘要： The testing-based approach is a fundamental tool for establishing posterior contraction rates. Although the Hellinger metric is attractive owing to the existence of a desirable test function, it is not directly applicable in Gaussian models, because translating the Hellinger metric into more intuitive metrics typically requires strong boundedness conditions. When the variance is known, this issue can be addressed by directly constructing a test function relative to the $L_2$-metric using the likelihood ratio test. However, when the variance is unknown, existing results are limited and rely on restrictive assumptions. To overcome this limitation, we derive a test function tailored to an unknown variance setting with respect to the $L_2$-metric and provide sufficient conditions for posterior contraction based on the testing-based approach. We apply this result to analyze high-dimensional regression and nonparametric regression.