自适应上确界范数后验收缩：小波Spike-and-Slab和各向异性Besov空间
Adaptive Supremum Norm Posterior Contraction: Wavelet Spike-and-Slab and Anisotropic Besov Spaces
摘要：
在统计学中，上确界范数损失对函数估计误差的量化更有直观意义。在误差未知的多元非参数回归的背景下，我们提出了一个贝叶斯过程的基础上的尖峰和板条先验和小波投影估计的回归函数和所有的混合偏导数。我们证明了它们的后验分布在上模损失下最优地自适应地收缩到真值。在贝叶斯非参数化理论中，通过指数误差检验的主定理不足以处理这个问题，我们提出了一种新的思想，将回归模型下的后验系统地简化为由准白噪声模型产生的后验，后者大大简化了我们的速率计算。因此，本文采取的第一步，明确显示如何可以将结果从白噪声回归模型在贝叶斯设置。
英文摘要：
Supremum norm loss is intuitively more meaningful to quantify function estimation error in statistics. In the context of multivariate nonparametric regression with unknown error, we propose a Bayesian procedure based on spike-and-slab prior and wavelet projections to estimate the regression function and all its mixed partial derivatives. We show that their posterior distributions contract to the truth optimally and adaptively under supremum-norm loss. The master theorem through tests with exponential errors used in Bayesian nonparametrics was not adequate to deal with this problem, and we developed a new idea such that posterior under the regression model is systematically reduced to a posterior arising from some quasi-white noise model, where the latter model greatly simplifies our rate calculations. Hence, this paper takes the first step in showing explicitly how one can translate results from white noise to regression model in a Bayesian setting.
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