We propose a new Bayesian strategy for adaptation to smoothness in nonparametric models based on heavy tailed series priors. We illustrate it in a variety of settings, showing in particular that the corresponding Bayesian posterior distributions achieve adaptive rates of contraction in the minimax sense (up to logarithmic factors) without the need to sample hyperparameters. Unlike many existing procedures, where a form of direct model (or estimator) selection is performed, the method can be seen as performing a soft selection through the prior tail. In Gaussian regression, such heavy tailed priors are shown to lead to (near-)optimal simultaneous adaptation both in the $L^2$- and $L^\infty$-sense. Results are also derived for linear inverse problems, for anisotropic Besov classes, and for certain losses in more general models through the use of tempered posterior distributions. We present numerical simulations corroborating the theory.

翻译：我们提出了一种基于重尾序列先验的新型贝叶斯策略，用于实现非参数模型中对光滑性的自适应。通过在不同场景中的实例化，我们特别展示了对应的贝叶斯后验分布能够达到极小极大意义下的自适应收缩速率（仅相差对数因子），且无需对超参数进行采样。现有方法通常执行直接模型（或估计量）选择，而本方法可视为通过先验尾部实现“软选择”。在高斯回归中，此类重尾先验在$L^2$和$L^\infty$范数意义下均能实现（近乎）最优的同步自适应。此外，本文还针对线性逆问题、各向异性贝索夫类，以及基于温度调控后验分布的一般模型中的特定损失函数推导了相关结论。数值模拟结果与理论分析高度吻合。