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级数先验的非参数模型中平滑性自适应的贝叶斯新策略。通过多种设定下的实例验证，我们特别证明了相应的贝叶斯后验分布能在无需超参数采样的条件下，实现极小化意义下的自适应收缩速率（可达对数因子）。与许多需执行直接模型（或估计量）选择的现有方法不同，本方法可视为通过先验尾部实现一种软选择。在高斯回归中，此类重尾先验被证明能在$L^2$和$L^\infty$范数意义下实现（近）最优同步自适应。文中还推导了线性反问题、各向异性贝索夫类的结果，并通过使用温化后验分布，推广至更一般模型中的特定损失函数。我们提供了数值模拟以佐证理论。