  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.
