Adaptation I: Smoothness

This chapter considers more flexible choices of prior distributions that allow, at least for certain loss functions, to automatically attain minimax optimal posterior convergence rates, up sometimes to slowly varying factors, without assuming any knowledge of underlying smoothness parameters. Two approaches to achieve this goal are hierarchical Bayes and empirical Bayes. We illustrate this adaptation-to-smoothness feature through a variety of prior classes: random histograms, randomly rescaled Gaussian processes, intrinsic Gaussian processes on manifolds, and heavy-tailed series priors. We briefly discuss other classes as well, including spike-and-slab priors and tree priors such as Bayesian CART. We conclude by a brief discussion on constructing adaptive confidence sets using posterior distributions.