  Many real-life applications involve estimation of curves that exhibit
complicated shapes including jumps or varying-frequency oscillations. Practical
methods have been devised that can adapt to a locally varying complexity of an
unknown function (e.g. variable-knot splines, sparse wavelet reconstructions,
kernel methods or trees/forests). However, the overwhelming majority of
existing asymptotic minimaxity theory is predicated on homogeneous smoothness
assumptions. Focusing on locally Holderian functions, we provide new locally
adaptive posterior concentration rate results under the supremum loss for
widely used Bayesian machine learning techniques in white noise and
non-parametric regression. In particular, we show that popular spike-and-slab
priors and Bayesian CART are uniformly locally adaptive. In addition, we
propose a new class of repulsive partitioning priors which relate to variable
knot splines and which are exact-rate adaptive. For uncertainty quantification,
we construct locally adaptive confidence bands whose width depends on the local
smoothness and which achieve uniform asymptotic coverage under local
self-similarity. To illustrate that spatial adaptation is not at all automatic,
we provide lower-bound results showing that popular hierarchical Gaussian
process priors fall short of spatial adaptation.
