Adaptive Supremum Norm Posterior Contraction: Spike-and-Slab Priors and Anisotropic Besov Spaces

Abstract

Supremum norm loss is intuitively more meaningful to quantify estimation error in statistics. In the context of multivariate nonparametric regression with unknown error, we propose a Bayesian procedure based on spike-and-slab prior and wavelet projections to estimate the regression function f and its derivatives. We show that their posteriors contract to the truth optimally and adaptively under supremum norm loss. We discovered that there is a lower limit in the range of smoothness that we can adapt to and this limit grows with dimension of the function's domain. The master theorem through exponential error test used in Bayesian nonparametrics was not adequate to deal with this problem, and we developed a new idea by bounding posterior under the regression model with a posterior arising from some quasi-white noise model, where the latter model greatly simplifies our calculations.