Sharp sup-norm Bayesian curve estimation

Sup-norm curve estimation is a fundamental statistical problem and, in principle, a premise for the construction of confidence bands for infinite-dimensional parameters. In a Bayesian framework, the issue of whether the sup-norm-concentration- of-posterior-measure approach proposed by Gin\'e and Nickl (2011), which involves solving a testing problem exploiting concentration properties of kernel and projection-type density estimators around their expectations, can yield minimax-optimal rates is herein settled in the affirmative beyond conjugate-prior settings obtaining sharp rates for common prior-model pairs like random histograms, Dirichlet Gaussian or Laplace mixtures, which can be employed for density, regression or quantile estimation.