  Building on ideas from Castillo and Nickl [Ann. Statist. 41 (2013)
1999-2028], a method is provided to study nonparametric Bayesian posterior
convergence rates when "strong" measures of distances, such as the sup-norm,
are considered. In particular, we show that likelihood methods can achieve
optimal minimax sup-norm rates in density estimation on the unit interval. The
introduced methodology is used to prove that commonly used families of prior
distributions on densities, namely log-density priors and dyadic random density
histograms, can indeed achieve optimal sup-norm rates of convergence. New
results are also derived in the Gaussian white noise model as a further
illustration of the presented techniques.
