  We investigate the problem of deriving posterior concentration rates under
different loss functions in nonparametric Bayes. We first provide a lower bound
on posterior coverages of shrinking neighbourhoods that relates the metric or
loss under which the shrinking neighbourhood is considered, and an intrinsic
pre-metric linked to frequentist separation rates. In the Gaussian white noise
model, we construct feasible priors based on a spike and slab procedure
reminiscent of wavelet thresholding that achieve adaptive rates of contraction
under $L^2$ or $L^{\infty}$ metrics when the underlying parameter belongs to a
collection of H\"{o}lder balls and that moreover achieve our lower bound. We
analyse the consequences in terms of asymptotic behaviour of posterior credible
balls as well as frequentist minimax adaptive estimation. Our results are
appended with an upper bound for the contraction rate under an arbitrary loss
in a generic regular experiment. The upper bound is attained for certain sieve
priors and enables to extend our results to density estimation.
