Adaptive Bayesian semiparametric density estimation in sup-norm

We investigate the problem of deriving adaptive posterior rates of contraction on $L^{\infty}$ balls in density estimation. Although it is known that log-density priors can achieve optimal rates when the true density is sufficiently smooth, adaptive rates were still to be proven. Recent works have shown that the so called spike-and-slab priors can achieve optimal rates of contraction under $L^{\infty}$ loss in white-noise regression and multivariate regression with normal errors. Here we show that a spike-and-slab prior on the log-density also allows for (nearly) optimal rates of contraction in density estimation under $L^{\infty}$ loss. Interestingly, our results hold without lower bound on the smoothness of the true density.