Adaptive Bayesian 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. Here we establish a generic $L^{\infty}$-contraction result for log-density priors with independent wavelet coefficients. The result is then applied to the so called spike-and-slab prior to obtain adaptive and minimax rates. Interestingly, our approach is different from previous works on $L^{\infty}$-contraction and is reminiscent to the classical test-based approach used in Bayesian nonparametrics. Moreover, we require no lower bound on the smoothness of the true density, albeit the rates are deteriorated by an extra $\log(n)$ factor in the case of low smoothness.