Adaptive Bayesian density estimation in sup-norm

We investigate the problem of deriving adaptive posterior rates contraction on L∞ balls in density estimation. Although it is known that log-density priors can achieve optimal when true sufficiently smooth, were still to be proven. Here we establish so-called spike-and-slab prior and rates. Along way, prove a generic result for with independent wavelet coefficients. Interestingly, our approach different from previous works reminiscent classical test-based used Bayesian nonparametrics. Moreover, require no lower bound smoothness density, albeit are deteriorated by an extra log(n) factor case low smoothness.