Adaptive Bayesian density estimation in sup-norm

Abstract

We investigate the problem of deriving adaptive posterior rates of contraction on L∞ 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 that the so-called spike-and-slab prior can achieve adaptive and optimal posterior contraction rates. Along the way, we prove a generic L∞ contraction result for log-density priors with independent wavelet coefficients. Interestingly, our approach is different from previous works on L∞ contraction and is reminiscent of 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.