Adaptive Bayesian density estimation in sup-norm

Abstract

We investigate the problem of deriving adaptive posterior rates of contraction on ${\mathbb{L}}_{\infty }$ 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 that the so-called spike-and-slab prior can achieve adaptive and optimal posterior contraction rates. Along the way, we prove a generic ${\mathbb{L}}_{\infty }$ contraction result for log-density priors with independent wavelet coefficients. Interestingly, our approach is different from previous works on ${\mathbb{L}}_{\infty }$ 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 $log(\mathit{n})$ factor in the case of low smoothness.