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 $\log(n)$ factor in the case of low smoothness.

翻译：我们调查了在密度估计中以$mathbb{L ⁇ infty}美元球得出适应性后继收缩率的问题。 虽然已知在真正密度足够平稳的情况下,日志密度前期可以实现最佳收缩率, 但适应性收缩率还有待证明。 在这里, 我们确定所谓的峰值和稀释率可以达到适应性和最佳后继收缩率。 沿途, 我们证明对于具有独立波盘系数的日志密度前期,我们有一个通用的 $mathb{L ⁇ infty}收缩率。 有趣的是, 我们的方法不同于以前关于 $\ mathb{L ⁇ infty} $ 的工程, 缩缩缩率是巴伊西亚非参数中使用的经典测试性方法的类似。 此外, 我们并不要求对真实密度的平滑度有更低的约束, 尽管在平滑度的情况下, 降速率会因额外的 $(n) 系数而恶化 。