Adaptive Bayesian density estimation in sup-norm

TL;DR

This paper demonstrates that spike-and-slab priors can adaptively and optimally estimate densities in the supremum norm, even without assuming a minimum smoothness level, using a novel wavelet-based approach.

Contribution

It proves adaptive posterior contraction rates for density estimation using spike-and-slab priors in the sup-norm, introducing a new wavelet coefficient analysis method.

Findings

Spike-and-slab priors achieve adaptive optimal rates in sup-norm density estimation.

The approach does not require a lower bound on the true density's smoothness.

Rates are slightly slower with an additional log factor for less smooth densities.

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