  Deep learning methods continue to have a decided impact on machine learning,
both in theory and in practice. Statistical theoretical developments have been
mostly concerned with approximability or rates of estimation when recovering
infinite dimensional objects (curves or densities). Despite the impressive
array of available theoretical results, the literature has been largely silent
about uncertainty quantification for deep learning. This paper takes a step
forward in this important direction by taking a Bayesian point of view. We
study Gaussian approximability of certain aspects of posterior distributions of
sparse deep ReLU architectures in non-parametric regression. Building on tools
from Bayesian non-parametrics, we provide semi-parametric Bernstein-von Mises
theorems for linear and quadratic functionals, which guarantee that implied
Bayesian credible regions have valid frequentist coverage. Our results provide
new theoretical justifications for (Bayesian) deep learning with ReLU
activation functions, highlighting their inferential potential.
