  In this article we consider Bayesian inference associated to deep neural
networks (DNNs) and in particular, trace-class neural network (TNN) priors
which were proposed by Sell et al. [39]. Such priors were developed as more
robust alternatives to classical architectures in the context of inference
problems. For this work we develop multilevel Monte Carlo (MLMC) methods for
such models. MLMC is a popular variance reduction technique, with particular
applications in Bayesian statistics and uncertainty quantification. We show how
a particular advanced MLMC method that was introduced in [4] can be applied to
Bayesian inference from DNNs and establish mathematically, that the
computational cost to achieve a particular mean square error, associated to
posterior expectation computation, can be reduced by several orders, versus
more conventional techniques. To verify such results we provide numerous
numerical experiments on model problems arising in machine learning. These
include Bayesian regression, as well as Bayesian classification and
reinforcement learning.
