  A Bayesian treatment of deep learning allows for the computation of
uncertainties associated with the predictions of deep neural networks. We show
how the concept of Errors-in-Variables can be used in Bayesian deep regression
to also account for the uncertainty associated with the input of the employed
neural network. The presented approach thereby exploits a relevant, but
generally overlooked, source of uncertainty and yields a decomposition of the
predictive uncertainty into an aleatoric and epistemic part that is more
complete and, in many cases, more consistent from a statistical perspective. We
discuss the approach along various simulated and real examples and observe that
using an Errors-in-Variables model leads to an increase in the uncertainty
while preserving the prediction performance of models without
Errors-in-Variables. For examples with known regression function we observe
that this ground truth is substantially better covered by the
Errors-in-Variables model, indicating that the presented approach leads to a
more reliable uncertainty estimation.
