  End-to-end learners for autonomous driving are deep neural networks that
predict the instantaneous steering angle directly from images of the
ahead-lying street. These learners must provide reliable uncertainty estimates
for their predictions in order to meet safety requirements and initiate a
switch to manual control in areas of high uncertainty. Yet end-to-end learners
typically only deliver point predictions, since distributional predictions are
associated with large increases in training time or additional computational
resources during prediction. To address this shortcoming we investigate
efficient and scalable approximate inference for the implicit copula neural
linear model of Klein, Nott and Smith (2021) in order to quantify uncertainty
for the predictions of end-to-end learners. The result are densities for the
steering angle that are marginally calibrated, i.e.~the average of the
estimated densities equals the empirical distribution of steering angles. To
ensure the scalability to large $n$ regimes, we develop efficient estimation
based on variational inference as a fast alternative to computationally
intensive, exact inference via Hamiltonian Monte Carlo. We demonstrate the
accuracy and speed of the variational approach in comparison to Hamiltonian
Monte Carlo on two end-to-end learners trained for highway driving using the
comma2k19 data set. The implicit copula neural linear model delivers accurate
calibration, high-quality prediction intervals and allows to identify
overconfident learners. Our approach also contributes to the explainability of
black-box end-to-end learners, since predictive densities can be used to
understand which steering actions the end-to-end learner sees as valid.
