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End-to-end learners for autonomous driving are deep neural networks thatpredict the instantaneous steering angle directly from images of theahead-lying street. These learners must provide reliable uncertainty estimatesfor their predictions in order to meet safety requirements and initiate aswitch to manual control in areas of high uncertainty. Yet end-to-end learnerstypically only deliver point predictions, since distributional predictions areassociated with large increases in training time or additional computationalresources during prediction. To address this shortcoming we investigateefficient and scalable approximate inference for the implicit copula neurallinear model of Klein, Nott and Smith (2021) in order to quantify uncertaintyfor the predictions of end-to-end learners. The result are densities for thesteering angle that are marginally calibrated, i.e.~the average of theestimated densities equals the empirical distribution of steering angles. Toensure the scalability to large $n$ regimes, we develop efficient estimationbased on variational inference as a fast alternative to computationallyintensive, exact inference via Hamiltonian Monte Carlo. We demonstrate theaccuracy and speed of the variational approach in comparison to HamiltonianMonte Carlo on two end-to-end learners trained for highway driving using thecomma2k19 data set. The implicit copula neural linear model delivers accuratecalibration, high-quality prediction intervals and allows to identifyoverconfident learners. Our approach also contributes to the explainability ofblack-box end-to-end learners, since predictive densities can be used tounderstand which steering actions the end-to-end learner sees as valid.

Marginally calibrated response distributions for end-to-end learning in autonomous driving