Deterministic neural networks (NNs) are increasingly being deployed in safety critical domains, where calibrated, robust and efficient measures of uncertainty are crucial. While it is possible to train regression networks to output the parameters of a probability distribution by maximizing a Gaussian likelihood function, the resulting model remains oblivious to the underlying confidence of its predictions. In this paper, we propose a novel method for training deterministic NNs to not only estimate the desired target but also the associated evidence in support of that target. We accomplish this by  placing evidential priors over our original Gaussian likelihood function and training our NN to infer the hyperparameters of our evidential distribution. We impose priors during training such that the model is penalized when its predicted evidence is not aligned with the correct output. Thus the model estimates not only the probabilistic mean and variance of our target but also the underlying uncertainty associated with each of those parameters. We observe that our evidential regression method learns well-calibrated measures of uncertainty on various benchmarks, scales to complex computer vision tasks, and is robust to adversarial input perturbations.
 Uncertainty estimation has a long history in neural networks, from modeling probability distribution parameters over outputs (Bishop, 1994) to Bayesian deep learning (Kendall & Gal, 2017) . Our work builds on this foundation and presents a scalable representation for inferring the parameters of an evidential uncertainty distribution while simultaneously learning regression tasks via MLE. In Bayesian deep learning, priors are placed over network weights and estimated using variational inference (Kingma et al., 2015) . Dropout (Gal & Ghahramani, 2016; Molchanov et al., 2017) and BBB (Blundell et al., 2015) rely on multiple samples to estimate predictive variance. Ensembles (Lakshminarayanan et al., 2017) provide a tangential approach where sampling occurs over multiple trained instances. In contrast, we place uncertainty priors over the likelihood function and thus only need a single forward pass to evaluate both prediction and uncertainty. Additionally, our approach of uncertainty estimation proved to be better calibrated and capable of predicting where the model fails. A large topic of research in Bayesian inference focuses on placing prior distributions over hierarchical models to estimate uncertainty (Gelman et al., 2006; 2008) . Our methodology falls under the class of evidential deep learning which models higher-order distribution priors over neural network predictions to interpret uncertainty. Prior works in this field (Sensoy et al., 2018; Malinin & Gales, 2018) have focused exclusively on modeling uncertainty in the classification domain with Dirichlet prior distributions. Our work extends this field into the broad range of regression learning tasks (e.g. depth estimation, forecasting, robotic control learning, etc.) and demonstrates generalizability to out-of-distribution test samples and complex learning problems. In this paper, we develop a novel method for training deterministic NNs that both estimates a desired target and evaluates the evidence in support of the target to generate robust metrics of model uncertainty. We formalize this in terms of learning evidential distributions, and achieve stable training by penalizing our model for prediction errors that scale with the available evidence. Our approach for evidential regression is validated on a benchmark regression task. We further demonstrate that this method robustly scales to a key task in computer vision, depth estimation, and that the predictive uncertainty increases with increasing out-of-distribution adversarial perturbation. This framework for evidential representation learning provides a means to achieve the precise uncertainty metrics required for robust neural network deployment in safety-critical domains. For convenience, define τ = 1/σ 2 be the precision of a Gaussian distribution. The change of variables transforms the Normal Inverse-Gamma distribution p(µ, σ 2 |γ, λ, α, β) to the equivalent Normal Gamma distribution p(µ, τ |γ, λ, α, β), parameterized by precision τ ∈ (0, ∞) instead of variance σ 2 ,
Fast, calibrated uncertainty estimation for neural networks without sampling This paper proposes a novel approach to estimate the confidence of predictions in a regression setting, opening the door to online applications with fully integrated uncertainty estimates. This paper proposed deep evidential regression, a method for training neural networks to not only estimate the output but also the associated evidence in support of that output.