Obtaining high-quality uncertainty estimates is essential for many applications of deep neural networks. In this paper, we theoretically justify a scheme for estimating uncertainties, based on sampling from a prior distribution. Crucially, the uncertainty estimates are shown to be conservative in the sense that they never underestimate a posterior uncertainty obtained by a hypothetical Bayesian algorithm. We also show concentration, implying that the uncertainty estimates converge to zero as we get more data. Uncertainty estimates obtained from random priors can be adapted to any deep network architecture and trained using standard supervised learning pipelines. We provide experimental evaluation of random priors on calibration and out-of-distribution detection on typical computer vision tasks, demonstrating that they outperform deep ensembles in practice. Deep learning has achieved huge success in many applications. In particular, increasingly often, it is used as a component in decision-making systems. In order to have confidence in decisions made by such systems, it is necessary to obtain good uncertainty estimates, which quantify how certain the network is about a given output. In particular, if the cost of failure is large, for example where the automated system has the capability to accidentally hurt humans, the availability and quality of uncertainty estimates can determine whether the system is safe to deploy at all (Carvalho, 2016; Leibig et al., 2017; Michelmore et al., 2018) . Moreover, when decisions are made sequentially, good uncertainty estimates are crucial for achieving good performance quickly (Bellemare et al., 2016; Houthooft et al., 2016; Ostrovski et al., 2017; Burda et al., 2018) . Because any non-Bayesian inference process is potentially sub-optimal (De Finetti, 1937) , these uncertainty estimates should ideally be relatable to Bayesian inference with a useful prior. Deep ensembles (Lakshminarayanan et al., 2017) , one of the most popular methods available for uncertainty estimation in deep networks today, struggle with this requirement. While deep ensembles can be related (Rubin, 1981) to Bayesian inference in settings where the individual models are trained on subsets of the data, this is not how they are used in practice. In order to improve data efficiency, all ensembles are typically trained using the same data (Lakshminarayanan et al., 2017) , resulting in a method which does not have a theoretical justification. Moreover, deep ensembles can give overconfident uncertainty estimates in practice. On the other hand, Monte-Carlo dropout can be viewed (Gal & Ghahramani, 2016) as a certain form of Bayesian inference. However, doing so requires requires either a limit to be taken or a generalization of variational inference to a quasi-KL divergence . In practice, MC dropout can give arbitrarily overconfident estimates (Foong et al., 2019) . More broadly, a category of approaches, known as Bayesian Neural Networks (Blundell et al., 2015; Welling & Teh, 2011; Neal, 1996) , maintains a distribution over the weights of the neural network. These methods have a sound Bayesian justification, but training them is both difficult and carries an accuracy penalty, particularly for networks with convolutional architectures (Osawa et al., 2019) . Moreover, tuning BNNs is hard and achieving a good approximation to the posterior is difficult (Brosse et al., 2018) . We use another way of obtaining uncertainties for deep networks, based on fitting random priors (Osband et al., 2018; 2019) . Random priors are easy to train and were found to work very well in practice (Burda et al., 2018) . To obtain the uncertainty estimates, we first train a predictor network to fit a prior. Two examples of prior-predictor pairs are shown in the top two plots of Figure 1 . On top, two predictors (green) were trained to fit two randomlygenerated priors (red). On the bottom, we obtain uncertainties from the difference between predictors and priors. Dots correspond to training points x i . Faced with a novel input point, we obtain an uncertainty ( Figure 1 , bottom plot) by measuring the error of the predictor network against this pattern. Intuitively, these errors will be small close to the training points, but large far from them. The patterns themselves are drawn from randomly initialized (and therefore untrained) neural networks. While this way of estimating uncertainties was known before (Osband et al., 2019) , it did not have a theoretical justification beyond Bayesian linear regression, which is too limiting for modern applications. Contributions We provide a sound theoretical framework for obtaining uncertainty estimates by fitting random priors, a method previously lacking a principled justification. Specifically, we justify estimates in the uncertainty of the output of neural networks with any architecture. In particular, we show in Lemma 1 and Proposition 1 that these uncertainty estimates are conservative, meaning they are never more certain than a Bayesian algorithm would be. Moreover, in Proposition 2 we show concentration, i.e. that the uncertainties become zero with infinite data. Empirically, we evaluate the calibration and out-of-distribution performance of our uncertainty estimates on typical computer vision tasks, showing a practical benefit over deep ensembles and MC dropout. We now re-visit the algorithm we defined in Section 3, with the aim of using the theory above to obtain practical improvements in the quality of the uncertainty estimates. Architecture and Choosing the Number of Bootstraps Our conservatism guarantee in Proposition 1 holds for any architecture for the predictor h Xf . In theory, the predictor could be completely arbitrary and does not even have to be a deep network. In particular, there is no formal requirement for the predictor architecture to be the same as the prior. On the other hand, to show concentration in Proposition 2, we had to ensure that the prior networks are representable by the predictor. In practice, we use the architecture shown in Figure 2 , where the predictor mirrors the prior, but has additional layers, giving it more representational power. Moreover, the architecture requires choosing the number of bootstraps B. Our experiments in Section 7 show that even using B = 1, i.e. one bootstrap, produces uncertainty estimates of high quality in practice. Modeling Epistemic and Aleatoric Uncertainty Proposition 1 and Proposition 2 hold for any Gaussian Process prior. By choosing the process appropriately, we can model both epistemic and aleatoric uncertainty. Denote by {n(x)} a stochastic process obtained by randomly initializing neural networks and denote by { (x)σ 2 A } the noise term, modeling the aleatoric (observation) noise, where samples are obtained from (x) ∼ N (0, 1) at each x independently (see Appendix D for more background on aleatoric noise). We can now choose the prior process as a sum {f (x)} = {n(x) + (x)σ 2 A } of epistemic component {n(x)} and the noise term. The amount of aleatoric uncertainty can be adjusted by choosing σ 2 A . Prior Choice, Weight Copying and Conservatism One question that can be asked about our architecture (Figure 2) is whether it is possible for the predictor to exactly copy the prior weights, giving zero uncertainty everywhere. A useful edge case to consider here is when we are solving a one-dimensional regression problem, σ 2 A = 0 and the both the priors and predictors are linear functions. In this case, after training on two points, the predictors will agree with the priors everywhere and uncertainty estimates will be zero. However, this is still consistent with our conservatism guarantee The reason for this is once we assume such a linear prior, we are comparing to a GP with a linear kernel. But a GP with that kernel will also have zero uncertainty after seeing two samples. In practice, this means that we have to choose the architecture of the prior networks be expressive enough, which is no different from choosing a reasonable prior for Bayesian inference. Empirically, the tested network architecture did not show weight copying. We provided a theoretical justification for the use of random priors for obtaining uncertainty estimates in the context of deep learning. We have shown that the obtained uncertainties are conservative and that they concentrate for any neural network architecture. We performed an extensive empirical comparison, showing that random priors perform similarly to deep ensembles in a typical supervised training setting, while outperforming them in a regime where we are able to accomplish near-zero training loss for the predictors. For the 1D regression experiment on synthetic data (Fig 1) , we used feed-forward neural networks with 2 layers of 128 units each and a 1-dimensional output layer. We used an ensemble size of 5. The network was trained on 20 points sampled from the negative domain of a sigmoid function and tested on 20 points sampled from the positive domain.
We provide theoretical support to uncertainty estimates for deep learning obtained fitting random priors.
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