Weprovidestrong empirical evidence thatexactinference does not have this pathology, hence it is due to the approximation and not the model.

R4."Lack of novelty": We respectfully disagree. To the best of our knowledge, Thms 1-3 are the first theo-33 retical results on the quality of BNN approximate inference in terms ofestimating function space uncertainty.34 The derivations are non-trivial, and the results applyregardless of the inference algorithm (not just VI, see35 lines 73-75, 118 & 130-131). This includes methods which are usually not expected to be over-confident,36 e.g. EP and Rényi VI, as long as factorised Gaussians/dropout distributions are used as approximate posteriors.37

A.1 Proof of Proposition 1 and 3 To prove Proposition 1, we first need the following lemma: Readers may refer to [47] for the proof of this lemma. Let's first consider the left handside, The first inequality is due to information processing inequality. The compactness assumption in Proposition 2 seems restrictive, since BNNs with Gaussian priors on weights will break the compactness assumption. Indeed, the assumptions in proposition 2 are merely sufficient conditions. In this section, we discuss the non-parametric counter part of Proposition 2, i.e., is the grid functional KL between a parametric model and a Gaussian process is still finite?

While Bayesian neural networks (BNNs) hold the promise of being flexible, well-calibrated statistical models, inference often requires approximations whose consequences are poorly understood.

Neural Linear Models (NLM) are deep Bayesian models that produce predictive uncertainty by learning features from the data and then performing Bayesian linear regression over these features. Despite their popularity, few works have focused on formally evaluating the predictive uncertainties of these models. Furthermore, existing works point out the difficulties of encoding domain knowledge in models like NLMs, making them unsuitable for applications where interpretability is required. In this work, we show that traditional training procedures for NLMs can drastically underestimate uncertainty in data-scarce regions. We identify the underlying reasons for this behavior and propose a novel training method that can both capture useful predictive uncertainties as well as allow for incorporation of domain knowledge.

Neural Linear Models (NLM) are deep models that produce predictive uncertainty by learning features from the data and then performing Bayesian linear regression over these features. Despite their popularity, few works have focused on formally evaluating the predictive uncertainties of these models. In this work, we show that traditional training procedures for NLMs can drastically underestimate uncertainty in data-scarce regions. We identify the underlying reasons for this behavior and propose a novel training procedure for capturing useful predictive uncertainties.

We describe a limitation in the expressiveness of the predictive uncertainty estimate given by mean-field variational inference (MFVI), a popular approximate inference method for Bayesian neural networks. In particular, MFVI fails to give calibrated uncertainty estimates in between separated regions of observations. This can lead to catastrophically overconfident predictions when testing on out-of-distribution data. Avoiding such overconfidence is critical for active learning, Bayesian optimisation and out-of-distribution robustness. We instead find that a classical technique, the linearised Laplace approximation, can handle 'in-between' uncertainty much better for small network architectures.