Isotropic Gaussian priors are the de facto standard for modern Bayesian neural network inference. However, such simplistic priors are unlikely to either accurately reflect our true beliefs about the weight distributions, or to give optimal performance. We study summary statistics of neural network weights in different networks trained using SGD. We find that fully connected networks (FCNNs) display heavytailed weight distributions, while convolutional neural network (CNN) weights display strong spatial correlations. Building these observations into the respective priors leads to improved performance on a variety of image classification datasets. Moreover, we find that these priors also mitigate the cold posterior effect in FCNNs, while in CNNs we see strong improvements at all temperatures, and hence no reduction in the cold posterior effect.

Deep kernel learning and related techniques promise to combine the representational power of neural networks with the reliable uncertainty estimates of Gaussian processes. One crucial aspect of these models is an expectation that, because they are treated as Gaussian process models optimized using the marginal likelihood, they are protected from overfitting. However, we identify pathological behavior, including overfitting, on a simple toy example. We explore this pathology, explaining its origins and considering how it applies to real datasets. Through careful experimentation on UCI datasets, CIFAR-10, and the UTKFace dataset, we find that the overfitting from overparameterized deep kernel learning, in which the model is "somewhat Bayesian", can in certain scenarios be worse than that from not being Bayesian at all. However, we find that a fully Bayesian treatment of deep kernel learning can rectify this overfitting and obtain the desired performance improvements over standard neural networks and Gaussian processes.

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Data augmentation is often used to incorporate inductive biases into models. Traditionally, these are hand-crafted and tuned with cross validation. The Bayesian paradigm for model selection provides a path towards end-to-end learning of invariances using only the training data, by optimising the marginal likelihood. We work towards bringing this approach to neural networks by using an architecture with a Gaussian process in the last layer, a model for which the marginal likelihood can be computed. Experimentally, we improve performance by learning appropriate invariances in standard benchmarks, the low data regime and in a medical imaging task. Optimisation challenges for invariant Deep Kernel Gaussian processes are identified, and a systematic analysis is presented to arrive at a robust training scheme. We introduce a new lower bound to the marginal likelihood, which allows us to perform inference for a larger class of likelihood functions than before, thereby overcoming some of the training challenges that existed with previous approaches.