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We conduct a thorough analysis of the relationship between the out-of-sample performance and the Bayesian evidence (marginal likelihood) of Bayesian neural networks (BNNs), as well as looking at the performance of ensembles of BNNs, both using the Boston housing dataset. Using the state-of-the-art in nested sampling, we numerically sample the full (non-Gaussian and multimodal) network posterior and obtain numerical estimates of the Bayesian evidence, considering network models with up to 156 trainable parameters. The networks have between zero and four hidden layers, either $tanh$ or $ReLU$ activation functions, and with and without hierarchical priors. The ensembles of BNNs are obtained by determining the posterior distribution over networks, from the posterior samples of individual BNNs re-weighted by the associated Bayesian evidence values. There is good correlation between out-of-sample performance and evidence, as well as a remarkable symmetry between the evidence versus model size and out-of-sample performance versus model size planes. Networks with $ReLU$ activation functions have consistently higher evidences than those with $tanh$ functions, and this is reflected in their out-of-sample performance. Ensembling over architectures acts to further improve performance relative to the individual BNNs.
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We develop variational Laplace for Bayesian neural networks (BNNs) which exploits a local approximation of the curvature of the likelihood to estimate the ELBO without the need for stochastic sampling of the neural-network weights. The Variational Laplace objective is simple to evaluate, as it is (in essence) the log-likelihood, plus weight-decay, plus a squared-gradient regularizer. Variational Laplace gave better test performance and expected calibration errors than maximum a-posteriori inference and standard sampling-based variational inference, despite using the same variational approximate posterior. Finally, we emphasise care needed in benchmarking standard VI as there is a risk of stopping before the variance parameters have converged. We show that early-stopping can be avoided by increasing the learning rate for the variance parameters.
In federated learning problems, data is scattered across different servers and exchanging or pooling it is often impractical or prohibited. We develop a Bayesian nonparametric framework for federated learning with neural networks. Each data server is assumed to provide local neural network weights, which are modeled through our framework. We then develop an inference approach that allows us to synthesize a more expressive global network without additional supervision, data pooling and with as few as a single communication round. We then demonstrate the efficacy of our approach on federated learning problems simulated from two popular image classification datasets.
Ensembles of geophysical models improve projection accuracy and express uncertainties. We develop a novel data-driven ensembling strategy for combining geophysical models using Bayesian Neural Networks, which infers spatiotemporally varying model weights and bias while accounting for heteroscedastic uncertainties in the observations. This produces more accurate and uncertainty-aware projections without sacrificing interpretability. Applied to the prediction of total column ozone from an ensemble of 15 chemistry-climate models, we find that the Bayesian neural network ensemble (BayNNE) outperforms existing ensembling methods, achieving a 49.4% reduction in RMSE for temporal extrapolation, and a 67.4% reduction in RMSE for polar data voids, compared to a weighted mean. Uncertainty is also well-characterized, with 90.6% of the data points in our extrapolation validation dataset lying within 2 standard deviations and 98.5% within 3 standard deviations.
In recent times, neural networks have become a powerful tool for the analysis of complex and abstract data models. However, their introduction intrinsically increases our uncertainty about which features of the analysis are model-related and which are due to the neural network. This means that predictions by neural networks have biases which cannot be trivially distinguished from being due to the true nature of the creation and observation of data or not. In order to attempt to address such issues we discuss Bayesian neural networks: neural networks where the uncertainty due to the network can be characterised. In particular, we present the Bayesian statistical framework which allows us to categorise uncertainty in terms of the ingrained randomness of observing certain data and the uncertainty from our lack of knowledge about how data can be created and observed. In presenting such techniques we show how errors in prediction by neural networks can be obtained in principle, and provide the two favoured methods for characterising these errors. We will also describe how both of these methods have substantial pitfalls when put into practice, highlighting the need for other statistical techniques to truly be able to do inference when using neural networks.
Current approaches in approximate inference for Bayesian neural networks minimise the Kullback-Leibler divergence to approximate the true posterior over the weights. However, this approximation is without knowledge of the final application, and therefore cannot guarantee optimal predictions for a given task. To make more suitable task-specific approximations, we introduce a new loss-calibrated evidence lower bound for Bayesian neural networks in the context of supervised learning, informed by Bayesian decision theory. By introducing a lower bound that depends on a utility function, we ensure that our approximation achieves higher utility than traditional methods for applications that have asymmetric utility functions. Furthermore, in using dropout inference, we highlight that our new objective is identical to that of standard dropout neural networks, with an additional utility-dependent penalty term. We demonstrate our new loss-calibrated model with an illustrative medical example and a restricted model capacity experiment, and highlight failure modes of the comparable weighted cross entropy approach. Lastly, we demonstrate the scalability of our method to real world applications with per-pixel semantic segmentation on an autonomous driving data set.
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