Bayesian Neural Networks (BNN) have emerged as a crucial approach for interpreting ML predictions. By sampling from the posterior distribution, data scientists may estimate the uncertainty of an inference. Unfortunately many inference samples are often needed, the overhead of which greatly hinder BNN's wide adoption. To mitigate this, previous work proposed propagating the first and second moments of the posterior directly through the network. However, on its own this method is even slower than sampling, so the propagated variance needs to be approximated such as assuming independence between neural nodes. The resulting trade-off between quality and inference time did not match even plain Monte Carlo sampling. Our contribution is a more principled variance propagation framework based on "spiked covariance matrices", which smoothly interpolates between quality and inference time. This is made possible by a new fast algorithm for updating a diagonal-plus-low-rank matrix approximation under various operations. We tested our algorithm against sampling based MC Dropout and Variational Inference on a number of downstream uncertainty themed tasks, such as calibration and out-of-distribution testing. We find that Favour is as fast as performing 2-3 inference samples, while matching the performance of 10-100 samples. In summary, this work enables the use of BNN in the realm of performance critical tasks where they have previously been out of reach.

Given the importance of getting calibrated predictions and reliable uncertainty estimations, various post-hoc calibration methods have been developed for neural networks on standard multi-class classification tasks. However, these methods are not well suited for calibrating graph neural networks (GNNs), which presents unique challenges such as accounting for the graph structure and the graph-induced correlations between the nodes. In this work, we conduct a systematic study on the calibration qualities of GNN node predictions. In particular, we identify five factors which influence the calibration of GNNs: general under-confident tendency, diversity of nodewise predictive distributions, distance to training nodes, relative confidence level, and neighborhood similarity. Furthermore, based on the insights from this study, we design a novel calibration method named Graph Attention Temperature Scaling (GATS), which is tailored for calibrating graph neural networks. GATS incorporates designs that address all the identified influential factors and produces nodewise temperature scaling using an attention-based architecture. GATS is accuracy-preserving, data-efficient, and expressive at the same time. Our experiments empirically verify the effectiveness of GATS, demonstrating that it can consistently achieve state-of-the-art calibration results on various graph datasets for different GNN backbones.