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Large-scale linear models are ubiquitous throughout machine learning, withcontemporary application as surrogate models for neural network uncertaintyquantification; that is, the linearised Laplace method. Alas, the computationalcost associated with Bayesian linear models constrains this method'sapplication to small networks, small output spaces and small datasets. Weaddress this limitation by introducing a scalable sample-based Bayesianinference method for conjugate Gaussian multi-output linear models, togetherwith a matching method for hyperparameter (regularisation) selection.Furthermore, we use a classic feature normalisation method (the g-prior) toresolve a previously highlighted pathology of the linearised Laplace method.Together, these contributions allow us to perform linearised neural networkinference with ResNet-18 on CIFAR100 (11M parameters, 100 outputs x 50kdatapoints), with ResNet-50 on Imagenet (50M parameters, 1000 outputs x 1.2Mdatapoints) and with a U-Net on a high-resolution tomographic reconstructiontask (2M parameters, 251k output~dimensions).

Sampling-based inference for large linear models, with application to linearised Laplace