In this thesis, we disentangle the generalized Gauss-Newton and approximate inference for Bayesian deep learning. The generalized Gauss-Newton method is an optimization method that is used in several popular Bayesian deep learning algorithms. Algorithms that combine the Gauss-Newton method with the Laplace and Gaussian variational approximation have recently led to state-of-the-art results in Bayesian deep learning. While the Laplace and Gaussian variational approximation have been studied extensively, their interplay with the Gauss-Newton method remains unclear. Recent criticism of priors and posterior approximations in Bayesian deep learning further urges the need for a deeper understanding of practical algorithms. The individual analysis of the Gauss-Newton method and Laplace and Gaussian variational approximations for neural networks provides both theoretical insight and new practical algorithms. We find that the Gauss-Newton method simplifies the underlying probabilistic model significantly. In particular, the combination of the Gauss-Newton method with approximate inference can be cast as inference in a linear or Gaussian process model. The Laplace and Gaussian variational approximation can subsequently provide a posterior approximation to these simplified models. This new disentangled understanding of recent Bayesian deep learning algorithms also leads to new methods: first, the connection to Gaussian processes enables new function-space inference algorithms. Second, we present a marginal likelihood approximation of the underlying probabilistic model to tune neural network hyperparameters. Finally, the identified underlying models lead to different methods to compute predictive distributions. In fact, we find that these prediction methods for Bayesian neural networks often work better than the default choice and solve a common issue with the Laplace approximation.
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