  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.
