  Bayesian learning using Gaussian processes provides a foundational framework
for making decisions in a manner that balances what is known with what could be
learned by gathering data. In this dissertation, we develop techniques for
broadening the applicability of Gaussian processes. This is done in two ways.
Firstly, we develop pathwise conditioning techniques for Gaussian processes,
which allow one to express posterior random functions as prior random functions
plus a dependent update term. We introduce a wide class of efficient
approximations built from this viewpoint, which can be randomly sampled once in
advance, and evaluated at arbitrary locations without any subsequent
stochasticity. This key property improves efficiency and makes it simpler to
deploy Gaussian process models in decision-making settings. Secondly, we
develop a collection of Gaussian process models over non-Euclidean spaces,
including Riemannian manifolds and graphs. We derive fully constructive
expressions for the covariance kernels of scalar-valued Gaussian processes on
Riemannian manifolds and graphs. Building on these ideas, we describe a
formalism for defining vector-valued Gaussian processes on Riemannian
manifolds. The introduced techniques allow all of these models to be trained
using standard computational methods. In total, these contributions make
Gaussian processes easier to work with and allow them to be used within a wider
class of domains in an effective and principled manner. This, in turn, makes it
possible to potentially apply Gaussian processes to novel decision-making
settings.
