Stochastic Representations with Gaussian Processes and Geometry

This thesis consists of 4 independent pieces of work and each of these have a dedicated chapter in the manuscript. The first chapter investigates contemporary methodologies for estimating predictive variance networks in regression neural networks. The second chapter goes beyond regression task, and studies Gaussian processes to present a Bayesian non-parametric way of inferring stochastic differential equations for both regression and continuous-time dynamical modelling. The third chapter unifies theory of geometry and Gaussian processes to present a latent variable model that respects both the distances and the topology of unlabelled data. The fourth, and last, chapter shortly reviews current methodologies for bivariate causal invariance and propose an algorithm using a non-parametric estimator robust towards a causal invariant: changes in the marginal distributions.