Many phenomena in fusion plasmas exhibit stochasticity, such as fluctuations due to turbulent transport of heat and particles, characteristics of plasma instabilities, etc. Inference of physical quantities and model parameters needs to account for additional sources of uncertainty due to measurement error and model uncertainty. In the face of these various levels of uncertainty, a probabilistic approach for analyzing the data is appropriate. In this PhD research, data from fusion experiments will be modeled through probability distributions on a statistical manifold. Information geometry provides tools for classification and regression on these manifolds, enabling a form of pattern recognition with advantages compared to traditional approaches in Euclidean data spaces. Both local and global plasma quantities will be studied, pertaining to the energy confinement, the edge transport barrier in the high confinement mode and magnetohydrodynamic instabilities, called edge-localized modes, occurring near the plasma boundary. To investigate the dependence of these phenomena on plasma conditions, new regression techniques on statistical manifolds will be applied and further developed. One such technique, called geodesic least squares regression (GLS), is a generalization of ordinary least squares, but allows errors in all variables and enjoys remarkable robustness properties. The aim of this PhD research is, on the one hand, to make progress in the description of the properties of GLS regression, its performance in the presence of data and model uncertainty and its extension toward a fully Bayesian technique operating on statistical manifolds. On the other hand, together with standard Bayesian techniques, the powerful methods developed in this framework will be used to increase the reliability and robustness of models describing the stochastic behavior and dependencies of plasma quantities that are key for the physics understanding of fusion plasmas and the design of new fusion devices. Applications in other domains, such as astronomy and climatology, may also be explored.