Abstract
Statistical inference often involves models which are non-linear in the parameters and which therefore typically exhibit non-Gaussian posterior distributions. These non-Gaussianities can be prominent especially when data is limited or not constraining enough. Many computational and analytical tools exist that can deal with non-Gaussian distributions, and empirical Gaussianisation transforms can be constructed that can reduce the amount of non-Gaussianity in a distribution. In this work, we employ methods from information geometry, which considers a set of probability distributions for some given model to be a manifold with a metric Riemannian structure, given by the Fisher information. In this framework we study the differential geometrical meaning of non-Gaussianities in a higher order Fisher approximation, and their respective transformation behaviour under re-parameterisation, which corresponds to a chart transition on the statistical manifold. While weak non-Gaussianities vanish in normal coordinates in a first order approximation, one can in general not find transformations that discard non-Gaussianities globally. As a topical application in cosmology we consider the likelihood of the supernovae distance-redshift relation for the parameter pair (Ωm0, w). We show that the corresponding manifold is non-flat and demonstrate the connection between confidence intervals and geodesic length, determine the curvature of that likelihood and quantify degeneracies by means of Lie-derivatives.