Information geometry in cosmology

May 29, 2020 - 3:00 pm to 4:00 pm
Nick Kokron (Stanford) via zoom

Join us this Friday, May 29th at 3pm exclusively on zoom for the next meeting of the Stats and ML Journal Club. This week, Nick Kokron will lead a discussion on Information Geometry. See you then! 

Title: Information geometry in cosmology

Abstract: Statistical inference more often than not involves models which are non-linear in the parameters thus leading to non-Gaussian posteriors. Many computational and analytical tools exist that can deal with non-Gaussian distributions, and empirical Gaussianisation transforms can reduce the amount of non-Gaussianity in a distribution. Alternatively, in this work, we employ methods from information geometry. The latter formulates a set of probability distributions for some given model as a manifold employing a Riemannian structure, equipped with a metric, 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 an application we consider the likelihood of the supernovae distance-redshift relation in cosmology for the parameter pair (Ωm0, w). We demonstrate the connection between confidence intervals and geodesic length and demonstrate how the Lie-derivative along the degeneracy directions gives hints at possible isometries of the Fisher metric.