Likelihood-based Inference on Nonlinear spaces

Abstract

When data in higher dimensions with a certain constraint on it, say a set of locations on a sphere, is encountered, some classical statistical analysis methods fail, as the data no longer assumes its values in a linear space. In this thesis we consider such datasets and aim to do likelihood-based inference on the center of the data. To model the nonlinearity, we consider the data to be a set of points on a Riemannian manifold. The general approach in this thesis comes from the classical result where the center can be repre- sented as the maximum likelihood estimator for the true mean of the dataset. To model an underlying distribution we will model the data as observations of realizations of Brownian motion on the manifold observed at a fixed time and use the transition density of the Brownian motion to construct a likelihood. The likelihood can then be approximated using diffusion bridges. This thesis thus first focuses on differential geometry as well as Itô and Stratonovich calculus. After that, we will introduce methods to construct a likelihood for the center of the dataset on a manifold before using simulated diffusion bridges to approximate this likelihood. We finish the thesis with some numerical experiments in Julia that demonstrate the results on the sphere.