On the global geometry of parametric models and information recovery

Abstract

We examine the question of which statistic or statistics should be used in order to recover information important for inference. We take a global geometric viewpoint, developing the local geometry of Amari. By examining the behaviour of simple geometric models, we show how not only the local curvature properties of parametric families but also the global geometric structure can be of crucial importance in finite-sample analysis. The tool we use to explore this global geometry is the Karhunen-Loève decomposition. Using global geometry, we show that the maximum likelihood estimate is the most important one-dimensional summary of information, but that traditional methods of information recovery beyond the maximum likelihood estimate can perform poorly. We also use the global geometry to construct better information summaries to be used with the maximum likelihood estimate.