Statistical Information

This book is a monograph that builds on the path-breaking work of Arthur Dempster and Glenn Shafer, and before them R. A. Fisher, in the field of statistical inference. The main thrust of the book lies around the idea of statistical information. The inferential mechanism that is used in this book to derive information about a parameter in a statistical experiment is new, which is what sets it apart from other books on the topic. This inference principle, which we call assumption-based reasoning, is based on a sound combination of logic and classical probability theory. Some traces of it can already be found in the original writings of Jacob Bernoulli, but this book presents for the first time a much more complete and elaborate descripton of it. In particular, it is shown that assumption-based inference on functional models is a generalization of both Bayesian inference and Fisher's fiducial inference. This is an interesting result regarding the old controversy between these two theories. Our approach provides a new and clear meaning to post-data probabilistic statements about an unknown parameter, for example statements based on the likelihood function. In particular, it is shown that this function cannot, in general, be considered to carry the entire statistical information contained in the experiment.

Information about statistical experiments is described in this monograph by functional models. They indicate how observations are generated in functions of an unknown parameter and stochastic disturbances. In the first part of the book we examine discrete functional models. These models are used to present the basic ideas of assumption-based reasoning in a form that is unhampered by technical difficulties. It is shown how several pieces of information can be combined and how the result can be focused on a question of interest. These operations provide an algebraic flavor to the analysis of statistical information, which is a perspective that is presented here for the first time. Some new preliminary results regarding a decision rule for hypothesis selection are also presented. In the second part of the book we treat several types of continuous models from the standpoint of assumption-based reasoning. This allows us to review and clarify several concepts and difficulties of Fisher's fiducial theory, for example the relation between traditional confidence intervals and fiducial intervals. Our approach also permits to determine the exact role and nature of improper priors in Bayesian inference. Finally, the third part of the book is dedicated to the analysis of linear models with Gaussian perturbations using assumption-based reasoning.