Information Geometry: Near Randomness and Near Independence

The main motivation for this book lies in the breadth of applications in which a statistical model is used to represent small departures from, for example, a Poisson process. Our approach uses information geometry to provide a common context but we need only rather elementary material from differential geometry, information theory and mathematical statistics. Introductory sections serve together to help those interested from the applications side in making use of our methods and results. Reported in this monograph is a body of results, and computer-algebraic methods that seem to have quite general applicability to statistical models admitting representation through parametric families of probability density functions. Some illustrations are given from a variety of contexts for geometric characterization of statistical states near to the three important standard basic reference states: (Poisson) randomness, uniformity, independence. The individual applications are somewhat heuristic models from various fields and we incline more to terminology and notation from the applications rather than from formal statistics. However, a common thread is a geometrical representation for statistical perturbations of the basic standard states, and hence results gain qualitative stability. Moreover, the geometry is controlled by a metric structure that owes its heritage through maximum likelihood to information theory so the quantitative features---lengths of curves, geodesics, scalar curvatures etc.---have some respectable authority. We see in the applications simple models for galactic void distributions and galaxy clustering, amino acid clustering along protein chains, cryptographic protection, stochastic fibre networks, coupled geometric features in hydrology and quantum chaotic behaviour.