Geometrical Methods for Statistical Inference and Decision

The important problems of statistics concern what we can learn from empirical data, what might happen next, and what is the best course of action. Statistical inference is the process of extracting information about the underlying nature of the data, to allow us to make predictions about future events. Statistical decision theory searches for strategies that will lead to optimal outcomes, taking into account the intrinsic uncertainty in our predictions. For instance, data on the response of patients to a pharmaceutical drug enable us to infer the effectiveness of the drug in the general population. Given a desirable social goal, such as maximising health benefits while minimising adverse reactions, we may then decide what treatment allocation and dosage levels would be optimal.It is remarkable fact that such statistical questions can be reframed in the mathematical language of geometry. To be more precise, geometric descriptions of objects, involving e.g. distances between points, can be applied to statistical models. However, instead of thinking of an object as a collection of points in 3-dimensional space, the points are now the various different probability distributions that could generate the data. To take a simple example, optimal estimation becomes the process of finding the geometric point which represents the best fitting distribution, and of quantifying how close it is to the true distribution generating the data. More sophisticated applications utilise e.g. the geometric curvature of the statistical model to quantify the uncertainty in our inferential conclusions.The geometric approach to statistical inference has been intensively studied, but there has been little attempt to apply it to statistical decision theory. Building on theoretical foundations recently laid down by Dawid and Lauritzen, this project will develop new theory and applications of geometric decision analysis. In particular it will introduce geometric concepts and techniques originating in Physics and Cosmology to the study of problems of statistical inference.