A mathematical theory of arguments for statistical evidence
The book, clearly written, is structured in two parts comprising two and six chapters, respectively. The first part addresses the problem of modeling reasoning through the use of functional models involving discrete spaces. The underlying theory of such models provides insight for the weighing of evidence. The second part treats the case of special linear functional models, termed Gaussian linear models in continuous spaces. The theory of hints has a central role throughout the book and the presented material aims at providing hints permitting the evaluation of some hypotheses of interest. Chapter 1 contains a presentation of the theory of generalized functional models, which through several examples are demonstrated to be useful tools in modeling reasoning under uncertainty. The reader is first provided with a concise introduction to the theory of hints and its most prominent concepts and properties appropriate to measuring the effect of assumption-based reasoning on functional models. In chapter 2, the notion of likelihood is extended to composite hypotheses in terms of the plausibility function of some hint obtained from a generalized functional model allowing the definition of a general weight of evidence applicable to both simple and composite hypotheses. The chapters that follow comprising the second part of the book, are devoted to special linear functional models involving continuous spaces. Starting with chapter 3, a theory of continuous hints is presented through a formal exposition. These have all the characteristics of hints on finite frames only adapted for continuous ones. A class of Gaussian linear models is also defined. These are generalized functional models of a special form. Chapter 4 treats the principle of assumption-based reasoning through the prism of a special class of Gaussian linear systems, which coincide with classical regression models with known error term when expressed in canonical coordinates. The information generated about the parameters can be employed in determining the degree of support and the degree of plausibility of the various hypotheses about the parameter. The results parallel those obtained by least squares or maximum likelihood methods. Chapter 5 looks again into assumption-based reasoning in the frame of any of the kinds of general Gaussian linear systems given in chapter 3. The inferences drawn are expressed as Gaussian hints that enable computation of the degrees of support and plausibility for hypotheses about a parameter. Chapter 6 looks at Gaussian hints put in an axiomatic framework for the valuation calculus as described by \textit{P. Shenoy} and \textit{G. Shafer} [Axioms for probability and belief function propagation. R. D. Shachter, et al. (eds.), Uncertainty in Artificial Intelligence, 169--198 (1990)] by defining the concept of a marginal hint. In chapter 7, it is demonstrated that this general framework enables the application of an efficient local propagation procedure for determining the Gaussian hint representing one's knowledge regarding a set of variables inferred from the available information. This is done by means of an algorithm described in detail. Finally, in chapter 8, the theory of Gaussian hints is looked into in a control context. It is shown in particular, that the well-known Kalman filter can be derived from the perspective of the assumption-based reasoning principle by locally propagating Gaussian hints in a Markov tree. The overall quality of the book is very good, the material is well organized and notations and terminology are unified. It is a very good presentation of the state of research in the area of modeling reasoning and an insightful reference for the statistician.
scientific article; zbMATH DE number 1868963