A Mathematical Theory of Arguments for Statistical Evidence

1. Front Matter

   Pages I-XIII

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2. The Theory of Generalized Functional Models

   • Paul-André Monney

   Pages 1-37

3. The Plausibility and Likelihood Functions

   • Paul-André Monney

   Pages 39-57

4. Hints on Continuous Frames and Gaussian Linear Systems

   • Paul-André Monney

   Pages 59-70

5. Assumption-Based Reasoning with Classical Regression Models

   • Paul-André Monney

   Pages 71-95

6. Assumption-Based Reasoning with General Gaussian Linear Systems

   • Paul-André Monney

   Pages 97-108

7. Gaussian Hints as a Valuation System

   • Paul-André Monney

   Pages 109-127

8. Local Propagation of Gaussian Hints

   • Paul-André Monney

   Pages 129-135

9. Application to the Kalman Filter

   • Paul-André Monney

   Pages 137-148

10. Back Matter

    Pages 149-154

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The subject of this book is the reasoning under uncertainty based on sta­ tistical evidence, where the word reasoning is taken to mean searching for arguments in favor or against particular hypotheses of interest. The kind of reasoning we are using is composed of two aspects. The first one is inspired from classical reasoning in formal logic, where deductions are made from a knowledge base of observed facts and formulas representing the domain spe­ cific knowledge. In this book, the facts are the statistical observations and the general knowledge is represented by an instance of a special kind of sta­ tistical models called functional models. The second aspect deals with the uncertainty under which the formal reasoning takes place. For this aspect, the theory of hints [27] is the appropriate tool. Basically, we assume that some uncertain perturbation takes a specific value and then logically eval­ uate the consequences of this assumption. The original uncertainty about the perturbation is then transferred to the consequences of the assumption. This kind of reasoning is called assumption-based reasoning. Before going into more details about the content of this book, it might be interesting to look briefly at the roots and origins of assumption-based reasoning in the statistical context. In 1930, R. A. Fisher [17] defined the notion of fiducial distribution as the result of a new form of argument, as opposed to the result of the older Bayesian argument.

• regression
• valuation
• value-at-risk

From the reviews:

"The book, clearly written, is structured in two parts comprising two and six chapters, respectively. … 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." (Evdokia Xekalaki, Zentralblatt MATH, Vol. 1100 (2), 2007)

• Department of Statistics, Purdue University, West Lafayette, USA

  Paul-André Monney

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