A mathematical theory of hints. An approach to the Dempster-Shafer theory of evidence
The authors present the theory of hints, a variant of the Dempster-Shafer theory of evidence. They keep the Dempster's original structure \((\Omega, P, \Gamma, \Theta)\), where \(\Omega\) and \(\Gamma\) denote spaces, \(P\) a probability measure defined on \(\Omega\), and \(\Gamma\) a multivalued mapping assigning a subset \(\Gamma(\omega) \subseteq \Theta\) to every \(\omega\in \Omega\), but they give it the following meaning: Suppose a certain question, whose answer is unknown. The elements of \(\Theta\) represent the possible answers to the question, exactly one of them being correct. The goal is to make assertions about the answer in the light of the available information. This information allows for several different interpretations, which constitute the set \(\Omega\). Not all interpretations from \(\Omega\) are equally likely and \(P\) is the known probability measure on \(\Omega\). If the interpretation \(\omega \in \Omega\) is the correct one, then the answer is in the subset \(\Gamma(\omega) \subseteq \Theta\). The structure \((\Omega,P, \Gamma, \Theta)\), called hint, can be used to judge the hypothesis that the answer is in a subset \(H\) of \(\Theta\) by assigning to it two quantities: a degree of support and a degree of plausibility. This monograph is divided into three Parts. In the Part I, the elementary theory of hints is given including the related concepts of support, credibility and plausibility. Part II discusses some possible applications of this theory in statistical inference, describing uncertainty in dynamical systems, diagnostics, and temporal and spatial reasoning. Finally, Part III develops the rudiments of a general theory of hints, where the sets of interpretations are not necessarily finite. The reader should have basic courses in probability theory and mathematical statistics. This book is intented for all who are interested in new ideas on evidence theory and its applications.
probabilistic assumption-based reasoning
scientific article; zbMATH DE number 776154