The principal aim of this book is to introduce to the widest
possible audience an original view of belief calculus and
uncertainty theory. In this geometric approach to uncertainty,
uncertainty measures can be seen as points of a suitably complex
geometric space, and manipulated in that space, for example,
combined or conditioned. In the chapters in Part I, Theories of
Uncertainty, the author offers an extensive recapitulation of the
state of the art in the mathematics of uncertainty. This part of
the book contains the most comprehensive summary to date of the
whole of belief theory, with Chap. 4 outlining for the first time,
and in a logical order, all the steps of the reasoning chain
associated with modelling uncertainty using belief functions, in an
attempt to provide a self-contained manual for the working
scientist. In addition, the book proposes in Chap. 5 what is
possibly the most detailed compendium available of all theories of
uncertainty. Part II, The Geometry of Uncertainty, is the core of
this book, as it introduces the author's own geometric approach to
uncertainty theory, starting with the geometry of belief functions:
Chap. 7 studies the geometry of the space of belief functions, or
belief space, both in terms of a simplex and in terms of its
recursive bundle structure; Chap. 8 extends the analysis to
Dempster's rule of combination, introducing the notion of a
conditional subspace and outlining a simple geometric construction
for Dempster's sum; Chap. 9 delves into the combinatorial
properties of plausibility and commonality functions, as equivalent
representations of the evidence carried by a belief function; then
Chap. 10 starts extending the applicability of the geometric
approach to other uncertainty measures, focusing in particular on
possibility measures (consonant belief functions) and the related
notion of a consistent belief function. The chapters in Part III,
Geometric Interplays, are concerned with the interplay of
uncertainty measures of different kinds, and the geometry of their
relationship, with a particular focus on the approximation problem.
Part IV, Geometric Reasoning, examines the application of the
geometric approach to the various elements of the reasoning chain
illustrated in Chap. 4, in particular conditioning and decision
making. Part V concludes the book by outlining a future, complete
statistical theory of random sets, future extensions of the
geometric approach, and identifying high-impact applications to
climate change, machine learning and artificial intelligence. The
book is suitable for researchers in artificial intelligence,
statistics, and applied science engaged with theories of
uncertainty. The book is supported with the most comprehensive
bibliography on belief and uncertainty theory.
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