DE

2021 | Book

# The Geometry of Uncertainty

## The Geometry of Imprecise Probabilities

Author: Dr. Fabio Cuzzolin

Publisher:

Book Series : Artificial Intelligence: Foundations, Theory, and Algorithms

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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.

##### 1. Introduction
Abstract
The mainstream mathematical theory of uncertainty is measure-theoretical probability, and is mainly due to the Russian mathematician Andrey Kolmogorov [1030].
Fabio Cuzzolin

#### Theories of uncertainty

##### 2. Belief functions
Abstract
The theory of evidence [1583] was introduced in the 1970s by Glenn Shafer as a way of representing epistemic knowledge, starting from a sequence of seminal papers [415, 417, 418] by Arthur Dempster [424].
Fabio Cuzzolin
##### 3. Understanding belief functions
Abstract
From the previous chapter’s summary of the basic notions of the theory of evidence, it is clear that belief functions are rather complex objects.
Fabio Cuzzolin
##### 4. Reasoning with belief functions
Abstract
Belief functions are fascinating mathematical objects, but most of all they are useful tools designed to achieve a more natural, proper description of one’s uncertainty concerning the state of the external world.
Fabio Cuzzolin
##### 5. A toolbox for the working scientist
Abstract
Machine learning is possibly the core of artificial intelligence, as it is concerned with the design of algorithms capable of allowing machines to learn from observations.
Fabio Cuzzolin
##### 6. The bigger picture
Abstract
As we have seen in the Introduction, several different mathematical theories of uncertainty are competing to be adopted by practitioners in all fields of applied science [1874, 2057, 1615, 784, 1229, 665].
Fabio Cuzzolin

#### The geometry of uncertainty

##### 7. The geometry of belief functions
Abstract
Belief measures are complex objects, even when classically defined on finite sample spaces. Because of this greater complexity, manipulating them, as we saw in Chapter 4, opens up a wide spectrum of options, whether we consider the issues of conditioning and combination, we seek to map belief measures onto different types of measures, or we are concerned with decision making, and so on.
Fabio Cuzzolin
##### 8. Geometry of Dempster’s rule
Abstract
As we have seen in Chapter 7, belief functions can be seen as points of a simplex B called the ‘belief space’. It is therefore natural to wonder whether the orthogonal sum operator (2.6), a mapping from a pair of prior belief functions to a posterior belief function on the same frame, can also be interpreted as a geometric operator in B. The answer is positive, and in this chapter we will indeed understand this property of Dempster’s rule in this geometric setting.
Fabio Cuzzolin
##### 9. Three equivalent models
Abstract
In this chapter, we show that we can indeed represent the same evidence in terms of a basic plausibility (or commonality) assignment on the power set, and compute the related plausibility (or commonality) set function by integrating the basic assignment over similar intervals [347].
Fabio Cuzzolin
##### 10. The geometry of possibility
Abstract
Possibility measures [531], just like probability measures, are a special case of belief functions when defined on a finite frame of discernment.
Fabio Cuzzolin

#### Geometric interplays

##### 11. Probability transforms: The affine family
Abstract
As we have seen in Chapter 4, the relation between belief and probability in the theory of evidence has been and continues to be an important subject of study.
Fabio Cuzzolin
##### 12. Probability transforms: The epistemic family
Abstract
Chapter 11 has taught us that the pignistic transform is just a member of what we have called the ‘affine’ group of probability transforms, characterised by their commutativity with affine combination (a property called ‘linearity’ in the TBM).
Fabio Cuzzolin
##### 13. Consonant approximation
Abstract
As we have learned in the last two chapters, probability transforms are a very wellstudied topic in belief calculus, as they are useful as a means to reduce the computational complexity of the framework (Section 4.7), they allow us to reduce decision making with belief functions to the classical utility theory approach (Section 4.8.1) and they are theoretically interesting for understanding the relationship between Bayesian reasoning and belief theory [291].
Fabio Cuzzolin
##### 14. Consistent approximation
Abstract
As we know, belief functions are complex objects, in which different and sometimes contradictory bodies of evidence may coexist, as they describe mathematically the fusion of possibly conflicting expert opinions and/or imprecise/corrupted measurements, among other things.
Fabio Cuzzolin

#### Geometric reasoning

##### 15. Geometric conditioning
Abstract
As we have seen in Chapter 4, Section 4.5, various distinct definitions of conditional belief functions have been proposed in the past.
Fabio Cuzzolin
##### 16. Decision making with epistemic transforms
Abstract
As we learned in Chapter 4, decision making with belief functions has been extensively studied.
Fabio Cuzzolin

#### The future of uncertainty

##### 17. An agenda for the future
Abstract
As we have seen in this book, the theory of belief functions is a modelling language for representing and combining elementary items of evidence, which do not necessarily come in the form of sharp statements, with the goal of maintaining a mathematical representation of an agent’s beliefs about those aspects of the world which the agent is unable to predict with reasonable certainty.
Fabio Cuzzolin
##### Backmatter
Title
The Geometry of Uncertainty
Author
Dr. Fabio Cuzzolin