The Geometry of Uncertainty: The Geometry of Imprecise Probabilities

Preface
Uncertainty
Probability
Beyond probability
Belief functions
Aim(s) of the book
Structure and topics
Acknowledgements
Table of Contents
1 Introduction
1.1 Mathematical probability
1.2 Interpretations of probability
1.2.1 Does probability exist at all?
1.2.2 Competing interpretations
1.2.3 Frequentist probability
1.2.4 Propensity
1.2.5 Subjective and Bayesian probability
1.2.6 Bayesian versus frequentist inference
1.3 Beyond probability
1.3.1 Something is wrong with probability Flaws of the frequentistic setting
1.3.2 Pure data: Beware of the prior
1.3.3 Pure data: Designing the universe?
1.3.4 No data: Modelling ignorance
1.3.5 Set-valued observations: The cloaked die
1.3.6 Propositional data
1.3.7 Scarce data: Beware the size of the sample
1.3.8 Unusual data: Rare events
1.3.9 Uncertain data
1.3.10 Knightian uncertainty
1.4 Mathematics (plural) of uncertainty
1.4.1 Debate on uncertainty theory
1.4.2 Belief, evidence and probability
Part I Theories of uncertainty
2 Belief functions
Chapter outline
2.1 Arthur Dempster’s original setting
2.2 Belief functions as set functions
2.2.1 Basic definitions Basic probability assignments Definition 4.
2.2.2 Plausibility and commonality functions
2.2.3 Bayesian belief functions
2.3 Dempster’s rule of combination
2.3.1 Definition
2.3.2 Weight of conflict
2.3.3 Conditioning belief functions
2.4 Simple and separable support functions
2.4.1 Heterogeneous and conflicting evidence
2.4.2 Separable support functions
2.4.3 Internal conflict
2.4.4 Inverting Dempster’s rule: The canonical decomposition
2.5 Families of compatible frames of discernment
2.5.1 Refinings
2.5.2 Families of frames
2.5.3 Consistent and marginal belief functions
2.5.4 Independent frames
2.5.5 Vacuous extension
2.6 Support functions
2.6.1 Families of compatible support functions in the evidential language
2.6.2 Discerning the relevant interaction of bodies of evidence
2.7 Quasi-support functions
2.7.1 Limits of separable support functions
2.7.2 Bayesian belief functions as quasi-support functions
2.7.3 Bayesian case: Bayes’ theorem
2.7.4 Bayesian case: Incompatible priors
2.8 Consonant belief functions
3 Understanding belief functions
Chapter outline
3.1 The multiple semantics of belief functions
3.1.1 Dempster’s multivalued mappings, compatibility relations
3.1.2 Belief functions as generalised (non-additive) probabilities
3.1.3 Belief functions as inner measures
3.1.4 Belief functions as credal sets
3.1.5 Belief functions as random sets
3.1.6 Behavioural interpretations
3.1.7 Common misconceptions Belief
3.2 Genesis and debate
3.2.1 Early support
3.2.2 Constructive probability and Shafer’s canonical examples
3.2.3 Bayesian versus belief reasoning
3.2.4 Pearl’s criticism
3.2.5 Issues with multiple interpretations
3.2.6 Rebuttals and justifications
3.3 Frameworks
3.3.1 Frameworks based on multivalued mappings
3.3.2 Smets’s transferable belief model
3.3.3 Dezert–Smarandache theory (DSmT)
3.3.4 Gaussian (linear) belief functions
3.3.5 Belief functions on generalised domains
3.3.7 Intervals and sets of belief measures
3.3.8 Other frameworks
4 Reasoning with belief functions
Chapter outline
4.1 Inference
4.1.1 From statistical data
4.1.2 From qualitative data
4.1.3 From partial knowledge
4.1.4 A coin toss example
4.2 Measuring uncertainty
4.2.1 Order relations
4.2.2 Measures of entropy
4.2.3 Principles of uncertainty
4.3 Combination
4.3.1 Dempster’s rule under fire
4.3.2 Alternative combination rules
4.3.3 Families of combination rules
4.3.4 Combination of dependent evidence
4.3.5 Combination of conflicting evidence
4.3.6 Combination of (un)reliable sources of evidence: Discounting
4.4 Belief versus Bayesian reasoning: A data fusion example
4.4.1 Two fusion pipelines
4.4.2 Inference under partially reliable data
4.5 Conditioning
4.5.1 Dempster conditioning
4.5.2 Lower and upper conditional envelopes
4.5.3 Suppes and Zanotti’s geometric conditioning
4.5.4 Smets’s conjunctive rule of conditioning
4.5.5 Disjunctive rule of conditioning
4.5.6 Conditional events as equivalence classes: Spies’s definition
4.5.7 Other work
4.5.8 Conditioning: A summary
4.6 Manipulating (conditional) belief functions
4.6.1 The generalised Bayes theorem
4.6.2 Generalising total probability
4.6.3 Multivariate belief functions
4.6.4 Graphical models
4.7 Computing
4.7.1 Efficient algorithms
4.7.2 Transformation approaches
4.7.3 Monte Carlo approaches
4.7.4 Local propagation
4.8 Making decisions
4.8.1 Frameworks based on utilities
4.8.2 Frameworks not based on utilities
4.8.3 Multicriteria decision making
4.9 Continuous formulations
4.9.1 Shafer’s allocations of probabilities
4.9.2 Belief functions on random Borel intervals
4.9.3 Random sets
4.9.4 Kramosil’s belief functions on infinite spaces
4.9.5 MV algebras
4.9.6 Other approaches
4.10 The mathematics of belief functions
4.10.1 Distances and dissimilarities
4.10.2 Algebra
4.10.3 Integration
4.10.4 Category theory
4.10.5 Other mathematical analyses
5 A toolbox for the working scientist
Chapter outline
5.1 Clustering
5.1.1 Fuzy, evidential and belief C-means
5.1.2 EVCLUS and later developments
5.1.3 Clustering belief functions
5.2 Classification
5.2.1 Generalised Bayesian classifier
5.2.2 Evidential k-NN
5.2.3 TBM model-based classifier
5.2.4 SVM classification
5.2.5 Classification with partial training data
5.2.6 Decision trees
5.2.7 Neural networks
5.2.8 Other classification approaches
5.3 Ensemble classification
5.3.1 Distance-based classification fusion
5.3.2 Empirical comparison of fusion schemes
5.3.3 Other classifier fusion schemes
5.4 Ranking aggregation
5.5 Regression
5.5.1 Fuzzy-belief non-parametric regression
5.5.2 Belief-modelling regression
5.6 Estimation, prediction and identification
5.6.1 State estimation
5.6.2 Time series analysis
5.6.3 Particle filtering
5.6.4 System identification
5.7 Optimisation
6 The bigger picture
Chapter outline
6.1 Imprecise probability
6.1.2 Gambles and behavioural interpretation
6.1.3 Lower and upper previsions
6.1.4 Events as indicator gambles
6.1.5 Rules of rational behaviour
6.1.6 Natural and marginal extension
6.1.7 Belief functions and imprecise probabilities
6.2 Capacities (a.k.a. fuzzy measures)
6.2.1 Special types of capacities
6.3 Probability intervals (2-monotone capacities)
6.3.1 Probability intervals and belief measures
6.4 Higher-order probabilities
6.4.1 Second-order probabilities and belief functions
6.4.2 Gaifman’s higher-order probability spaces
6.4.3 Kyburg’s analysis
6.4.4 Fung and Chong’s metaprobability
6.5 Fuzzy theory
6.5.1 Possibility theory
6.5.2 Belief functions on fuzzy sets
6.5.3 Vague sets
6.5.4 Other fuzzy extensions of the theory of evidence
6.6 Logic
6.6.1 Saffiotti’s belief function logic
6.6.2 Josang’s subjective logic
6.6.3 Fagin and Halpern’s axiomatisation
6.6.4 Probabilistic argumentation systems
6.6.5 Default logic
6.6.6 Ruspini’s logical foundations
6.6.7 Modal logic interpretation
6.6.8 Probability of provability
6.6.9 Other logical frameworks
6.7 Rough sets
6.7.1 Pawlak’s algebras of rough sets
6.7.2 Belief functions and rough sets
6.8 Probability boxes
6.8.1 Probability boxes and belief functions
6.8.2 Approximate computations for random sets
6.8.3 Generalised probability boxes
6.9 Spohn’s theory of epistemic beliefs
6.9.1 Epistemic states
6.9.2 Disbelief functions and Spohnian belief functions
6.9.3 α-conditionalisation
6.10 Zadeh’s generalised theory of uncertainty
6.11 Baoding Liu’s uncertainty theory
6.12 Info-gap decision theory
6.12.1 Info-gap models
6.12.2 Robustness of design
6.13 Vovk and Shafer’s game-theoretical framework
6.13.1 Game-theoretic probability
6.13.2 Ville/Vovk game-theoretic testing
6.13.3 Upper price and upper probability
6.14 Other formalisms
6.14.1 Endorsements
6.14.2 Fril-fuzzy theory
6.14.3 Granular computing
6.14.4 Laskey’s assumptions
6.14.5 Harper’s Popperian approach to rational belief change
6.14.6 Shastri’s evidential reasoning in semantic networks
6.14.7 Evidential confirmation theory
6.14.8 Groen’s extension of Bayesian theory
6.14.9 Padovitz’s unifying model
6.14.10 Similarity-based reasoning
6.14.11 Neighbourhoods systems
6.14.12 Comparative belief structures
Part II The geometry of uncertainty
7 The geometry of belief functions
Outline of Part II
Chapter outline
7.1 The space of belief functions
7.1.1 The simplex of dominating probabilities
7.1.2 Dominating probabilities and L1 norm
7.1.3 Exploiting the M¨obius inversion lemma
7.1.4 Convexity of the belief space
7.2 Simplicial form of the belief space
7.2.1 Faces of B as classes of belief functions
7.3 The differential geometry of belief functions
7.3.1 A case study: The ternary case
7.3.2 Definition of smooth fibre bundles
7.3.3 Normalised sum functions
7.4 Recursive bundle structure
7.4.1 Recursive bundle structure of the space of sum functions
7.4.2 Recursive bundle structure of the belief space
7.4.3 Bases and fibres as simplices
7.5 Research questions
Appendix: Proofs
8 Geometry of Dempster’s rule
8.1 Dempster combination of pseudo-belief functions
8.2 Dempster sum of affine combinations
8.3 Convex formulation of Dempster’s rule
8.4 Commutativity
8.4.1 Affine region of missing points
8.4.2 Non-combinable points and missing points: A duality
8.4.3 The case of unnormalised belief functions
8.5 Conditional subspaces
8.5.1 Definition
8.5.2 The case of unnormalised belief functions
8.5.3 Vertices of conditional subspaces
8.6 Constant-mass loci
8.6.1 Geometry of Dempster’s rule in B2
8.6.2 Affine form of constant-mass loci
8.6.3 Action of Dempster’s rule on constant-mass loci
8.7 Geometric orthogonal sum
8.7.1 Foci of conditional subspaces
8.7.2 Algorithm
8.8 Research questions
Appendix: Proofs
9 Three equivalent models
Chapter outline
9.1 Basic plausibility assignment
9.1.1 Relation between basic probability and plausibility assignments
9.2 Basic commonality assignment
9.2.1 Properties of basic commonality assignments
9.3 The geometry of plausibility functions
9.3.1 Plausibility assignment and simplicial coordinates
9.3.2 Plausibility space
9.4 The geometry of commonality functions
9.5 Equivalence and congruence
9.5.1 Congruence of belief and plausibility spaces
9.5.2 Congruence of plausibility and commonality spaces
9.6 Pointwise rigid transformation
9.6.1 Belief and plausibility spaces
9.6.2 Commonality and plausibility spaces
Appendix: Proofs
10 The geometry of possibility
Chapter outline
10.1 Consonant belief functions as necessity measures
10.2 The consonant subspace
10.2.1 Chains of subsets as consonant belief functions
10.2.2 The consonant subspace as a simplicial complex
10.3 Properties of the consonant subspace
10.3.1 Congruence of the convex components of CO
10.3.2 Decomposition of maximal simplices into right triangles
10.4 Consistent belief functions
10.4.1 Consistent knowledge bases in classical logic
10.4.2 Belief functions as uncertain knowledge bases
10.4.3 Consistency in belief logic
10.5 The geometry of consistent belief functions
10.5.1 The region of consistent belief functions
10.5.2 The consistent complex
10.5.3 Natural consistent components
10.6 Research questions
Appendix: Proofs
Part III Geometric interplays
11 Probability transforms: The affine
family
Chapter outline
11.1 Affine transforms in the binary case
11.2 Geometry of the dual line
11.2.1 Orthogonality of the dual line
11.2.2 Intersection with the region of Bayesian normalised sum functions
11.3 The intersection probability
11.3.1 Interpretations of the intersection probability
11.3.2 Intersection probability and affine combination
11.3.3 Intersection probability and convex closure
11.4 Orthogonal projection
11.4.1 Orthogonality condition
11.4.2 Orthogonality flag
11.4.3 Two mass redistribution processes
11.4.4 Orthogonal projection and affine combination
11.4.5 Orthogonal projection and pignistic function
11.5 The case of unnormalised belief functions
11.6 Comparisons within the affine family
Appendix: Proofs
12 Probability transforms: The epistemic family
Chapter outline
12.1 Rationale for epistemic transforms
12.1.1 Semantics within the probability-bound interpretation
12.1.2 Semantics within Shafer’s interpretation
12.2 Dual properties of epistemic transforms
12.2.1 Relative plausibility, Dempster’s rule and pseudo-belief functions
12.2.2 A (broken) symmetry
12.2.3 Dual properties of the relative belief operator
12.2.4 Representation theorem for relative beliefs
12.2.5 Two families of Bayesian approximations
12.3 Plausibility transform and convex closure
12.4 Generalisations of the relative belief transform
12.4.1 Zero mass assigned to singletons as a singular case
12.4.2 The family of relative mass probability transformations
12.4.3 Approximating the pignistic probability and relative plausibility
12.5 Geometry in the space of pseudo-belief functions
12.5.1 Plausibility of singletons and relative plausibility
12.5.2 Belief of singletons and relative belief
12.5.3 A three-plane geometry
12.5.4 A geometry of three angles
12.5.5 Singular case
12.6 Geometry in the probability simplex
12.7 Equality conditions for both families of approximations
12.7.1 Equal plausibility distribution in the affine family
12.7.2 Equal plausibility distribution as a general condition
Appendix: Proofs
13 Consonant approximation
The geometric approach to approximation
Chapter content
Summary of main results
Chapter outline
13.1 Geometry of outer consonant approximations in the consonant simplex
13.1.1 Geometry in the binary case
13.1.2 Convexity
13.1.3 Weak inclusion and mass reassignment
13.1.4 The polytopes of outer approximations
13.1.5 Maximal outer approximations
13.1.6 Maximal outer approximations as lower chain measures
13.2 Geometric consonant approximation
13.2.1 Mass space representation
13.2.2 Approximation in the consonant complex
13.2.3 Möbius inversion and preservation of norms, induced orderings
13.3 Consonant approximation in the mass space
13.3.1 Results of Minkowski consonant approximation in the mass space
13.3.2 Semantics of partial consonant approximations in the mass space
13.3.3 Computability and admissibility of global solutions
13.3.4 Relation to other approximations
13.4 Consonant approximation in the belief space
13.4.1 L1 approximation
13.4.2 (Partial) L2 approximation
13.4.3 L∞ approximation
13.4.4 Approximations in the belief space as generalised maximal outer approximations
13.5 Belief versus mass space approximations
13.5.1 On the links between approximations in M and B
13.5.2 Three families of consonant approximations
Appendix: Proofs
14 Consistent approximation
Chapter content
Chapter outline
14.1 The Minkowski consistent approximation problem
14.2 Consistent approximation in the mass space
14.2.1 L1 approximation
14.2.2 L∞ approximation
14.2.3 L2 approximation
14.3 Consistent approximation in the belief space
14.3.1 L1/L2 approximations
14.3.2 L∞ consistent approximation
14.4 Approximations in the belief versus the mass space
Appendix: Proofs
Part IV Geometric reasoning
15 Geometric conditioning
Chapter content
Chapter outline
15.1 Geometric conditional belief functions
15.2 Geometric conditional belief functions in M
15.2.1 Conditioning by L1 norm
15.2.2 Conditioning by L2 norm
15.2.3 Conditioning by L∞ norm
15.2.4 Features of geometric conditional belief functions in M
15.2.5 Interpretation as general imaging for belief functions
15.3 Geometric conditioning in the belief space
15.3.1 L2 conditioning in B
15.3.2 L1 conditioning in B
15.3.3 L∞ conditioning in B
15.4 Mass space versus belief space conditioning
15.4.1 Geometric conditioning: A summary
15.5 An outline of future research
Appendix: Proofs
16 Decision making with epistemic
transforms
Chapter content
Chapter outline
16.1 The credal set of probability intervals
16.1.1 Lower and upper simplices
16.1.2 Simplicial form
16.1.3 Lower and upper simplices and probability intervals
16.2 Intersection probability and probability intervals
16.3 Credal interpretation of Bayesian transforms: Ternary case
16.4 Credal geometry of probability transformations
16.4.1 Focus of a pair of simplices
16.4.2 Probability transformations as foci
16.4.3 Semantics of foci and a rationality principle
16.4.4 Mapping associated with a probability transformation
16.4.5 Upper and lower simplices as consistent probabilities
16.5 Decision making with epistemic transforms
16.5.1 Generalisations of the TBM
16.5.2 A game/utility theory interpretation
Appendix: Proofs
Part V
17 An agenda for the future
Open issues
A research programme
17.1 A statistical random set theory
17.1.1 Lower and upper likelihoods
17.1.2 Generalised logistic regression
17.1.3 The total probability theorem for random sets
17.1.5 Frequentist inference with random sets
17.1.6 Random-set random variables
17.2 Developing the geometric approach
17.2.1 Geometry of general combination
17.2.2 Geometry of general conditioning
17.2.3 A true geometry of uncertainty
17.2.4 Fancier geometries
17.2.5 Relation to integral and stochastic geometry
17.3 High-impact developments
17.3.1 Climate change
17.3.2 Machine learning
17.3.3 Generalising statistical learning theory
References