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