Intro
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