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The approach to probability theory followed in this book (which differs radically from the usual one, based on a measure-theoretic framework) characterizes probability as a linear operator rather than as a measure, and is based on the concept of coherence, which can be framed in the most general view of conditional probability. It is a `flexible' and unifying tool suited for handling, e.g., partial probability assessments (not requiring that the set of all possible `outcomes' be endowed with a previously given algebraic structure, such as a Boolean algebra), and conditional independence, in a way that avoids all the inconsistencies related to logical dependence (so that a theory referring to graphical models more general than those usually considered in bayesian networks can be derived). Moreover, it is possible to encompass other approaches to uncertain reasoning, such as fuzziness, possibility functions, and default reasoning.
The book is kept self-contained, provided the reader is familiar with the elementary aspects of propositional calculus, linear algebra, and analysis.
List of contents
1 Introduction.- 1.1 Aims and motivation.- 1.2 A brief historical perspective.- 2 Events as Propositions.- 2.1 Basic concepts.- 2.2 From "belief" to logic?.- 2.3 Operations.- 2.4 Atoms (or "possible worlds").- 2.5 Toward probability.- 3 Finitely Additive Probability.- 3.1 Axioms.- 3.2 Sets (of events) without structure.- 3.3 Null probabilities.- 4 Coherent probability.- 4.1 Coherence.- 4.2 Null probabilities (again).- 5 Betting Interpretation of Coherence.- 6 Coherent Extensions of Probability Assessments.- 6.1 de Finetti's fundamental theorem.- 6.2 Probabilistic logic and inference.- 7 Random Quantities.- 8 Probability Meaning and Assessment: a Reconciliation.- 8.1 The "subjective" view.- 8.2 Methods of evaluation.- 9 To Be or not To Be Compositional?.- 10 Conditional Events.- 10.1 Truth values.- 10.2 Operations.- 10.3 Toward conditional probability.- 11 Coherent Conditional Probability.- 11.1 Axioms.- 11.2 Assumed or acquired conditioning?.- 11.3 Coherence.- 11.4 Characterization of a coherent conditional probability.- 11.5 Related results.- 11.6 The role of probabilities 0 and 1.- 12 Zero-Layers.- 12.1 Zero-layers induced by a coherent conditional probability.- 12.2 Spohn's ranking function.- 12.3 Discussion.- 13 Coherent Extensions of Conditional Probability.- 14 Exploiting Zero Probabilities.- 14.1 The algorithm.- 14.2 Locally strong coherence.- 15 Lower and Upper Conditional Probabilities.- 15.1 Coherence intervals.- 15.2 Lower conditional probability.- 15.3 Dempster's theory.- 16 Inference.- 16.1 The general problem.- 16.2 The procedure at work.- 16.3 Discussion.- 16.4 Updating probabilities 0 and 1.- 17 Stochastic Independence in a Coherent Setting.- 17.1 "Precise" probabilities.- 17.2 "Imprecise" probabilities.- 17.3 Discussion.- 17.4Concluding remarks.- 18 A Random Walk in the Midst of Paradigmatic Examples.- 18.1 Finite additivity.- 18.2 Stochastic independence.- 18.3 A not coherent "Radon-Nikodym" conditional probability.- 18.4 A changing "world".- 18.5 Frequency vs. probability.- 18.6 Acquired or assumed (again).- 18.7 Choosing the conditioning event.- 18.8 Simpson's paradox.- 18.9 Belief functions.- 19 Fuzzy Sets and Possibility as Coherent Conditional Probabilities.- 19.1 Fuzzy sets: main definitions.- 19.2 Fuzziness and uncertainty.- 19.3 Fuzzy subsets and coherent conditional probability.- 19.4 Possibility functions and coherent conditional probability.- 19.5 Concluding remarks.- 20 Coherent Conditional Probability and Default Reasoning.- 20.1 Default logic through conditional probability equal to 1.- 20.2 Inferential rules.- 20.3 Discussion.- 21 A Short Account of Decomposable Measures of Uncertainty.- 21.1 Operations with conditional events.- 21.2 Decomposable measures.- 21.3 Weakly decomposable measures.- 21.4 Concluding remarks.
Summary
The approach to probability theory followed in this book (which differs radically from the usual one, based on a measure-theoretic framework) characterizes probability as a linear operator rather than as a measure, and is based on the concept of coherence, which can be framed in the most general view of conditional probability. It is a `flexible' and unifying tool suited for handling, e.g., partial probability assessments (not requiring that the set of all possible `outcomes' be endowed with a previously given algebraic structure, such as a Boolean algebra), and conditional independence, in a way that avoids all the inconsistencies related to logical dependence (so that a theory referring to graphical models more general than those usually considered in bayesian networks can be derived). Moreover, it is possible to encompass other approaches to uncertain reasoning, such as fuzziness, possibility functions, and default reasoning.
The book is kept self-contained, provided the reader is familiar with the elementary aspects of propositional calculus, linear algebra, and analysis.
