Ebook: Probabilistic Logic in a Coherent Setting

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.

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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.

---

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.
Content:
Front Matter....Pages i-6
Introduction....Pages 7-15
Events as Propositions....Pages 17-24
Finitely Additive Probability....Pages 25-29
Coherent probability....Pages 31-35
Betting Interpretation of Coherence....Pages 37-42
Coherent Extensions of Probability Assessments....Pages 43-48
Random Quantities....Pages 49-51
Probability Meaning and Assessment: a Reconciliation....Pages 53-56
To Be or not To Be Compositional?....Pages 57-59
Conditional Events....Pages 61-72
Coherent Conditional Probability....Pages 73-97
Zero-Layers....Pages 99-108
Coherent Extensions of Conditional Probability....Pages 109-115
Exploiting Zero Probabilities....Pages 117-126
Lower and Upper Conditional Probabilities....Pages 127-136
Inference....Pages 137-161
Stochastic Independence in a Coherent Setting....Pages 163-190
A Random Walk in the Midst of Paradigmatic Examples....Pages 191-213
Fuzzy Sets and Possibility as Coherent Conditional Probabilities....Pages 215-240
Coherent Conditional Probability and Default Reasoning....Pages 241-255
Back Matter....Pages 271-291
A Short Account of Decomposable Measures of Uncertainty....Pages 257-270

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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.
Content:
Front Matter....Pages i-6
Introduction....Pages 7-15
Events as Propositions....Pages 17-24
Finitely Additive Probability....Pages 25-29
Coherent probability....Pages 31-35
Betting Interpretation of Coherence....Pages 37-42
Coherent Extensions of Probability Assessments....Pages 43-48
Random Quantities....Pages 49-51
Probability Meaning and Assessment: a Reconciliation....Pages 53-56
To Be or not To Be Compositional?....Pages 57-59
Conditional Events....Pages 61-72
Coherent Conditional Probability....Pages 73-97
Zero-Layers....Pages 99-108
Coherent Extensions of Conditional Probability....Pages 109-115
Exploiting Zero Probabilities....Pages 117-126
Lower and Upper Conditional Probabilities....Pages 127-136
Inference....Pages 137-161
Stochastic Independence in a Coherent Setting....Pages 163-190
A Random Walk in the Midst of Paradigmatic Examples....Pages 191-213
Fuzzy Sets and Possibility as Coherent Conditional Probabilities....Pages 215-240
Coherent Conditional Probability and Default Reasoning....Pages 241-255
Back Matter....Pages 271-291
A Short Account of Decomposable Measures of Uncertainty....Pages 257-270
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