Probabilistic Logic in a Coherent Setting

by Coletti, Giulianella; Scozzafava, Romano

• ISBN13: 9781402009174
• ISBN10: 1402009178
• Format: Hardcover
• Copyright: 2002-12-01
• Publisher: Kluwer Academic Pub

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.

Table of Contents

Introduction	7	(10)
Aims and motivation	7	(5)
A brief historical perspective	12	(5)
Events as Propositions	17	(8)
Basic concepts	17	(1)
From ``belief'' to logic?	18	(2)
Operations	20	(1)
Atoms (or ``possible worlds'')	21	(3)
Toward probability	24	(1)
Finitely Additive Probability	25	(6)
Axioms	25	(1)
Sets (of events) without structure	26	(1)
Null probabilities	27	(4)
Coherent probability	31	(6)
Coherence	31	(3)
Null probabilities (again)	34	(3)
Betting Interpretation of Coherence	37	(6)
Coherent Extensions of Probability Assessments	43	(6)
de Finetti's fundamental theorem	43	(2)
Probabilistic logic and inference	45	(4)
Random Quantities	49	(4)
Probability Meaning and Assessment: a Reconciliation	53	(4)
The ``subjective'' view	53	(2)
Methods of evaluation	55	(2)
To Be or not To Be Compositional?	57	(4)
Conditional Events	61	(12)
Truth values	63	(2)
Operations	65	(5)
Toward conditional probability	70	(3)
Coherent Conditional Probability	73	(26)
Axioms	73	(1)
Assumed or acquired conditioning?	74	(2)
Coherence	76	(4)
Characterization of a coherent conditional probability	80	(10)
Related results	90	(4)
The role of probabilities 0 and 1	94	(5)
Zero-Layers	99	(10)
Zero-layers induced by a coherent conditional probability	99	(2)
Spohn's ranking function	101	(1)
Discussion	102	(7)
Coherent Extensions of Conditional Probability	109	(8)
Exploiting Zero Probabilities	117	(10)
The algorithm	117	(5)
Locally strong coherence	122	(5)
Lower and Upper Conditional Probabilities	127	(10)
Coherence intervals	127	(1)
Lower conditional probability	128	(6)
Dempster's theory	134	(3)
Inference	137	(26)
The general problem	137	(2)
The procedure at work	139	(12)
Discussion	151	(4)
Updating probabilities 0 and 1	155	(8)
Stochastic Independence in a Coherent Setting	163	(28)
``Precise'' probabilities	164	(15)
``Imprecise'' probabilities	179	(7)
Discussion	186	(4)
Concluding remarks	190	(1)
A Random Walk in the Midst of Paradigmatic Examples	191	(24)
Finite additivity	191	(2)
Stochastic independence	193	(1)
A not coherent ``Radon-Nikodym'' conditional probability	194	(3)
A changing ``world''	197	(1)
Frequency vs. probability	198	(4)
Acquired or assumed (again)	202	(1)
Choosing the conditioning event	202	(2)
Simpson's paradox	204	(2)
Belief functions	206	(9)
Fuzzy Sets and Possibility as Coherent Conditional Probabilities	215	(26)
Fuzzy sets: main definitions	216	(3)
Fuzziness and uncertainty	219	(6)
Fuzzy subsets and coherent conditional probability	225	(7)
Possibility functions and coherent conditional probability	232	(8)
Concluding remarks	240	(1)
Coherent Conditional Probability and Default Reasoning	241	(16)
Default logic through conditional probability equal to 1	243	(4)
Inferential rules	247	(4)
Discussion	251	(6)
A Short Account of Decomposable Measures of Uncertainty	257	(14)
Operations with conditional events	258	(4)
Decomposable measures	262	(4)
Weakly decomposable measures	266	(4)
Concluding remarks	270	(1)
Bibliography	271	(14)
Index	285