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.
Author(s): Giulianella Coletti, Romano Scozzafava
Series: Trends in Logic 15
Publisher: Kluwer Academic Publishers
Year: 2002
Language: English
Pages: 296
Cover......Page 1
Series......Page 3
Volumes of this Series......Page 296
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 8
1.1 Aims and motivation......Page 12
1.2 A brief historical perspective......Page 17
2.1 Basic concepts......Page 22
2.2 From “belief” to logic?......Page 23
2.3 Operations......Page 25
2.4 Atoms (or “possible worlds”)......Page 26
2.5 Toward probability......Page 29
3.1 Axioms......Page 30
3.2 Sets {of events) without structure......Page 31
3.3 Null probabilities......Page 32
4.1 Coherence......Page 36
4.2 Null probabilities (again)......Page 39
5 – Betting Interpretation of Coherence......Page 42
6.1 de Finetti’s fundamental theorem......Page 48
6.2 Probabilistic logic and inference......Page 50
7 – Random Quantities......Page 54
8.1 The “subjective” view......Page 58
8.2 Methods of evaluation......Page 60
9 – To Be or not To Be Compositional?......Page 62
10 – Conditional Events......Page 66
10.1 Truth values......Page 68
10.2 Operations......Page 70
10.3 Toward conditional probability......Page 75
11.1 Axioms......Page 78
11.2 Assumed or acquired conditioning?......Page 79
11.3 Coherence......Page 81
11.4 Characterization of a coherent conditional probability......Page 85
11.5 Related results......Page 95
11.6 The role of probabilities 0 and 1......Page 99
12.1 Zero-layers induced by a coherent conditional probability......Page 104
12.2 Spohn’s ranking function......Page 106
12.3 Discussion......Page 107
13 – Coherent Extensions of Conditional Probability......Page 114
14.1 The algorithm......Page 122
14.2 Locally strong coherence......Page 127
15.1 Coherence intervals......Page 132
15.2 Lower conditional probability......Page 133
15.3 Dempster’s theory......Page 139
16.1 The general problem......Page 142
16.2 The procedure at work......Page 144
16.3 Discussion......Page 156
16.4 Updating probabilities 0 and 1......Page 160
17 – Stochastic Independence in a Coherent Setting......Page 168
17.1 “Precise” probabilities......Page 169
17.2 ”Imprecise” probabilities......Page 184
17.3 Discussion......Page 191
17.4 Concluding remarks......Page 195
18.1 Finite additivity......Page 196
18.2 Stochastic independence......Page 198
18.3 A not coherent “Radon-Nikodym” conditional probability......Page 199
18.4 A changing “world”......Page 202
18.5 Frequency vs. probability......Page 203
18.7 Choosing the conditioning event......Page 207
18.8 Simpson’s paradox......Page 209
18.9 Belief functions......Page 211
19 – Fuzzy Sets and Possibility as Coherent Conditional Probabilities......Page 220
19.1 Fuzzy sets: main definitions......Page 221
19.2 Fuzziness and uncertainty......Page 224
19.3 Fuzzy subsets and coherent conditional probability......Page 230
19.4 Possibility functions and coherent conditional probability......Page 237
19.5 Concluding remarks......Page 245
20 – Coherent Conditional Probability and Default Reasoning......Page 246
20.1 Default logic through conditional probability equal to 1......Page 248
20.2 Inferential rules......Page 252
20.3 Discussion......Page 256
21 – A Short Account of Decomposable Measures of Uncertainty......Page 262
21.1 Operations with conditional events......Page 263
21.2 Decomposable measures......Page 267
21.3 Weakly decomposable measures......Page 271
21.4 Concluding remarks......Page 275
A-B......Page 276
C......Page 278
D......Page 281
F-G......Page 283
H-J-K......Page 285
L-N......Page 286
P-R-S......Page 287
V-W-Z......Page 289
A-B-C......Page 290
D......Page 291
E-F-G-H-I-J-K-L......Page 292
M-N-O-P-Q-R......Page 293
S-T-U-W-Y-Z......Page 294
