Author | Kramosil, Ivan. author |
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Title | Probabilistic Analysis of Belief Functions [electronic resource] / by Ivan Kramosil |
Imprint | Boston, MA : Springer US : Imprint: Springer, 2001 |
Connect to | http://dx.doi.org/10.1007/978-1-4615-0587-7 |
Descript | XVII, 214 p. online resource |
1 Introduction -- 1.1 Uncertainty in the World Around -- 1.2 Classical Probability Theory - Uncertainty as Randomness -- 1.3 Dempster-Shafer Approach to Uncertainty Processing -- 1.4 Relations to the Theory of General Systems -- 2 Preliminaries on Axiomatic Probability Theory -- 2.1 Probability Spaces and Random Variables -- 2.2 Some Intuition and Motivation Behind -- 2.3 Conditional Probabilities and Stochastic Independence -- 2.4 The Most Simple Setting of the Strong Law of Large Numbers -- 3 Probabilistic Model of Decision Making under Uncertainty -- 3.1 An Intuitive Background to Decision Making Theory -- 3.2 Decision Making under Uncertainty -- 3.3 Statistical Decision Functions -- 3.4 The Bayesian and the Minimax Principles -- 3.5 An Example and Some Problems Involved -- 4 Basic Elements of Dempster-Shafer Theory -- 4.1 From Intuition to Compatibility Relations -- 4.2 From Compatibility Relations to Belief Functions -- 4.3 Some Remarks and Comments -- 4.4 Semantical Consistence and Correctness of Belief Functions -- 5 Elementary Properties of Belief Functions -- 5.1 Plausibility Functions -- 5.2 Basic Probability Assignments and Belief Functions -- 5.3 Super-Additivity of Belief Functions -- 5.4 Some Particular Cases of Belief Functions -- 5.5 Belief Functions and the Case of Total Ignorance -- 6 Probabilistic Analysis of Dempster Combination Rule -- 6.1 Knowledge Acquisition as Dynamical Process -- 6.2 Combining Compatibility Relations -- 6.3 Towards Dempster Combination Rule -- 6.4 Elementary Properties of Dempster Combination Rule -- 6.5 Dual Combination Rule -- 7 Nonspecificity Degrees of Basic Probability Assignments -- 7.1 The Most Simple Case of Nonspecificity Degrees -- 7.2 Nonspecificity Degrees of Dempster Products -- 7.3 Quasi-Deconditioning -- 7.4 The Case of Dual Combination Rule -- 8 Belief Functions Induced by Partial Compatibility Relations -- 8.1 Compatibility Relations over Sets of States and Sets of Empirical Values -- 8.2 Partial Generalized Compatibility Relations -- 8.3 Belief Functions Defined by Partial Generalized Compatibility Relations -- 8.4 Partial Generalized Compatibility Relations with the Same Compatibility Relation -- 8.5 Approximations of Belief Functions by the Partial Generalized Ones -- 9 Belief Functions over Infinite State Spaces -- 9.1 Towards Infinite Basic Spaces -- 9.2 Definability of Degrees of Belief for Subsets of Infinite Spaces -- 9.3 Extensions of Degrees of Belief to Non-Regular Subsets -- 9.4 Elementary Properties of Extended Belief Functions -- 9.5 Bounds of Application of Extended Belief Functions -- 9.6 Survey of Approximations of Degrees of Belief over Infinite Spaces -- 10 Boolean Combinations of Set-Valued Random Variables -- 10.1 Combining Set-Valued Random Variables -- 10.2 Belief Functions Defined by Unions of Set-Valued Random Variables -- 10.3 Belief Functions Defined by Intersections of Set-Valued Random Variables -- 11 Belief Functions with Signed and Nonstandard Values -- 11.1 The Inversion Problem for Degrees of Belief and Belief Functions -- 11.2 Signed Measures -- 11.3 Degrees of Belief Are Leaving the Unit Interval of Reals -- 11.4 Dempster Combination Rule for Basic Signed Measure Assignments -- 11.5 Inversion Rule for Basic Signed Measure Assignments -- 11.6 Almost Invertibility of Basic Signed Measure Assignments -- 11.7 Degrees of Belief with Nonstandard Values -- 11.8 An Abstract Algebraic Approach to the Inversion Problem -- 12 Jordan Decomposition of Signed Belief Functions -- 12.1 Hahn Decomposition Theorem for Signed Measures -- 12.2 Jordan Decomposition of Signed Belief Functions -- 12.3 Generalizing Conditioned Belief Functions -- 13 Monte-Carlo Estimations for Belief Functions -- 13.1 Strong Law of Large Numbers Applied to Belief Functions -- 13.2 Towards Monte-Carlo Algorithms for Belief Functions -- 13.3 Asymptotic Properties of Monte-Carlo Estimations of Belief Functions -- 13.4 Chebyshev Inequality for Monte-Carlo Estimations of Belief Functions -- 14 Boolean-Valued and Boolean-Like Processed Belief Functions -- 14.1 Intuition, Motivation and Preliminaries on Boolean Algebras -- 14.2 Boolean-Valued Probability Measures -- 14.3 Boolean-Valued Belief and Plausibility Functions -- 14.4 Boolean-Like Structure over The Unit Interval of Real Numbers -- 14.5 Probability Measures with Values in Boolean-Like Structured Unit Interval of Real Numbers -- 14.6 Basic Nonstandard Probability Assignments -- 15 References -- 16 Index