Probabilistic Analysis of Belief Functions

Inspired by the eternal beauty and truth of the laws governing the run of stars on heavens over his head, and spurred by the idea to catch, perhaps for the smallest fraction of the shortest instant, the Eternity itself, man created such masterpieces of human intellect like the Platon's world of ideas manifesting eternal truths, like the Euclidean geometry, or like the Newtonian celestial me chanics. However, turning his look to the sub-lunar world of our everyday efforts, troubles, sorrows and, from time to time but very, very seldom, also our successes, he saw nothing else than a world full of uncertainty and tem porariness. One remedy or rather consolation was that of the deep and sage resignation offered by Socrates: I know, that I know nothing. But, happy or unhappy enough, the temptation to see and to touch at least a very small por tion of eternal truth also under these circumstances and behind phenomena charged by uncertainty was too strong. Probability theory in its most sim ple elementary setting entered the scene. It happened in the same, 17th and 18th centuries, when celestial mechanics with its classical Platonist paradigma achieved its greatest triumphs. The origins of probability theory were inspired by games of chance like roulettes, lotteries, dices, urn schemata, etc. and probability values were simply defined by the ratio of successful or winning results relative to the total number of possible outcomes.

Contenu
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.