Soft Methods for Handling Variability and Imprecision

by Dubois, Didier; Lubiano, M. Asuncion; Prade, Henri; Gil, Maria Angeles; Grzegorzewski, Przemyslaw

• ISBN13: 9783540850267
• ISBN10: 3540850260
• Format: Paperback
• Copyright: 2008-09-01
• Publisher: Springer Nature

Summary

Probability theory has been the only well-founded theory of uncertainty for a long time. It was viewed either as a powerful tool for modelling random phenomena, or as a rational approach to the notion of degree of belief. During the last thirty years, in areas centered around decision theory, artificial intelligence and information processing, numerous approaches extending or orthogonal to the existing theory of probability and mathematical statistics have come to the front. The common feature of those attempts is to allow for softer or wider frameworks for taking into account the incompleteness or imprecision of information. Many of these approaches come down to blending interval or fuzzy interval analysis with probabilistic methods. This book gathers contributions to the 4th International Conference on Soft methods in Probability and Statistics. Its aim is to present recent results illustrating such new trends that enlarge the statistical and uncertainty modeling traditions, towards the handling of incomplete or subjective information. It covers a broad scope ranging from philosophical and mathematical underpinnings of new uncertainty theories, with a stress on their impact in the area of statistics and data analysis, to numerical methods and applications to environmental risk analysis and mechanical engineering. A unique feature of this collection is to establish a dialogue between fuzzy random variables and imprecise probability theories.

Table of Contents

Invited Papers
Imprecise Probabilistic Prediction for Categorical Data: From Bayesian Inference to the Imprecise Dirichlet-Multinomial Model	p. 3
Fuzzy Bayesian Inference	p. 10
Soft Methods for Treating Uncertainties: Applications in the Field of Environmental Risks	p. 16
Robert Feron: A Pioneer in Soft Methods for Probability and Statistics	p. 27
Foundations
Relating Prototype Theory and Label Semantics	p. 35
Fuzzy Probabilities Based on the Likelihood Function	p. 43
Possibility Measures in Probabilistic Inference	p. 51
Non-well-Founded Probabilities on Streams	p. 59
Relating Epistemic Irrelevance to Event Trees	p. 66
Probability Revision, the Uniformity Rule, and the Chan-Darwiche Metric	p. 74
Statistical Methods
Statistical Inference
On Nonparametric Predictive Inference for Bernoulli Quantities with Set-Valued Data	p. 85
Inferring a Possibility Distribution from Very Few Measurements	p. 92
Statistics with Vague Data and the Robustness to Data Representation	p. 100
Statistical Tests
On a Linear Independence Test for Interval-Valued Random Sets	p. 111
A Small-Sample Nonparametric Independence Test for the Archimedean Family of Bivariate Copulas	p. 118
Defuzzification of Fuzzy p-Values	p. 126
Testing 'Two-Sided' Hypothesis about the Mean of an Interval-Valued Random Set	p. 133
Asymptotic Tests for the Variance of a Fuzzy Random Variable Using the D[subscript K]-Metric	p. 140
Empirical Results Concerning a Fuzzy-Based Measure of Symmetry of Real Random Variables	p. 147
Fuzzy Kendall [tau] Statistic for Autocorrelated Data	p. 155
Estimation
Mixture Model Estimation with Soft Labels	p. 165
Imprecise Functional Estimation: The Cumulative Distribution Case	p. 175
Non-parametric Density Estimation Based on Label Semantics	p. 183
Empirical Comparisons of Goodness-of-Fit Tests for Binomial Distributions Based on Fuzzy Representations	p. 190
Mathematical Aspects
Invited Session: Fuzzy Set-Valued Analysis
A Generalization of Hukuhara Difference	p. 203
A Note about Bobylev's Differential	p. 211
On Boundary Value Problems for Fuzzy Differential Equations	p. 218
On Fuzzy Sets Convolution, Fuzzy Lipschitz Sets and Triangular Fuzzy Sets of Rank p	p. 226
Measure Theory
Generalised p-Boxes on Totally Ordered Spaces	p. 235
The F. Riesz Representation Theorem and Finite Additivity	p. 243
Set-Valued Stochastic Integrals with Respect to a Real Valued Martingale	p. 253
Differential Equations
On Stochastic Differential Equations with Fuzzy Set Coefficients	p. 263
Strong Solution of Set-Valued Stochastic Differential Equation	p. 271
Convergence Results
Convergences of Random Variables with Respect to Coherent Upper Probabilities Defined by Hausdorff Outer Measures	p. 281
On Convergence in Necessity and Its Laws of Large Numbers	p. 289
Combinatorics of Imprecise Probabilities
The Omnipresence of Cycle-Transitivity in the Comparison of Random Variables	p. 299
Geometry of Cores of Submodular Coherent Upper Probabilities and Possibility Measures	p. 306
On Transformations between Belief Spaces	p. 313
Variance in Generalized Probability Representations
Lower and Upper Covariance	p. 323
Some Properties of the d[subscript K]-Variance for Interval-Valued Random Sets	p. 331
Aggregation Functions
Triangular Conorms on the Space of Non-decreasing Lists of Non-negative Real Numbers	p. 341
On Patchwork Techniques for 2-Increasing Aggregation Functions and Copulas	p. 349
Engineering
Computational Methods
A Hierarchical Fusion of Expert Opinion in the TBM	p. 361
Application of Non-convex Fuzzy Variables to Fuzzy Structural Analysis	p. 369
Handling Uncertainty in Higher Dimensions with Potential Clouds towards Robust Design Optimization	p. 376
Reliability of Structures under Consideration of Uncertain Time-Dependent Material Behaviour	p. 383
Data Analysis
A New Insight into the Linguistic Summarization of Time Series Via a Degree of Support: Elimination of Infrequent Patterns	p. 393
An Autoregressive Model with Fuzzy Random Variables	p. 401
Tackling Multiple-Instance Problems in Safety-Related Domains by Quasilinear SVM	p. 409
Decision-Making
An Efficient Normal Form Solution to Decision Trees with Lower Previsions	p. 419
Group Decision Making with Soft AHP Based on the Random Set View of Fuzzy Sets	p. 427
Author Index	p. 435
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