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 Content
Invited Papers.- Imprecise Probabilistic Prediction for Categorical Data: From Bayesian Inference to the Imprecise Dirichlet-Multinomial Model.- Fuzzy Bayesian Inference.- Soft Methods for Treating Uncertainties: Applications in the Field of Environmental Risks.- Robert Féron: A Pioneer in Soft Methods for Probability and Statistics.- Foundations.- Relating Prototype Theory and Label Semantics.- Fuzzy Probabilities Based on the Likelihood Function.- Possibility Measures in Probabilistic Inference.- Non-well-Founded Probabilities on Streams.- Relating Epistemic Irrelevance to Event Trees.- Probability Revision, the Uniformity Rule, and the Chan–Darwiche Metric.- Statistical Methods.- On Nonparametric Predictive Inference for Bernoulli Quantities with Set-Valued Data.- Inferring a Possibility Distribution from Very Few Measurements.- Statistics with Vague Data and the Robustness to Data Representation.- On a Linear Independence Test for Interval-Valued Random Sets.- A Small-Sample Nonparametric Independence Test for the Archimedean Family of Bivariate Copulas.- Defuzzification of Fuzzy p-Values.- Testing ‘Two-Sided’ Hypothesis about the Mean of an Interval-Valued Random Set.- Asymptotic Tests for the Variance of a Fuzzy Random Variable Using the D K -Metric.- Empirical Results Concerning a Fuzzy-Based Measure of Symmetry of Real Random Variables.- Fuzzy Kendall Statistic for Autocorrelated Data.- Mixture Model Estimation with Soft Labels.- Imprecise Functional Estimation: The Cumulative Distribution Case.- Non-parametric Density Estimation Based on Label Semantics.- Empirical Comparisons of Goodness-of-Fit Tests for Binomial Distributions Based on Fuzzy Representations.- Mathematical Aspects.- A Generalization of Hukuhara Difference.- A Note about Bobylev’s Differential.- On Boundary Value Problems for Fuzzy Differential Equations.- On Fuzzy Sets Convolution, Fuzzy Lipschitz Sets and Triangular Fuzzy Sets of Rank p.- Generalised p-Boxes on Totally Ordered Spaces.- The F. Riesz Representation Theorem and Finite Additivity.- Set-Valued Stochastic Integrals with Respect to a Real Valued Martingale.- On Stochastic Differential Equations with Fuzzy Set Coefficients.- Strong Solution of Set-Valued Stochastic Differential Equation.- Convergences of Random Variables with Respect to Coherent Upper Probabilities Defined by Hausdorff Outer Measures.- On Convergence in Necessity and Its Laws of Large Numbers.- The Omnipresence of Cycle-Transitivity in the Comparison of Random Variables.- Geometry of Cores of Submodular Coherent Upper Probabilities and Possibility Measures.- On Transformations between Belief Spaces.- Lower and Upper Covariance.- Some Properties of the d K -Variance for Interval-Valued Random Sets.- Triangular Conorms on the Space of Non-decreasing Lists of Non-negative Real Numbers.- On Patchwork Techniques for 2-Increasing Aggregation Functions and Copulas.- Engineering.- A Hierarchical Fusion of Expert Opinion in the TBM.- Application of Non-convex Fuzzy Variables to Fuzzy Structural Analysis.- Handling Uncertainty in Higher Dimensions with Potential Clouds towards Robust Design Optimization.- Reliability of Structures under Consideration of Uncertain Time-Dependent Material Behaviour.- A New Insight into the Linguistic Summarization of Time Series Via a Degree of Support: Elimination of Infrequent Patterns.- An Autoregressive Model with Fuzzy Random Variables.- Tackling Multiple-Instance Problems in Safety-Related Domains by Quasilinear SVM.- An Efficient Normal Form Solution to Decision Trees with Lower Previsions.- Group Decision Making with Soft AHP Based on the Random Set View of Fuzzy Sets.
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