This edited volume is the proceedings of the 2006 International Conference on Soft Methods in Probability and Statistics (SMPS 2006) hosted by the Artificial Intelligence Group at the University of Bristol, between 5-7 September 2006. This is the third of a series of biennial conferences organized in 2002 by the Systems Research Institute from the Polish Academy of Sciences in Warsaw, and in 2004 by the Department of Statistics and Operational Research at the University of Oviedo in Spain. These conferences provide a forum for discussion and research into the fusion of soft methods with probability and statistics, with the ultimate goal of integrated uncertainty modelling in complex systems involving human factors. In addition to probabilistic factors such as measurement error and other random effects, the modelling process often requires us to make qualitative and subject judgments that cannot easily be translated into precise probability values. Such judgments give rise to a number of different types of uncertainty including; fuzziness if they are based on linguistic information; epistemic uncertainty when their reliability is in question; ignorance when they are insufficient to identify or restrict key modelling parameters; imprecision when parameters and probability distributions can only be estimated within certain bounds. Statistical theory has not traditionally been concerned with modelling uncertainty arising in this manner but soft methods, a range of powerful techniques developed within AI, attempt to address those problems where the encoding of subjective information is unavoidable. These are mathematically sound uncertainty modelling methodologies which are complementary to conventional statistics and probability theory. Therefore, a more realistic modelling process providing decision makers with an accurate reflection of the true current state of our knowledge (and ignorance) requires an integrated framework incorporating both probability theory, statistics and soft methods. This fusion motivates innovative research at the interface between computer science (AI), mathematics and systems engineering.

Zusammenfassung
The idea of soft computing emerged in the early 1990s from the fuzzy systems c- munity, and refers to an understanding that the uncertainty, imprecision and ig- rance present in a problem should be explicitly represented and possibly even - ploited rather than either eliminated or ignored in computations. For instance, Zadeh de?ned 'Soft Computing' as follows: Soft computing differs from conventional (hard) computing in that, unlike hard computing, it is tolerant of imprecision, uncertainty and partial truth. In effect, the role model for soft computing is the human mind. Recently soft computing has, to some extent, become synonymous with a hybrid approach combining AI techniques including fuzzy systems, neural networks, and biologically inspired methods such as genetic algorithms. Here, however, we adopt a more straightforward de?nition consistent with the original concept. Hence, soft methods are understood as those uncertainty formalisms not part of mainstream s- tistics and probability theory which have typically been developed within the AI and decisionanalysiscommunity.Thesearemathematicallysounduncertaintymodelling methodologies which are complementary to conventional statistics and probability theory.

Inhalt
Keynote Papers.- Generalized Theory of Uncertainty (GTU) Principal Concepts and Ideas.- Reasoning with Vague Probability Assessments.- Soft Methods in Earth Systems Engineering.- Statistical Data Processing under Interval Uncertainty: Algorithms and Computational Complexity.- Soft Methods in Statistics and Random Information Systems.- On Testing Fuzzy Independence.- Variance Decomposition of Fuzzy Random Variables.- Fuzzy Histograms and Density Estimation.- Graded Stochastic Dominance as a Tool for Ranking the Elements of a Poset.- On Neyman-Pearson Lemma for Crisp, Random and Fuzzy Hypotheses.- Fuzzy Probability Distributions Induced by Fuzzy Random Vectors.- On the Identifiability of TSK Additive Fuzzy Rule-Based Models.- An Asymptotic Test for Symmetry of Random Variables Based on Fuzzy Tools.- Exploratory Analysis of Random Variables Based on Fuzzifications.- A Method to Simulate Fuzzy Random Variables.- Friedman's Test for Ambiguous and Missing Data.- Probability of Imprecisely-Valued Random Elements with Applications.- Measure-Free Martingales with Application to Classical Martingales.- A Note on Random Upper Semicontinuous Functions.- Optional Sampling Theorem and Representation of Set-Valued Amart.- On a Choquet Theorem for Random Upper Semicontinuous Functions.- A General Law of Large Numbers, with Applications.- Applications and Modelling of Imprecise Operators.- Fuzzy Production Planning Model for Automobile Seat Assembling.- Optimal Selection of Proportional Bounding Quantifiers in Linguistic Data Summarization.- A Linguistic Quantifier Based Aggregation for a Human Consistent Summarization of Time Series.- Efficient Evaluation of Similarity Quantified Expressions in the Temporal Domain.- Imprecise Probability Theory.- Conditional Lower Previsions forUnbounded Random Quantities.- Extreme Lower Probabilities.- Equivalence Between Bayesian and Credal Nets on an Updating Problem.- Varying Parameter in Classification Based on Imprecise Probabilities.- Comparing Proportions Data with Few Successes.- A Unified View of Some Representations of Imprecise Probabilities.- Possibility, Evidence and Interval Methods.- Estimating an Uncertain Probability Density.- Theory of Evidence with Imperfect Information.- Conditional IF-probability.- On Two Ways for the Probability Theory on IF-sets.- A Stratification of Possibilistic Partial Explanations.- Finite Discrete Time Markov Chains with Interval Probabilities.- Evidence and Compositionality.- High Level Fuzzy Labels for Vague Concepts.- Integrated Uncertainty Modelling in Applications.- Possibilistic Channels for DNA Word Design.- Transformation of Possibility Functions in a Climate Model of Intermediate Complexity.- Fuzzy Logic for Stochastic Modeling.- A CUSUM Control Chart for Fuzzy Quality Data.- A Fuzzy Synset-Based Hidden Markov Model for Automatic Text Segmentation.- Applying Fuzzy Measures for Considering Interaction Effects in Fine Root Dispersal Models.- Scoring Feature Subsets for Separation Power in Supervised Bayes Classification.- Interval Random Variables and Their Application in Queueing Systems with LongTailed Service Times.- Online Learning for Fuzzy Bayesian Prediction.