Other uncertainty theories based on capacities

This paper proposes a conceptually simple but expressive framework for handling propositional information stemming from several sources, namely a two-tiered propositional logic augmented with classical modal axioms (BC-logic), a fragment of the non-normal modal logic EMN, whose semantics is expressed in terms of two-valued monotonic set-functions called Boolean capacities. We present a theorem-preserving translation of Belnap logic in this setting. As special cases, we can recover previous translations of three-valued logics such as Kleene and Priest logics. Our translation bridges the gap between Belnap logic, epistemic logic, and theories of uncertainty like possibility theory or belief functions, and paves the way to a unified approach to various inconsistency handling methods.

This special issue contains a selection of articles from the Second International Symposium on Imprecise Probabilities and Their Applications (ISIPTA ’01) [4]. The symposium was held at Cornell University, USA, in June 2001, and was chaired by Gert de Cooman, Terrence L. Fine, and Teddy Seidenfeld. It was a successor of the first successful ISIPTA meeting [3]. This was held in 1999, and was intended to promote interaction and communication between researchers in a great diversity of fields but with common interests in imprecise probabilities. ISIPTA ’01 was no exception in this respect: the 47 papers presented there covered a wide and heterogeneous range of topics, such as algorithms, belief change, civil engineering, pattern recognition, coherence, combination of uncertainties, conditioning, credal nets, decision making, economics and finance, elicitation, foundations, independence, mathematical models for uncertainty, probabilistic logic, reliability, and statistical inference. This heterogeneity as well as lively discussions during the meeting, turned ISIPTA ’01 into successful attempt to bring together an interdisciplinary community of researchers actively involved in advancing and promoting a generalized view of probability theories. Shortly after ISIPTA ’01, in February 2002, a few active members in this community decided to found SIPTA, the Society for Imprecise Probability: Theories and Applications. This society manages the organization of the ISIPTA meetings, nowadays a well-established international forum for discussing imprecise probabilities. And, being aware that the field of imprecise probabilities is not always easily accessible, due to its heterogeneity and the related variety of different Flanguages_ used to talk about imprecise probabilities, the Society has also recently started organizing the SIPTA Schools in Imprecise Probabilities, the next of which is due to be held in July 2006 in Madrid, Spain. At this point it is useful to be more precise about the subject of the ISIPTA meetings, namely, imprecise probabilities. The name Fimprecise probability_ was intended from the beginnings in a very wide sense, as a generic term for the many mathematical or statistical models that measure chance or uncertainty without sharp numerical probabilities. These models can be qualitative, such as comparative probability models and partial preference orderings, or quantitative, such as interval probabilities, belief functions, upper and lower expectations or previsions, just to name a few. More important, though, is that imprecise probability models are needed in inference problems characterized by scarce, vague or conflicting information. These

With the rapid development of information technology, multi-sensor information fusion research has been made great progress, currently there are Bayesian test, the voting method, fuzzy sets, neural networks and other methods algorithm being used in multi-sensor information fusion commonly. However, there are many shortcomings which are difficult to overcome in the process of fusion by these methods, so now the international community has already proposed to apply the D-S evidence theory to information fusion. This method does not require the priori probability, and it's very useful when it is used in information fusion, but the information with the conflict can not be effectively fused by using D-S evidence theory. Larger conflict in the evidence, the fusion result is more easily prone to false consensus. Therefore, this paper presents an improved information fusion method based on original D-S evidence theory, and this method can deal with the evidences with conflict effectively. In this paper, the improvement is in the denominator of the D-S evidence theory. When the evidence is purely of conflict, the denominator is zero, the formula does not make sense. This improvement thinking is coupled with sum between the orthogonal sum of all of evidence and product, which is between the conflict factor and the inverse value of the standard deviation of the support degree, and then it subtract maximum difference of the basic probability of the evidence. If the conflict of the evidence is greater, this advantage of this method is more obvious. If there is no conflict in the evidence, the fusion result is consistent with item of original D-S evidence theory. Finally the simulation experiment gives a set of data with more conflict. The comparative data from the experiments show that the improved method of information fusion is essential to improved in resolving conflict issues, and it is effective.

The problem of soft information fusion is one of the key challenges in both fields of information fusion and situation awareness. However uncertainty modelling in the context of semantic information fusion is not well-researched.In this paper we present the current state-of-the-art regarding fusion of uncertain semantic information using simple conceptual graphs and propose methods to improve its handling of partial conflict, reliability and expressiveness regarding modal uncertainty. Particular attention is paid to the issues of scalability and tractability. Similarities to other uncertainty handling frameworks such as possibility theory and belief functions are discussed. Potential avenues are identified for further research regarding uncertainty representation in semantic information fusion.

Reasoning with fuzzy and uncertain evidence using epistemic random fuzzy sets: General framework and practical models

In a previous paper in this Journal, a “hybrid method” was proposed for the joint propagation of probability distributions (expressing variability) and possibility distributions (i.e., fuzzy numbers, expressing imprecision or partial ignorance) in the computation of risk. In order to compare the results of the hybrid computation (a random fuzzy set) to a tolerance threshold (a tolerable level of risk), a postprocessing method was proposed. Recent work has highlighted a shortcoming of this postprocessing step which yields overly conservative results. A postprocessing method based on Shafer’s theory of evidence provides a rigorous answer to the problem of comparing a random fuzzy set with a threshold. The principles behind the new postprocessing scheme are presented and illustrated with a synthetic example.

Previous work on logical systems and uncertainty modeling, especially using possibility distributions, is reviewed. The author emphasizes that there exists a natural relation between belief functions and random sets, and it is emphasized that Choquet's theorem can be used to obtain probabilistic interpretation for a large class of uncertainty measures, including possibility measures.

Joint exploration of regional importance of possibilistic and probabilistic uncertainty in stability analysis

The transformation between probability and possibility has been studied by many researchers. Most of those studies examined the principles that must be satisfied for the transformation, and devised an equation satisfying them in a heuristic way. They did not show that the proposed transformation was the only one satisfying the principles. The paper devises three new transformation methods based on evidence theory. Each starts from some principles in the same way as the previous research, but finds the only transformation satisfying the principles. One of them considers that possibility is an ordinal scale of uncertainty, and obtains the possibilistic order from a given probability distribution. The remaining two regard possibility as a ratio scale, and transforms a probability distribution into a possibility distribution. All of the three generate the same ordinal structure of possibility. However, two of them generate different possibility distributions. Thus, the paper examines which of the two is more valid from the point of view of the given principles.

Artificial Intelligence has emerged and been considerably developed in the last fifty years together with the advent of the computer age, as a new scientific area aiming at processing information in agreement with the way humans do. Inevitably, such a range of efforts, both theoretical and application-oriented, and motivated by new concerns, has led to modify and enlarge the way basic notions such as uncertainty, belief, knowledge, or evidence could be thought, represented and handled in practice. In particular, there has been a major trend in uncertainty (more specifically, partial belief) modelling, emphasizing the idea that the degree of confidence in an event is not totally determined by the confidence in the opposite event, as assumed in probability theory. Possibility theory (Dubois and Prade, 1988) belongs to this trend that describes partial belief in terms of certainty and plausibility, viewed as distinct concepts. Belief and plausibility functions (Shafer, 1976), or lower and upper probabilities (Walley, 1991) are other important representatives of this trend. The distinctive features of possibility theory are its computational simplicity, and its position as a bridge between numerical and symbolic theories of partial belief for practical reasoning. Possibility theory, based on a non-additive setting that contrasts with probability theory, provides a potentially more qualitative treatment of partial belief, since the operations “max” and “min” play a role somewhat analogous to the sum and the product in probability calculus. There are two different kinds of possibility theory: one is qualitative and the other is quantitative. They share the same kind of set-functions, but they differ when it comes to conditioning and combination

The increased availability of a wide range of sensing technologies over the last few decades has resulted in an equivalent increased need for reliable information fusion methods in machine learning applications. While existing theories such as the Dempster-Shafer theory and the possibility theory have been used for several years now, they do not provide guarantees of error calibration in information fusion settings. The Conformal Predictions (CP) framework is a new game-theoretic approach to reliable machine learning, which provides a methodology to obtain error calibration under classification and regression settings. In this work, we present a methodology to extend the Conformal Predictions framework to both classification and regression-based information fusion settings. This methodology is based on applying the CP framework to each data source as an independent hypothesis test, and subsequently using p-value combination methods as a test statistic for the combined hypothesis after fusion. The proposed methodology was studied in classification and regression settings within two real-world application contexts: person recognition using multiple modalities (classification), and head pose estimation using multiple image features (regression). Our experimental results showed that quantile methods of combining p-values (such as the Standard Normal Function and the Non-conformity Aggregation methods) provided the most statistically valid calibration results, and can be considered to extend the CP framework for information fusion settings.

The main goal of information fusion can be seen as improving human or automatic decision-making by exploiting diversities in information from multiple sources. High-level information fusion aims specifically at decision support regarding situations, often expressed as “achieving situation awareness”. A crucial issue for decision making based on such support is trust that can be defined as “accepted dependence”, where dependence or dependability is an overall term for many other concepts, e.g., reliability. This position paper reports on ongoing and planned research concerning imprecise probability as an approach to improved dependability in high-level information fusion. We elaborate on high-level information fusion from a generic perspective and a partial mapping from a taxonomy of dependability to high-level information fusion is presented. Three application domains: defense, manufacturing, and precision agriculture, where experiments are planned to be implemented are depicted. We conclude that high-level information fusion as an application-oriented research area, where precise probability (Bayesian theory) is commonly adopted, provides an excellent evaluation ground for imprecise probability.

The two known information theories, probability and possibility theory, are based on t-conorm decomposable fuzzy measures, so that bijective mappings exist between their set-valued measures and their point-valued distributions. Further, their random set (Dempster-Shafer evidence theoretical) interpretations have simple topological structures, with bijective mappings between the subset focal elements and the point singletons. We introduce the concepts of distributional and aggregable random sets and random set completion, and first use them as a model in which to cast probability and possibility measures and distributions. Then, towards the goal of deriving new forms of information theory, general Sugeno conorm decomposable fuzzy measures and ring-like aggregable random sets with set-intersection structural aggregation are examined, but it is shown that in these two cases no new information theories are forthcoming.

This article is devoted to applications of fuzzy set theory, possibility theory and evidence theory in civil engineering, presenting current work of a group or researchers at the University of Innsbruck. We argue that these methods are well suited for analyzing and processing the parameter uncertainties arising in soil mechanics and construction management. We address two specific applications here: finite element computations in foundation engineering and a queueing model in earth work.

Mathematical theories of uncertainty in which measures of uncertainty are now well established are fuzzy set theory, evidence theory, and possibility theory, in addition to classical set theory and probability theory. Some of these theories are connected. Classical set theory is subsumed under fuzzy set theory. Probability theory and possibility theory are branches of evidence theory, while evidence theory is, in turn, a branch of fuzzy measure theory. Although information aspects of fuzzy measure theory have not been investigated as yet, the theory is briefly introduced in this chapter because it represents a broad framework for future research.