Recently, the problem of measuring the conflict between two bodies of evidence represented by belief functions has known a regain of interest. In most works related to this issue, Dempster's rule plays a central role. In this paper, we propose to study the notion of conflict from a different perspective. We start by examining consistency and conflict on sets and extract from this settings basic properties that measures of consistency and conflict should have. We then extend this basic scheme to belief functions in different ways. In particular, we do not make any a priori assumption about sources (in)dependence and only consider such assumptions as possible additional information.

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A pair of lower and upper cumulative distribution functions, also called probability box or p-box, is among the most popular models used in imprecise probability theory. They arise naturally in expert elicitation, for instance in cases where bounds are specified on the quantiles of a random variable, or when quantiles are specified only at a finite number of points. Many practical and formal results concerning p-boxes already exist in the literature. In this paper, we provide new efficient tools to construct multivariate p-boxes and develop algorithms to draw inferences from them. For this purpose, we formalise and extend the theory of p-boxes using Walley's behavioural theory of imprecise probabilities, and heavily rely on its notion of natural extension and existing results about independence modeling. In particular, we allow p-boxes to be defined on arbitrary totally preordered spaces, hence thereby also admitting multivariate p-boxes via probability bounds over any collection of nested sets. We focus on the cases of independence (using the factorization property), and of unknown dependence (using the Fréchet bounds), and we show that our approach extends the probabilistic arithmetic of Williamson and Downs. Two design problems-a damped oscillator, and a river dikedemonstrate the practical feasibility of our results.

Abstract-There are many available methods to integrate information source reliability in an uncertainty representation, but there are only a few works focusing on the problem of evaluating this reliability. However, data reliability and confidence are essential components of a data warehousing system, as they influence subsequent retrieval and analysis. In this paper, we propose a generic method to assess data reliability from a set of criteria using the theory of belief functions. Customizable criteria and insightful decisions are provided. The chosen illustrative example comes from real-world data issued from the Sym'Previus predictive microbiology oriented data warehouse.

a b s t r a c tWhen conjunctively merging two belief functions concerning a single variable but coming from different sources, Dempster rule of combination is justified only when information sources can be considered as independent. When dependencies between sources are illknown, it is usual to require the property of idempotence for the merging of belief functions, as this property captures the possible redundancy of dependent sources. To study idempotent merging, different strategies can be followed. One strategy is to rely on idempotent rules used in either more general or more specific frameworks and to study, respectively, their particularization or extension to belief functions. In this paper, we study the feasibility of extending the idempotent fusion rule of possibility theory (the minimum) to belief functions. We first investigate how comparisons of information content, in the form of inclusion and least-commitment, can be exploited to relate idempotent merging in possibility theory to evidence theory. We reach the conclusion that unless we accept the idea that the result of the fusion process can be a family of belief functions, such an extension is not always possible. As handling such families seems impractical, we then turn our attention to a more quantitative criterion and consider those combinations that maximize the expected cardinality of the joint belief functions, among the least committed ones, taking advantage of the fact that the expected cardinality of a belief function only depends on its contour function.

Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. Many such models, sometimes referred to as distortion models, have been proposed in the literature, such as the pari mutuel model, the linear vacuous mixtures or the constant odds ratio model. This paper is the rst part of a two paper series where we study the sets of probabilities induced by such models, regarding them as neighbourhoods dened over specic metrics or premetrics. We also compare them in terms of a number of properties: precision, number of extreme points, n-monotonicity, behaviour under conditioning, etc. This rst part tackles this study on some of the most popular distortion models in the literature, while the second part studies less known neighbourhood models and summarises our ndings.

Learning from uncertain data has been drawing increasing attention in recent years. In this paper, we propose a tree induction approach which can not only handle uncertain data, but also furthermore reduce epistemic uncertainty by querying the most valuable uncertain instances within the learning procedure. We extend classical decision trees to the framework of belief functions to deal with a variety of uncertainties in the data. In particular, we use entropy intervals extracted from the evidential likelihood to query selected uncertain querying training instances when needed, in order to improve the selection of the splitting attribute. Our experiments show the good performances of proposed active belief decision trees under different conditions.

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