Uncertainty quantification is a key aspect in many tasks such as model selection/regularization, or quantifying prediction uncertainties to perform active learning or OOD detection. Within credal approaches that consider modeling uncertainty as probability sets, upper entropy plays a central role as an uncertainty measure. This paper is devoted to the computational aspect of upper entropies, providing an exhaustive algorithmic and complexity analysis of the problem. In particular, we show that the problem has a strongly polynomial solution, and propose many significant improvements over past algorithms proposed for 2-monotone lower probabilities and their specific cases.

In this article, an extension of the Evidential Neural Network (ENN) introduced by Denœux is proposed by replacing the classical discounting operation with a refinement of the latter, called the contextual discounting (CD) operation, which allows for class-specific reliability degrees and yields a more expressive representation of uncertainty. This extension, named CD-ENN, is trained using the cross-entropy of the normalized contour function, making it efficient to learn its additional parameters. Experimental results on UCI datasets show that CD-ENN outperforms the original ENN in terms of predictive performance, as measured by the recently introduced generalized negative log-likelihood criterion.

Given past observations of an ordinal variable, we want to predict a future observation. This paper provides the solution, according to the likelihood-based evidential method for statistical inference and prediction, of this problem, in an algebraic form. This result is obtained after establishing that the prediction of an ordinal variable can be computed, under some conditions on the possibility distribution representing the estimation uncertainty in this method, by integrating the marginals of this distribution.

We study a general optimization problem in which coefficients in the objective are uncertain. We use capacities (lower probabilities) to model such uncertainty. Two popular criteria in imprecise probability, namely maximality and E-admissibility, are employed to compare solutions. We characterize non-dominated solutions with respect to these criteria in terms of well-known notions in multi-objective optimization. These characterizations are novel and make it possible to derive several interesting results. Specially, for convex problems, maximality and E-admissibility are equivalent for any capacities even though the set of associated acts is not convex, and in case of 2-monotone capacities, finding an arbitrary non-dominated solution and checking if a given solution is non-dominated are both tractable. For combinatorial problems, we show a general result: in case of 2-monotone capacities, if the deterministic version of the problem can be solved in polynomial time, checking E-admissibility can also be done in polynomial time. Lastly, for the matroid optimization problem, more refined results are also obtained thanks to these characterizations, namely the connectedness of E-admissible solutions and an outer approximation based on the greedy algorithm for non-dominated solutions with respect to maximality.

We study a general optimization problem with an uncertain linear objective. We address the uncertainty using two models: belief functions and, more generally, capacities. In the former model, we use the generalized minimax regret criterion introduced by Yager, while in the latter one, we extend this criterion, to find optimal solutions. This paper identifies some tractable cases for the resulting problem. Furthermore, when focal sets of the considered belief functions are Cartesian products of intervals, we develop a 2-approximation method that mirrors the well-known midpoint scenario method used for minimax regret optimization problems with interval data.

Given past observations of an ordinal variable, we want to predict a future observation. This paper provides the solution, according to the relative likelihood-based evidential method for statistical inference and prediction, of this problem, in an algebraic form. This result is obtained after establishing that the prediction of an ordinal variable can be computed, under some conditions, by integrating the marginals of the possibility distribution representing the estimation uncertainty in this method.

Recently, it was shown that a radial basis function network (RBFN) with a softmax output layer amounts to pooling by Dempster's rule positive and negative evidence for each class, and approximating the resulting belief function by a probability distribution using the plausibility transform. This so-called latent belief function offers a richer uncertainty quantification than the probabilistic output of the RBFN. In this paper, we show that there exists actually a set of latent belief functions for a RBFN. This set is obtained by considering all possible dependence structures, which are described by correlations, between the positive and negative evidence for each class. Furthermore, we show that performance can be enhanced by optimizing the correlations brought to light.

Recent works have studied 0-1 combinatorial optimization problems where profits of items are measured on a qualitative scale such as “low”, “medium” and “high”. In this study, we extend this body of work by allowing these profits to be both qualitative and uncertain. In the first step, we use probability theory to handle uncertainty. In the second step, we use evidence theory to handle uncertainty. We combine their approaches with approaches in decision making under uncertainty that utilize the Maximum Expected Utility principle and generalized Hurwicz criterion, to compare solutions. We show that under probabilistic uncertainty and a special case of evidential uncertainty where the focal sets are rectangles, the task of identifying the non-dominated solutions can be framed as solving a multi-objective version of the considered problem. This result mirrors that of the case of qualitative profits with no uncertainty.

We investigate a general optimization problem with a linear objective in which the coefficients are uncertain and the uncertainty is represented by a belief function. We consider five common criteria to compare solutions in this setting: generalized Hurwicz, strong dominance, weak dominance, maximality and E-admissibility. We provide characterizations for the non-dominated solutions with respect to these criteria when the focal sets of the belief function are Cartesian products of compact sets. These characterizations correspond to established concepts in optimization. They make it possible to find non-dominated solutions by solving known variants of the deterministic version of the optimization problem or even, in some cases, simply by solving the deterministic version.

In this paper, we investigate the interest of learning a group of evidential sources using contextual corrections, which is equivalent to directly learning an optimized conjunctive combination instead of optimizing each source individually. Several experiments on synthetic and real UCI data demonstrates the interest of the approach.

We study the single source single destination shortest path problem in a graph where information about arc weights is modelled by a belief function. We consider three common criteria to compare paths with respect to their weights in this setting: generalized Hurwicz, strong dominance and weak dominance. We show that in the particular case where the focal sets of the belief function are Cartesian products of intervals, finding best, i.e. , non-dominated, paths according to these criteria amounts to solving known variants of the deterministic shortest path problem, for which exact resolution algorithms exist.

Adopting a general framework to faithfully represent uncertainty, such as belief function theory, usually comes at a cost. In many real-life applications, we are constrained to handle mass functions that have too many focal elements. Fortunately, one can resort to approximation techniques to bypass this issue. In this paper, we extend the classical approximation techniques, which are mainly specificity-based, to other belief function relations such as lattice dominance. This allows to overcome the limits of classical techniques in some applications.

This paper addresses a variant of the vehicle routing problem with time windows where service and travel times are modeled within the framework of belief function theory. This theory is general as it offers to model several facets of information imperfection, including uncertainty and imprecision. An extension of stochastic programming with recourse is used to tackle the problem. This approach aims to regain the feasibility of the routes that missed one or more of the customer time windows due to the uncertain nature of the problem. A memetic algorithm is devised to solve the problem on an adaptation of literature instances.

Information content is classically measured by entropy measures in probability theory, that can be interpreted as a measure of internal inconsistency of a probability distribution. While extensions of Shannon entropy have been proposed for quantifying information content of a belief function, other trends have been followed which rather focus on the notion of consistency between sets. Relying on previous general entropy measures of probability, we propose in this paper to establish some links between the different measures of internal inconsistency of a belief functions. We propose a general formulation which encompasses inconsistency measures derived from Shannon entropy as well as those derived from the N-consistency family of measures.

In this paper, an improvement of the quality of an evidential source of information is proposed using contextual corrections depending on partial decisions obtained from an interval dominance relation on the source outputs. Numerical experiments with the EkNN classifier and synthetic and real data allows us to illustrate the performances and the interest of this method.

Measuring inconsistency has been and is still an active research topic in both logic and evidence theory. However, the two fields have developed distinct notions and measures of inconsistency, following different paths. In this paper, we attempt to build some first bridges between the two trends, suggesting some first means for one to enrich the other, and vice-versa.