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International audience; Dealing with partially known or missing data is a common problem in machine learning. This work is interested in the problem of querying the true value of data to improve the quality of the learned model, when those data are only partially known. This study is in the line of active learning, since we consider that the precise value of some partial data can be queried to reduce the uncertainty in the learning process, yet can consider any kind of partial data (not only entirely missing one). We propose a querying strategy based on the concept of racing algorithms in which several models are competing. The idea is to identify the query that will help the most to quickly decide the winning model in the competition. After discussing and formalizing the general ideas of our approach, we study the particular case of decision trees in case of interval-valued features and setvalued labels. The experimental results indicate that, in comparison to other baselines, the proposed approach significantly outperforms simpler strategies in the case

International audience; The theory of belief functions (TBF) is now a widespread framework to deal and reason with uncertain and imprecise information, in particular to solve information fusion and clustering problems. Combination functions (rules) and distances are essential tools common to both the clustering and information fusion problems in the context of TBF, which have generated considerable literature. Distances and combination between evidence corpus of TBF are indeed often used within various clustering and classification algorithms, however their interplay and connections have seldom been investigated, which is the topic of this paper. More precisely, we focus on the problem of aggregating evidence corpus to obtain a representative one, and we show through an impossibility theorem that in this case, there is a fundamental contradiction between the use of conjunctive combination rules on the one hand, and the use of distances on the other hand. Rather than adding new methodologies, such results are instrumental in guiding the user among the many methodologies that already exist. To illustrate the interest of our results, we discuss different cases where they are at play. Impact Statement-Within the theory of belief functions, both distances and conjunctive combination rules can be used to achieve very similar purposes: evaluating the conflict between sources, performing supervised or unsupervised learning in presence of evidential information, or more simply obtaining a synthetic representation of multiple items of information. However, the results obtained by both approaches may show some inconsistency between them. This paper provides some insight as to why this may happen, showing that the two approaches are definitely at odds, and that using distances is, for instance, incompatible with some fundamental notions of the theory of belief functions, such as the least commitment principle. We illustrate the importance of the studied differences on problems such as k-centroid clustering, and discuss the importance of interpretations in such problems, which is rarely done in the literature.

International audience; Deep networks, like some other learning models, can associate high trust to unreliable predictions. Making these models robust and reliable is therefore essential, especially for critical decisions. This experimental paper shows that the conformal prediction approach brings a convincing solution to this challenge. Conformal prediction consists in predicting a set of classes covering the real class with a user-defined frequency. In the case of atypical examples, the conformal prediction will predict the empty set. Experiments show the good behavior of the conformal approach, especially when the data is noisy.

There are relatively few works dealing with conformal prediction for multi-task learning issues, and this is particularly true for multi-target regression. This paper focuses on the problem of providing valid (i.e., frequency calibrated) multi-variate predictions. To do so, we propose to use copula functions applied to deep neural networks for inductive conformal prediction. We show that the proposed method ensures efficiency and validity for multi-target regression problems on various data sets., Comment: 17 pages, 8 figures, under review

International audience; We present two different strategies to extend the classical multi-label chaining approach to handle imprecise probability estimates. These estimates use convex sets of distributions (or credal sets) in order to describe our uncertainty rather than a precise one. The main reasons one could have for using such estimations are (1) to make cautious predictions (or no decision at all) when a high uncertainty is detected in the chaining and (2) to make better precise predictions by avoiding biases caused in early decisions in the chaining. We adapt both strategies to the case of the naive credal classifier, showing that this adaptations are computationally efficient. Our experimental results on missing labels, which investigate how reliable these predictions are in both approaches, indicate that our approaches produce relevant cautiousness on those hard-to-predict instances where the precise models fail.

International audience; Occupancy grids are common tools used in robotics to represent the robot environment, and that may be used to plan trajectories, select additional measurements to acquire, etc. However, deriving information about those occupancy grids from sensor measurements often induce a lot of uncertainty, especially for grid elements that correspond to occluded or far away area from the robot. This means that occupancy information may be quite uncertain and imprecise at some places, while being very accurate at others. Modelling finely this occupancy information is essential to decide the optimal action the robot should take, but a refined modelling of uncertainty often implies a higher computational cost, a prohibitive feature for real-time applications. In this paper, we introduce the notion of credal occupancy grids, using the very general theory of imprecise probabilities to model occupancy uncertainty. We also show how one can perform efficient, real-time inferences with such a model, and show a use-case applying the model to an autonomous vehicle trajectory planning problem.

Learning ranking models is a difficult task, in which data may be scarce and cautious predictions desirable. To address such issues, we explore the extension of the popular parametric probabilistic Plackett–Luce model, often used to model rankings, to the imprecise setting where estimated parameters are set-valued. In particular, we study how to achieve cautious or conservative inference with it, and illustrate their application on label ranking problems, a specific supervised learning task.

International audience; It is well-known in computational social choice that the weighted average does not guarantee any equity or fairness in the share of goods. In a supervised learning problem, this translates into the fact that the empirical risk will lead to models that are good in average, but may have terrible performances for under-represented populations. Such a behaviour is quite damaging in some problems, such as the ones involving imbalanced data sets, in the inputs or the outputs (default prediction, ethical issues,. . .). On the other hand, the OWA operator is known in computational social choice to be able to correct this unfairness. This paper proposes a means to transpose this feature to the supervised learning setting.

International audience; Predictions from classification models are most often used as final decisions. Yet, there are situations where the prediction serves as an input for another constrained decision problem. In this paper, we consider such an issue where the classifier provides imprecise and/or uncertain predictions that need to be managed within the decision problem. More precisely, we consider the optimisation of a mix of material pieces of different types in different containers. Information about those pieces is modelled by a mass function provided by a cautious classifier. Our proposal concerns the statement of the optimisation problem within the framework of belief function. Finally, we give an illustration of this problem in the case of plastic sorting for recycling purposes.

Modelling the preferences of a decision maker about alternatives having multiple criteria usually starts by collecting preference information (comparisons of alternatives, importance of criteria, …), which are then used to fit a preference model issued from some set of hypothesis (weighted average, CP-net, lexicographic orderings, AHP, …). In practice, this process often leads to inconsistencies that may be due to inaccurate information provided by the decision maker, who can be unsure about the provided information, or to a poor choice of hypothesis set, which can be too restrictive or not well adapted to the decision process. In this paper, we propose to use belief functions as a way to quantify and resolve such inconsistencies, notably by allowing the decision maker to express her/his certainty about the provided preferential information. Our framework is generic, in the sense that it does not assume a given set of hypothesis a priori, and is consistent with precise methods, in the sense that in the absence of uncertainty and inconsistencies in the information, precise models are ultimately retrieved.

International audience; Idempotence is a desirable property when cautiousness is wanted in an information fusion process, since in this case combining identical information should not lead to the reinforcement of some hypothesis. Idempotent operators also guarantee that identical information items are not counted twice in the fusion process, a very important property in decentralized applications where the information origin cannot always be tracked (ad-hoc wireless networks are typical examples). In the theory of belief functions, a sound way to combine conjunctively multiple information items is to design a combination rule that selects the least informative element among a subset of belief functions more informative than each of the combined ones. In contrast, disjunctive rules can be retrieved by selecting the most informative element among a subset of belief functions less informative than each of the combined ones. One interest of such approaches is that they provide idempotent rules by construction. The notions of less and more informative are often formalized through partial orderings extending usual set-inclusion, yet the only two informative partial orders that provide a straightforward idempotent rule leading to a unique result are those based on the conjunctive and disjunctive weight functions. In this article, we show that other partial orders can achieve a similar goal when the problem is slightly relaxed into a distance optimization one. Building upon previous work, this paper investigates the use of distances compatible with informative partial orders to determine a unique solution to the combination problem. The obtained operators are conjunctive/disjunctive, idempotent and commutative, but lack associativity. They are, however, quasi-associative allowing sequential combinations at no extra complexity. Some experiments demonstrate interesting discrepancies as compared to existing approaches, notably with the aforementioned rules relying on weight functions.

International audience; Within imprecise probability theory, the extreme points of convex probability sets have an important practical role (to perform inference on graphical models, to compute expectation bounds,. . .). This is especially true for sets presenting specic features that make them easy to manipulate in applications. This easiness is the reason why extreme points of such models (probability intervals, possibility distributions,. . .) have been well studied. Yet, imprecise cumulative distributions (a.k.a. p-boxes) constitute an important exception, as the characterization of their extreme points remain to be studied. This is what we do in this paper, where we characterize the maximal number of extreme points of a p-box, give a family of p-boxes that attains this number and show an algorithm that allows to compute the extreme points of a given p-box. To achieve all this, we also provide what we think to be a new characterization of extreme points of a belief function.

When classifying an example on the basis of an observed population of (training) samples, at least three kinds of situations can arise where picking a single class may be difficult: high aleatory uncertainty due to the natural mixing of classes, high epistemic uncertainty due to the scarcity of training data, and non-conformity or atypicality of the example with respect to observations made so far. While the two first kinds of situations have been explored extensively, the last one still calls for a principled analysis. This paper is a first proposal to address this issue within the theory of belief function.

The problem of learning from imprecise data has recently attracted increasing attention, and various methods to tackle this problem have been proposed. In this paper, we discuss and compare two quite opposite approaches, an “optimistic” one that interprets imprecise data in a way that is most favourable for a candidate model, and a “pessimistic” one in which model choice is guided by the most unfavourable interpretation. To avoid an overly extreme behaviour, a modified version of the latter has recently been proposed, which we complement by an adjusted version of the optimistic approach. By presenting the various methods within a common (loss minimization) framework and discussing illustrative examples, we hope to provide some insight into important properties and differences, thereby paving the way for a more formal analysis.

Various strategies for active learning have been proposed in the machine learning literature. In uncertainty sampling, which is among the most popular approaches, the active learner sequentially queries the label of those instances for which its current prediction is maximally uncertain. The predictions as well as the measures used to quantify the degree of uncertainty, such as entropy, are almost exclusively of a probabilistic nature. In this paper, we advocate a distinction between two different types of uncertainty, referred to as epistemic and aleatoric, in the context of active learning. Roughly speaking, these notions capture the reducible and the irreducible part of the total uncertainty in a prediction, respectively. We conjecture that, in uncertainty sampling, the usefulness of an instance is better reflected by its epistemic than by its aleatoric uncertainty. This leads us to suggest the principle of "epistemic uncertainty sampling", which we instantiate by means of a concrete approach for measuring epistemic and aleatoric uncertainty. In experimental studies, epistemic uncertainty sampling does indeed show promising performance., Draft version of a paper to be published in the proceedings of DS 2019, 22nd International Conference on Discovery Science, Split, Croatia, 2019

International audience; In uncertainty theories, a common problem is to define how we can extend relations between sets (e.g., inclusion, ranking, consistency, …) to corresponding notions between uncertainty representations. Such definitions can then be used to perform the same operations as those that are done for sets: comparing information content, ordering alternatives or checking consistency, to name a few. In this paper, we propose a general way to extend set relations to belief functions, using constrained stochastic matrices to identify those belief functions in relation. We then study some properties of our proposal, as well as its connections with existing works focusing on specific relations.

International audience; Gaussian discriminant analysis is a popular classification model, that in the precise case can produce unreliable predictions in case of high uncertainty. While imprecise probability theory offer a nice theoretical framework to solve this issue, it has not been yet applied to Gaussian discriminant analysis. This work remedies this, by proposing a new Gaussian discrimi-nant analysis based on robust Bayesian analysis and near-ignorance priors. The model delivers cautious predictions, in form of set-valued class, in case of limited or imperfect available information. Experiments show that including an imprecise component in the Gaussian discriminant analysis produces reasonably cautious predictions, in the sense that the number of set-valued predictions is not too high, and that those predictions correspond to hard-to-classify instances, that is instances for which the precise classifier accuracy drops.

International audience; Preference elicitation is a key element of any multi-criteria decision analysis (MCDA) problem, and more generally of individual user preference learning. Existing efficient elicitation procedures in the literature mostly use either robust or Bayesian approaches. In this paper, we are interested in extending the former ones by allowing the user to express uncertainty in addition of her preferential information and by modelling it through belief functions. We show that doing this, we preserve the strong guarantees of robust approaches, while overcoming some of their drawbacks. In particular, our approach allows the user to contradict herself, therefore allowing us to detect inconsistencies or ill-chosen model, something that is impossible with more classical robust methods.

International audience; The pari-mutuel model is a betting scheme that has its origins in horse racing, and that has been applied in a number of contexts, mostly economics. In this paper, we consider the set of probability measures compatible with a pari-mutuel model, characterize its extreme points, and investigate the properties of the associated lower and upper probabilities. We show that the pari-mutuel model can be embedded within the theory of probability intervals, and prove necessary and sucient conditions for it to be a belief function or a minitive measure. In addition, we also investigate the combination of dierent pari-mutuel models and their denition on product spaces.

International audience; In this paper, we explore the problem of estimating lower and upper densities from imprecisely defined families of parametric kernels. Such estimations allow to rely on a single bandwidth value, and we show that it provides good results on classification tasks when extending the naive Bayesian classifier

International audience; In this paper, we are interested in the label ranking problem. We are more specifically interested in the recent trend consisting in predicting partial but more accurate (i. e. , making less incorrect statements) orders rather than complete ones. To do so, we propose a ranking method based on label-wise decomposition. We estimate an imprecise probabilistic model over each label rank and we infer a partial order from it using optimisation techniques. This leads to new results concerning a particular bilinear assignment problem. Finally, we provide some experiments showing the feasibility of our method.

In this paper, we study the conjunction of possibility measures when they are interpreted as coherent upper probabilities, that is, as upper bounds for some set of probability measures. We identify conditions under which the minimum of two possibility measures remains a possibility measure. We provide graphical way to check these conditions, by means of a zero-sum game formulation of the problem. This also gives us a nice way to adjust the initial possibility measures so their minimum is guaranteed to be a possibility measure. Finally, we identify conditions under which the minimum of two possibility measures is a coherent upper probability, or in other words, conditions under which the minimum of two possibility measures is an exact upper bound for the intersection of the credal sets of those two possibility measures., Comment: 23 pages, 2 figures

International audience; Two variables are called comonotone when there is an increasing relation between them, in the sense that when one of them increases (decreases), so does the other one. This notion has been widely investigated in probability theory, and is related to copulas. This contribution studies how the notion of comonotonicity can be extended to an imprecise setting on discrete spaces, where probabilities are only known to belong to a convex set. We define comonotonicity for such sets and investigate its characterizations in terms of lower probabilities, as well as its connection with copulas. As this theoretical characterization can be tricky to apply to general lower probabilities, we also investigate specific models of practical importance. In particular, we provide some sufficient conditions for a comonotone belief function with fixed marginals to exist, and characterize comonotone bivariate p-boxes.

We propose in this article a new relational clustering method that can return a partial answer (i.e., a set of clusterings) in some cases. Starting from relational or similarity data, we determine a partial equivalence relation defined on the set of objects (two objects are linked if they belong to the same cluster): the key idea is to allow the method to abstain on some pairwise links because they cannot be determined with enough certainty from the data. This cautious equivalence relation represents a set of possible hard clusterings which can be obtained by completing the partial relation. This formalization makes it possible to easily detect ambiguous links and to identify subsets of objects with uncertain relationship. We illustrate the potential interest of our approach as a tool for exploratory data analysis of synthetic and real data sets.

International audience; In this paper, we introduce a generic way to represent and manipulate pairwise information about partial orders (representing rankings, preferences, ...) with belief functions. We provide generic and practical tools to make inferences from this pairwise information and illustrate their use onthe machine learning problems that are label ranking and multi-label prediction. Our approach differs from most other quantitative approaches handling complete or partial orders, in the sense that partial orders are here considered as primary objects and not as incomplete specifications of ideal but unknown complete orders.

International audience; In some applications of machine learning and information retrieval (e.g. medical diagnosis, image recognition, pre-classification...), it can be preferable to provide less informative but more reliable predictions. This can be done by making partial predictions in the form of class subsets when the available information is insufficient to provide a reliable unique class. Imprecise probabilistic approaches offer nice tools to learn models from which such cautious predictions can be produced. However, the learning and inference processes of such models are computationally harder than their precise counterparts. In this paper, we introduce and study a particular binary decomposition strategy, nested dichotomies, that offer computational advantages in both the learning (due to the binarization process) and the inference (due to the decomposition strategy) processes. We show with experiments that these computational advantages do not lower the performances of the classifiers, and can even improve them when the class space has some structure.

Knowledge regarding complex systems are heterogeneous and fragmented.modelling dynamic complex systems in the framework of dynamic credal networks.practical methodology coupling Dirichlet distributions with interval probabilities to incrementally build and update model parameters whatever source and format of knowledge.enables to take into account (1) stochastic and epistemic uncertainties pertaining to the system; (2) the confidence level on the different sources of information.illustrate the application of the methodology to the modelling of a simplified industrial case study. Modeling complex dynamical systems from heterogeneous pieces of knowledge varying in precision and reliability is a challenging task. We propose the combination of dynamical Bayesian networks and of imprecise probabilities to solve it. In order to limit the computational burden and to make interpretation easier, we also propose to encode pieces of (numerical) knowledge as probability intervals, which are then used in an imprecise Dirichlet model to update our knowledge. The idea is to obtain a model flexible enough so that it can easily cope with different uncertainties (i.e., stochastic and epistemic), integrate new pieces of knowledge as they arrive and be of limited computational complexity.

International audience; So-called credal classifiers offer an interesting approach when the reliability or robustness of predictions have to be guaranteed. Through the use of convex probability sets, they can select multiple classes as prediction when information is insufficient and predict a unique class only when the available information is rich enough. The goal of this paper is to explore whether this particular feature can be used advantageously in the setting of co-training, in which a classifier strengthen another one by feeding it with new labeled data. We propose several co-training strategies to exploit the potential indeterminacy of credal classifiers and test them on several UCI datasets. We then compare the best strategy to the standard co-training process to check its efficiency.

International audience; Hidden Markov models are popular tools for gesture recognition. Once the generative processes of gestures have been identified, an observation sequence is usually classified as the gesture having the highest likelihood, thus ignoring possible prior information. In this paper, we consider two potential improvements of such methods: the inclusion of prior information, and the possibility of considering convex sets of probabilities (in the likelihoods and the prior) to infer imprecise, but more reliable, predictions when information is insufficient. We apply the proposed approach to technical gestures, typically characterized by severe class imbalance. By modelling such imbalances as a prior information, we achieve more accurate results, while the imprecise quantification is shown to produce more reliable estimates.

Distances between mass functions are instrumental tools in evidence theory, yet it is not always clear in which situation a particular distance should be used. Indeed, while the mathematical properties of distances have been well studied, how to interpret them is still a largely open issue. As a step towards answering this question, we propose to interpret distances by looking at their compatibility with partial orders. We formalize this compatibility through some mathematical properties thereby allowing to combine the advantages of both partial orders (clear semantics) and distances (richer structure and access to numerical tools). We explore in particular the case of informational partial orders, and how distances compatible with such orders can be used to approximate initial belief functions by simpler ones through the use of convex optimization. We finish by discussing some perspectives of the current work. Distances and partial orders in the theory of belief functions are formally related through a desirable property.The compatibility of several L k norm based distances with some informational partial orders is proved.These results are exploited to approximate initial belief functions by simpler ones through the use of convex optimization.

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 Fr\'echet bounds), and we show that our approach extends the probabilistic arithmetic of Williamson and Downs. Two design problems---a damped oscillator, and a river dike---demonstrate the practical feasibility of our results., Comment: 33 pages, 8 figures; v2: added keywords, added doi link

Contact: destercke@supagro.inra.fr, sdestercke@gmail.com; International audience; Information is said to be bipolar when it has a positive and a negative part. The problem of representing and processing such bipolar information has recently received a lot of attention in uncertainty theories. In this paper, we are concerned with the representation of asymmetric bipolarity, i.e., with situations where positive and negative information are unrelated and processed in parallel. In this latter case, positive information consists of observations of experiment results, showing what values are possible, whereas negative information consists of constraints (e.g., provided by an expert), restricting the range of possible variable values. Up to now, there are no proposition as to how such bipolar information can be treated in the framework of imprecise probability theory, i.e., when information is represented by convex sets of probabilities. In this paper, we propose the basis of such a framework and provide some illustrative examples.

International audience; This paper is a fine review of various aspects related to the statistical handling of "on-tic" random fuzzy sets by the means of appropriate distances. It is quite comprehensive and helpful, as it clarifies the status of fuzzy sets in such methods, explains the advan-tages of using a distance-based approach, specifies the pitfalls in which one should not fall when dealing with "ontic" random fuzzy sets and provides some illustration of practical computations. Not being a statistician but an occasional user of statistics, my discussion will mainly focus on this more practical aspect.

Given an imprecise probabilistic model over a continuous space, computing lower/upper expectations is often computationally hard to achieve, even in simple cases. Because expectations are essential in decision making and risk analysis, tractable methods to compute them are crucial in many applications involving imprecise probabilistic models. We concentrate on p-boxes (a simple and popular model), and on the computation of lower expectations of non-monotone functions. This paper is devoted to the univariate case, that is where only one variable has uncertainty. We propose and compare two approaches : the first using general linear programming, and the second using the fact that p-boxes are special cases of random sets. We underline the complementarity of both approaches, as well as the differences., Comment: 31 pages, 6 figures, constitute an extended version of a small paper accepted in ISIPTA conference, and a preprint version of a paper accepted in IJAR

Learning from uncertain data has attracted increasing attention in recent years. In this paper, we propose a decision tree learning method that can not only handle uncertain data, but also reduce epistemic uncertainty by querying the most valuable uncertain instances within the learning procedure. Specifically, we use entropy intervals extracted from the evidential likelihood to query uncertain training instances when needed, with the goal to improve the selection of the splitting attribute. Experimental results under various conditions confirm the interest of the proposed approach. HighlightsActive belief decision trees are learned from uncertain data modelled by belief functions.A query strategy is proposed to query the most valuable uncertain instances while learning decision trees.To deal with evidential data, entropy intervals are extracted from the evidential likelihood.Experiments with UCI data illustrate the robustness of proposed approach to various kinds of uncertain data.

International audience; The paper studies the problem of actively learning from instances characterized by imprecise features or imprecise class labels, where by actively learning we understand the possibility to query the precise value of imprecisely specified data. We differ from classical active learning by the fact that in the later, data are either fully precise or completely missing, while in our case they can be partially specified. Such situations can appear when sensor errors are important to encode, or when experts have only specified a subset of possible labels when tagging data. We provide a general active learning technique that can be applied in principle to any model. It is inspired from racing algorithms, in which several models are competing against each others. The main idea of our method is to identify the query that will be the most helpful in identifying the winning model in the competition. After discussing and formalizing the general ideas of our approach, we illustrate it by studying the particular case of binary SVM in the case of interval valued features and set-valued labels. The experimental results indicate that, in comparison to other baselines, racing algorithms provide a faster reduction of the uncertainty in the learning process, especially in the case of imprecise features.

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International audience; Within the fuzzy literature, the issue of ranking fuzzy intervals has been addressed by many authors, who proposed various solutions to the problem. Most of these solutions intend to find a total order on a given collection of fuzzy intervals. However, if one sees fuzzy intervals as descriptions of uncertain quantities, an alternative to rank them is to use ranking rules issued from the imprecise probabilistic literature. In this paper , we investigate ranking rules based on different statistical features (mean, median) and orderings, and relate the obtained (partial) orders to some classical proposals. In particular, we propose a generic expression of stochastic orderings, and then use it to systematically investigate extensions of the most usual stochastic orderings to fuzzy intervals. We also show some relations between those extensions, and explore their relation with existing fuzzy ranking proposals.

International audience; Combining pieces of information provided by several sources without or with little prior knowledge about the behavior of the sources is an old yet still important and rather open problem in the belief function theory. In this paper, we propose an approach to select the behavior of sources based on a very general and expressive fusion scheme, that has the important advantage of making clear the assumptions made about the sources. The selection process itself relies on two cornerstones that are the notions of specificity and consistency of a knowledge representation, and that we adapt to the considered fusion scheme. We illustrate our proposal on different examples and show that the proposed approach actually encompasses some important existing fusion strategies.

In this paper, we study how multilabel predictions can be obtained when our uncertainty is described by a convex set of probabilities. Such predictions, typically consisting of a set of potentially optimal decisions, are hard to make in large decision spaces such as the one considered in multilabel problems. However, we show that when considering the Hamming or the ranking loss, outer-approximating predictions can be efficiently computed from label-wise information, as in the precise case. We also perform some first experiments showing the behaviour of the partial predictions obtained through these approximations. Such experiments also confirm that predictions become partial on those labels where the precise prediction is likely to make an error. HighlightsWe consider the problem of multilabel prediction with probability sets.We show that efficient approximations can be obtained for the Hamming loss.We show that efficient approximations can be obtained for the ranking loss.We perform first experiments confirming the expected behaviour of the approximations.

International audience; The inclusion-exclusion principle is a well-known property in probability theory, and is instrumental in some computational problems such as the evaluation of system reliability or the calculation of the probability of a Boolean formula in diagnosis. However, in the setting of uncertainty theories more general than probability theory, this principle no longer holds in general. It is therefore useful to know for which families of events it continues to hold. This paper investigates this question in the setting of belief functions. After exhibiting original sufficient and necessary conditions for the principle to hold, we illustrate its use on the uncertainty analysis of Boolean and non-Boolean systems in reliability.

International audience; In this paper, we study how multilabel predictions can be obtained when our uncertainty is described by a convex set of probabilities. Such predictions, typically consisting of a set of potentially optimal decisions, are hard to make in large decision spaces such as the one considered in multilabel problems. However, we show that when considering the Hamming loss, an approximate prediction can be efficiently computed from label-wise information, as in the precise case. We also perform some first experiments showing the interest of performing partial predictions in the multilabel case.

International audience; In many application data are often imperfect, incomplete or more generally uncertain. Whereas imperfect data are often removed from samples or corrected into precise data, using their initial imperfect structures to learn classifiers remains a challenge. As data uncertainty can be expressed in many forms (missing data, fuzzy sets, probabilities), working within the belief function framework enables a large number of models. The E2M decision trees is a methodology that provide predictions from uncertain data modelled by belief functions. In this paper, the problem of rubber quality prediction is presented with a belief function modelling of some data uncertainties. Some resulting E2M decision trees are presented in order to improve the interpretation of the tree compared to standard decision trees.