Argumentation theory provides a formal framework to represent and analyse debates where participants propose arguments that attack or support others and assign scores expressing their opinions. Quantitative Bipolar Argumentation Frameworks model such debates by assigning initial weights to arguments and using semantics to compute final scores that reflect attackers’ and supporters’ influence. One of the major challenge is setting appropriate initial weights when debaters’ opinions are uncertain. In this paper, we introduce a formal approach to uncertainty propagation in Quantitative Bipolar argumentation frameworks by representing initial weights as belief mass functions over a discretized unit interval. We introduce two new propagation models: (i) an exact model that computes final mass functions by combining focal elements of initial weights with parent arguments via bipolar gradual semantics; (ii) a practical approximation that projects the exact mass onto a user-specified partition and reconstructs masses using the Moebius inverse. We prove mathematical properties of the projection and show that the baseline, while computationally efficient, can be overconfident by failing to preserve expectations. Our approximation reduces the exponential complexity of the exact model while satisfying Epistemic Cautiousness, yielding acceptability intervals that contain true theoretical expectations and balancing tractability with theoretical soundness.

Evidence Theory (a.k.a. Dempster-Shafer theory) offers a quite general and powerful setting to reason under uncertainty. However, most of the approaches allowing one to instantiate mass functions, the basic building block of the theory, rely on measured, objective data, and there are very few approaches to instantiate them from subjective, expert opinions. Here we suggest a viable solution, based on the concept of pairwise comparisons, for the elicitation of such subjective information. The approach is based on the well-established body of knowledge on the theory of pairwise comparisons and shows its flexibility as it can incorporate multiple representations for the subjective judgments and multiple experts, just to cite two possible extensions.

Abstract To generate accurate and reliable predictions, modern AI systems need to combine data from multiple modalities, such as text, images, audio, spreadsheets, and time series. However, collecting training and test data for many modalities is challenging and time-consuming, creating a need for cost-efficient multi-modal data acquisition. In this paper we advocate that this can be realized by disentangling epistemic and aleatoric uncertainty. It is commonly assumed in the machine learning community that epistemic uncertainty can be reduced by collecting more data, while aleatoric uncertainty is irreducible. We claim that this assumption can be challenged in modern multi-modal AI systems, and we introduces an innovative data acquisition framework where uncertainty disentanglement leads to actionable decisions, allowing cost-efficient sampling in two directions: sample size and data modality. The main hypothesis is that aleatoric uncertainty decreases as the number of modalities increases, while epistemic uncertainty decreases by collecting more observations. We provide a theoretical analysis and proof-of-concept implementations on various multi-modal datasets to prove the usefulness of our framework, which combines ideas from active learning, active feature acquisition and uncertainty quantification.

Online deliberation platforms allow people to exchange their opinions around a specified issue and to vote on these opinions in order to reach a collective decision. Argumentation allows to structure and analyse user input for these platforms. A debate can be represented by a quantitative bipolar argumentation framework where votes on each argument of the debate are aggregated into an initial weight. One of the main challenges these platforms face is sparse voting i.e. participants vote on a few number of arguments, leading to an imbalance of the number of votes between the arguments. In this paper, we propose a methodology that handles sparse voting in online debates, by introducing imprecise quantitative bipolar argumentation frameworks that incorporate uncertainty on the initial weights. Specifically, we leverage votes on arguments to initialize weight intervals that represent the uncertainty on the initial weights, using the imprecise Dirichlet model. We use four state-of-the-art bipolar gradual semantics to generate a final acceptability interval on each argument and we introduce several properties to study the effect of these semantics on the uncertainty on each argument’s final evaluation. Our methodology allows for a more robust representation of argument strength in the presence of limited data.

We propose a method for obtaining statistically guaranteed prediction sets for functional machine learning methods: surrogate models which map between function spaces, motivated by the need to build reliable PDE emulators. The method constructs nested prediction sets on a low-dimensional representation (an SVD) of the surrogate model’s error, and then maps these sets to the prediction space using set-propagation techniques. This results in prediction sets for functional surrogate models with conformal prediction coverage guarantees. We use zonotopes as basis of the set construction, which allow an exact linear propagation and are closed under Cartesian products, making them well-suited to this high-dimensional problem. The method is model agnostic and can thus be applied to complex Sci-ML models, including Neural Operators, but also in simpler settings. We also introduce a technique to capture the truncation error of the SVD, preserving the guarantees of the method.

Abstract In this paper, we present a formal framework to (1) aggregate probabilistic ensemble members into either a representative classifier or a credal classifier, and (2) perform various decision tasks based on this uncertainty quantification. We first elaborate on the aggregation problem under a class of distances between distributions. We then propose generic methods to robustify uncertainty quantification and decisions, based on the obtained ensemble and representative probability. To facilitate the scalability of the proposed framework, for all the problems and applications covered, we elaborate on their computational complexities from the theoretical aspects and leverage theoretical results to derive efficient algorithmic solutions. Finally, relevant sets of experiments are conducted to assess the usefulness of the proposed framework in uncertainty sampling, classification with a reject option, and set-valued prediction-making.

Ordered weighted averaging (OWA) functions, a.k.a. Generalised Gini Index, are routinely used to obtain fair solutions. However, while they ensure some level of fairness in the result, two remaining questions are why they recommend a given alternative, and how robust is this recommendation. We bring practical and theoretically grounded solutions to these questions, by providing an explanation engine for robust OWA that consists in a normative transitive chain of self-evident arguments, themselves based on the normative properties of the model. We provide a thorough theoretical study of the engine, showing that it is sound and complete with respect to the model, with a theoretical upper bound on the length of the explanation and a tractable algorithm (even though minimizing the length is NP-hard). We also provide experimental evidence that the engine performs well on synthetic data. Thus, we guarantee that an explanation can always be found, and that reasoning according to the provided scheme always produces a valid statement. Moreover, the explanations allow to probe the normative requirements of the model, so as to allow validation, accountability and recourse, that are key components of trustworthy AI.

CNES has developed the CARS pipeline to massively produce Digital Surface Models (DSM) for remote sensing applications. DSM are computed from pairs of very high resolution satellite imagery using multi-view stereo methods. In this paper, we compute robust elevation confidence intervals with more than 90% accuracy alongside the high resolution DSM. We first estimate disparity confidence intervals using possibility distributions during the dense matching step, where most errors usually occur. Disparity confidence intervals are then processed with caution throughout every step of the pipeline to be transformed into elevation confidence intervals. Intervals accuracy is evaluated on both urban and glacier high resolution images.

The Plackett–Luce model is a popular parametric probabilistic model to define distributions between rankings of objects, modelling for instance observed preferences of users or ranked performances of algorithms. Since such observations may be scarce (users may provide partial preferences, or not all algorithms are run for a given experiment), it may be useful to consider the case where the parameters of the Plackett–Luce model are imprecisely known. In this paper, we first introduce the imprecise Plackett–Luce model, induced by a set of parameters (for instance, parameters with a high relative likelihood). Given a set of possible parameters for the model, we then provide an efficient algorithm to make cautious inferences, returning sets of possible optimal rankings (for instance in the form of partial orders). We illustrate the use of our imprecise model on label ranking, a specific kind of supervised learning.

This paper proposes a tentative and original survey of meeting points between Knowledge Representation and Reasoning (KRR) and Machine Learning (ML), two areas which have been developed quite separately in the last four decades. First, some common concerns are identified and discussed such as the types of representation used, the roles of knowledge and data, the lack or the excess of information, or the need for explanations and causal understanding. Then, the survey is organised in seven sections covering most of the territory where KRR and ML meet. We start with a section dealing with prototypical approaches from the literature on learning and reasoning: Inductive Logic Programming, Statistical Relational Learning, and Neurosymbolic AI, where ideas from rule-based reasoning are combined with ML. Then we focus on the use of various forms of background knowledge in learning, ranging from additional regularisation terms in loss functions, to the problem of aligning symbolic and vector space representations, or the use of knowledge graphs for learning. Then, the next section describes how KRR notions may benefit to learning tasks. For instance, constraints can be used as in declarative data mining for influencing the learned patterns; or semantic features are exploited in low-shot learning to compensate for the lack of data; or yet we can take advantage of analogies for learning purposes. Conversely, another section investigates how ML methods may serve KRR goals. For instance, one may learn special kinds of rules such as default rules, fuzzy rules or threshold rules, or special types of information such as constraints, or preferences. The section also covers formal concept analysis and rough sets-based methods. Yet another section reviews various interactions between Automated Reasoning and ML, such as the use of ML methods in SAT solving to make reasoning faster. Then a section deals with works related to model accountability, including explainability and interpretability, fairness and robustness. Finally, a section covers works on handling imperfect or incomplete data, including the problem of learning from uncertain or coarse data, the use of belief functions for regression, a revision-based view of the EM algorithm, the use of possibility theory in statistics, or the learning of imprecise models. This paper thus aims at a better mutual understanding of research in KRR and ML, and how they can cooperate. The paper is completed by an abundant bibliography.

Many works within robust combinatorial optimisation consider interval-valued costs or constraints. While most of these works focus on finding a unique solution following a robust criteria such as minimax, a few consider the problem of characterising a set of possibly optimal solutions. This paper is situated within this line of work, and considers the problem of exactly enumerating the set of possibly optimal matroids under interval-valued costs. We show in particular that each solution in this set can be obtained through a polynomial procedure, and provide an efficient algorithm to achieve the enumeration.

This contribution presents a concrete example of uncertainty propagation in a stereo matching pipeline. It considers the problem of matching pixels between pairs of images whose radiometry is uncertain and modeled by possibility distributions. Copulas serve as dependency models between variables and are used to propagate the imprecise models. The propagation steps are detailed in the simple case of the Sum of Absolute Difference cost function for didactic purposes. The method results in an imprecise matching cost curve. To reduce computation time, a sufficient condition for conserving possibility distributions after the propagation is also presented. Finally, results are compared with Monte Carlo simulations, indicating that the method produces envelopes capable of correctly estimating the matching cost.

One way to make decisions under uncertainty is to select an optimal option from a possible range of options, by maximizing the expected utilities derived from a probability model. However, under severe uncertainty, identifying precise probabilities is hard. For this reason, imprecise probability models uncertainty through convex sets of probabilities, and considers decision rules that can return multiple options to reflect insufficient information. Many well-founded decision rules have been studied in the past, but none of those standard rules are able to control the number of returned alternatives. This can be a problem for large decision problems, due to the cognitive burden decision makers have to face when presented with a large number of alternatives. Our contribution proposes regret-based ideas to construct new decision rules which return a bounded number of options, where the limit on the number of options is set in advance by the decision maker as an expression of their cognitive limitation. We also study their consistency and numerical behaviour.

The representation and quantification of uncertainty has received increasing attention in machine learning in the recent past. The formalism of credal sets provides an interesting alternative in this regard, especially as it combines the representation of epistemic (lack of knowledge) and aleatoric (statistical) uncertainty in a rather natural way. In this paper, we elaborate on uncertainty measures for credal sets from the perspective of machine learning. More specifically, we provide an overview of proposals, discuss existing measures in a critical way, and also propose a new measure that is more tailored to the machine learning setting. Based on an experimental study, we conclude that theoretically well-justified measures also lead to better performance in practice. Besides, we corroborate the difficulty of the disaggregation problem, that is, of decomposing the amount of total uncertainty into aleatoric and epistemic uncertainty in a sound manner, thereby complementing theoretical findings with empirical evidence.

Identifying the preferences of a given user through elicitation is a central part of multi-criteria decision aid (MCDA) or preference learning tasks. Two classical ways to perform this elicitation is to use either a robust or a Bayesian approach. However, both have their shortcoming: the robust approach has strong guarantees through very strong hypotheses, but cannot integrate uncertain information. While the Bayesian approach can integrate uncertainties, but sacrifices the previous guarantees and asks for stronger model assumptions. In this paper, we propose and test a method based on possibility theory, which keeps the guarantees of the robust approach without needing its strong hypotheses. Among other things, we show that it can detect user errors as well as model misspecification.

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.

Belief functions are quite generic models when it comes to represent uncertain data, as it extends a wide range of uncertainty models (possiblity and probability distributions, among others). Usually, belief functions are defined over finite spaces, however many real word problems require to deal with beliefs over a continuous space while maintaining computational efficiency. This paper discusses the case of focal sets on the unit simplex, and proposes efficient inference tools to manipulate them. Such sets can be used to represent unknown proportions that one may face in various fields like soil contamination managing, plastic sorting or image reconstruction. In this paper, we illustrate their use on an industrial problem of plastic sorting, where the proportion of material impurities must not go over a limit while minimizing the rejection of sorted materials, whose nature is uncertain.

It is often the case in the applications of Multi-Criteria Decision Making that the values of alternatives are unknown on some attributes. An interesting situation arises when the attributes having missing values are actually not relevant and shall thus be removed from the model. Given a model that has been elicited on the complete set of attributes, we are looking thus for a way -- called restriction operator -- to automatically remove the missing attributes from this model. Axiomatic characterizations are proposed for three classes of models. For general quantitative models, the restriction operator is characterized by linearity, recursivity and decomposition on variables. The second class is the set of monotone quantitative models satisfying normalization conditions. The linearity axiom is changed to fit with these conditions. Adding recursivity and symmetry, the restriction operator takes the form of a normalized average. For the last class of models -- namely the Choquet integral, we obtain a simpler expression. Finally, a very intuitive interpretation is provided.

We propose a method for reliable prediction in multi-class classification, where reliability refers to the possibility of partial abstention in cases of uncertainty. More specifically, we allow for predictions in the form of preorder relations on the set of classes, thereby generalizing the idea of set-valued predictions. Our approach relies on combining learning by pairwise comparison with a recent proposal for modeling uncertainty in classification, in which a distinction is made between reducible (a.k.a. epistemic) uncertainty caused by a lack of information and irreducible (a.k.a. aleatoric) uncertainty due to intrinsic randomness. The problem of combining uncertain pairwise predictions into a most plausible preorder is then formalized as an integer programming problem. Experimentally, we show that our method is able to appropriately balance reliability and precision of predictions.

Indeterminate classifiers are cautious models able to predict more than one class in case of high uncertainty. A problem that arises when using such classifiers is how to evaluate their performances. This problem has already been considered in the case where all prediction errors have equivalent costs (that we will refer as the "0/1 costs" or accuracy setting). The purpose of this paper is to study the case of generic cost functions. We provide some properties that the costs of indeterminate predictions could or should follow, and review existing proposals in the light of those properties. This allows us to propose a general formula fitting our properties that can be used to produce and evaluate indeterminate predictions. Some experiments on the cost-sensitive problem of ordinal regression illustrate the behavior of the proposed evaluation criterion.

In decision tasks such as bioprocess efficiency comparison, scientific literature is a valuable source of data. This large number of scientific data is heterogeneously structured, mainly in textual format. Innovative tools able to integrate and treat constantly new information are required. In this context, the use of semantic web methods such as ontologies seems relevant to structure the experimental information. Imprecision and uncertainty can arise from data incompleteness and variability. This is particularly true for processes involving biological materials. Document reliability should also be considered. Soft computing methods have the potential to be the kingpin of specialized software that can be integrated in decision support systems (DSS) intended to solve these issues. This paper presents the implementation of a pipeline which permits to: (1) structure and integrate the experimental data of interest by using ontologies, (2) assess data source reliability, (3) compute and visualize indicators taking into account data imprecision.

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.

Binary decomposition techniques transform a multi-class problem into several simpler binary problems. In such techniques, a classical issue is to ensure the consistency between the binary assessments of conditional probabilities. Nested dichotomies, which consider tree-shaped decomposition, do not suffer from this issue. Yet, a wrong probability estimate in the tree can strongly biase the results and provide wrong predictions. To overcome this issue, we consider in this paper imprecise nested dichotomies, in which binary probabilities become imprecise. We show in experiments that the approach has many advantages: it provides cautious inferences when only little information is available, and allows to make efficient computations with imprecise probabilities even when considering generic cost functions.

In decision tasks, imprecision and uncertainty can arise from many sources such as data uncertainty, data reliability, or the necessity to use intermediate (non-fully reliable) models. User preferences may also be included into the decision process, making it even more complex. This is particularly true for processes involving biological materials. Soft computing methods have the potential to be the kingpin of specialized software that can be integrated in decision support systems intended to solve the mentioned issues. This work presents three such methods and their implementation: (i)-simulation module with uncertain inputs and interval analysis, (ii)-reliability module allowing to rank candidates according to expert criteria, and (iii)-bipolar flexible querying module, that makes the difference between constraints and wishes in a data base query. A decision support system architecture is proposed and illustrated on a case study, dealing with biodegradable and biosourced packaging design.

In real world applications, data are often uncertain or imperfect. In most classification approaches, they are transformed into precise data. However, this uncertainty is an information in itself which should be part of the learning process. Data uncertainty can take several forms: probabilities, (fuzzy)sets of possible values, expert assessments, etc. We therefore need a flexible and generic enough model to represent and treat this uncertainty, such as belief functions. Decision trees are well known classifiers which are usually learned from precise datasets. In this paper we propose a methodology to learn decision trees from uncertain data in the belief function framework. In the proposed method, the tree parameters are estimated through the maximization of an evidential likelihood function computed from belief functions, using the recently proposed E2M algorithm that extends the classical EM. Some promising experiments compare the obtained trees with classical CART decision trees.

As Dempster–Shafer theory spreads in different application fields, and as mass functions are involved in more and more complex systems, the need for algorithms randomly generating mass functions arises. Such algorithms can be used, for instance, to evaluate some statistical properties or to simulate the uncertainty in some systems (e.g., data base content, training sets). As such random generation is often perceived as secondary, most of the proposed algorithms use straightforward procedures whose sample statistical properties can be difficult to characterize. Thus, although such algorithms produce randomly generated mass functions, they do not always produce what could be expected from them (for example, uniform sampling in the set of all possible mass functions). In this paper, we briefly review some well-known algorithms, explaining why their statistical properties are hard to characterize. We then provide relatively simple algorithms and procedures to perform efficient random generation of mass functions whose sampling properties are controlled.

We explore the relationship between possibility measures (supremum preserving normed measures) and p-boxes (pairs of cumulative distribution functions) on totally preordered spaces, extending earlier work in this direction by De Cooman and Aeyels, among others. We start by demonstrating that only those p-boxes who have 0–1-valued lower or upper cumulative distribution function can be possibility measures, and we derive expressions for their natural extension in this case. Next, we establish necessary and sufficient conditions for a p-box to be a possibility measure. Finally, we show that almost every possibility measure can be modelled by a p-box, simply by ordering elements by increasing possibility. Whence, any techniques for p-boxes can be readily applied to possibility measures. We demonstrate this by deriving joint possibility measures from marginals, under varying assumptions of independence, using a technique known for p-boxes. Doing so, we arrive at a new rule of combination for possibility measures, for the independent case.

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.

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.

Many practical methods and different approaches have been proposed to assess Multi-State Systems (MSS) reliability measures. The universal generating function (UGF) method, introduced in 1986, is known to be a very efficient way of evaluating the availability of different types of MSSs. In this paper, we propose an extension of the UGF method considering epistemic uncertainties. This extended method allows one to model ill-known probabilities and transition rates, or to model both aleatory and epistemic uncertainty in a single model. It is based on the use of belief functions which are general models of uncertainty. We also compare this extension with UGF methods based on interval arithmetic operations performed on probabilistic bounds.

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. © 2011 Wiley Periodicals, Inc.

Many authors have studied fuzzy belief structures, that is belief functions having fuzzy sets as focal elements. One of the main reason for this is that this structure offers a convenient way to mix probabilistic and fuzzy information. Still, one point on which authors often disagree is how information represented by fuzzy belief structures should be processed, i.e., how should be defined fusion operations, decision rules, uncertainty measures, uncertainty propagation, etc. for such representations. In this paper, we consider that fuzzy belief structures are mapped into classical belief structures encoding the same information, and propose to manipulate these latter structures. From a practical standpoint, it has the benefit that processing tools proper to belief structures can be used together with their interpretation, rather than having to consider a mix of probabilistic and fuzzy calculi, which can be harder to interpret.

The belief structure resulting from the combination of consonant and independent marginal random sets is not, in general, consonant. Also, the complexity of such a structure grows exponentially with the number of combined random sets, making it quickly intractable for computations. In this paper, we propose a simple guaranteed consonant outer approximation of this structure. The complexity of this outer approximation does not increase with the number of marginal random sets (i.e., of dimensions), making it easier to handle in uncertainty propagation. Features and advantages of this outer approximation are then discussed, with the help of some illustrative examples.

When multiple sources provide information about the same unknown quantity, their fusion into a synthetic interpretable message is often a tricky problem, especially when sources are conflicting. In this paper, we propose to use possibility theory and the notion of maximal coherent subsets (MCSs), often used in logic-based representations, to build a fuzzy belief structure that will be instrumental both for extracting useful insight about various features of the information conveyed by the sources and for compressing this information into a unique possibility distribution. Extentions and properties of the basic fusion rule are also studied.

When multiple sources provide information about the same unknown quantity, their fusion into a synthetic interpretable message is often a tedious problem, especially when sources are conflicting. In this paper, we propose to use possibility theory and the notion of maximal coherent subsets, often used in logic-based representations, to build a fuzzy belief structure that will be instrumental both for extracting useful information about various features of the information conveyed by the sources and for compressing this information into a unique possibility distribution.

One of the main advantages of fuzzy modeling is the ability to yield interpretable results. Amongst these modeling methods, the OLS algorithm is a mathematically robust technique that allows to induce a fuzzy rule base from a set of training data. It does so by using linear regression to select the most important rules. However, the original OLS algorithm only relies upon numerical accuracy, and doesn't take interpretability matters into account. Thus, we propose some modifications to the original method so that it builds interpretable rule bases.