Belief Functions

The theory of belief functions, sometimes referred to as evidence theory or Dempster-Shafer theory, was first introduced by Arthur P. Dempster in the context of statistical inference, to be later developed by Glenn Shafer as a general framework for modelling epistemic uncertainty. Belief theory and the closely related random set theory form natural frameworks for modelling situations in which data are missing or scarce: think of extremely rare events such as volcanic eruptions or power plant meltdowns, problems subject to huge uncertainties due to the number and complexity of the factors involved (e.g. climate change), but also the all-important issue with generalisation from small training sets in machine learning. This tutorial is designed to introduce the principles and rationale of random sets and belief function theory to mainstream statisticians, mathematicians and working scientists, survey the key elements of the methodology and the most recent developments, make practitioners aware of the set of tools that have been developed for reasoning in the belief function framework on real-world problems. Attendees will acquire first-hand knowledge of how to apply these tools to significant problems in major application fields such as computer vision, climate change, and others. A research programme for the future of random set theory and high impact applications is eventually outlined.

The resolution of the Top-k queries is an important and a difficult problem in the large scale systems. Aggregation is the best way to solve this type of queries. It reduces substantially the amount of exchanged data and optimizes the search task. In this paper, we present a new algorithm with the aim of solving this type of problem efficiently. Our algorithm consists of three phases and defines only one threshold, unlike the other already existing algorithms. The simulations on real datasets and the comparison with some other algorithms show that our solution is very competitive. MOTS-CLÉS : agrégation, requêtes Top-k, fonction monotone, seuil, phase, listes triées.

WHY A MATHEMATICS OF UNCERTAINTY?

- probabilities do not represent well ignorance and lack of data;
- evidence is normally limited, rather than infinite as assumed by (frequentist) probability;
- expert knowledge needs often to be combined with hard evidence;
- in extreme cases (rare events or far-future predictions) very little data;
- bottom line: not enough evidence to determine the actual probability describing the problem.

This half-day tutorial on Belief function (random sets) for the working scientist was presented on July 9th 2016 at the latest International Joint Conference on Artificial Intelligence (IJCAI-16).

The tutorial is very comprehensive (468 slides), covering:

(i) a review of mathematical probability and its interpretations (Bayesian and frequentist);

(ii) the rational for going beyond standard probability: it's all about the data!

(iii) the basis notions of the theory of belief functions;

(iv) reasoning with belief functions: inference, combination/conditioning, graphical models, decision making;

(v) using belief functions for classification, regression, estimation, etc;

(vi) dealing with computational issues and extending belief measures to real numbers;

(vii) the main frameworks derived from belief theory, and its relationship with other theories of uncertainty;

(viii) a number of example applications;

(ix) new horizons, from the formulation of limit theorems for random sets, generalising the notion of likelihood and logistic regression for rare event estimation, climatic change modelling and new foundations for machine learning based on random set theory, a geometry of uncertainty.

Tutorial slides are downloadable at http://cms.brookes.ac.uk/staff/FabioCuzzolin/files/IJCAI2016.pdf

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned.

In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence.

The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.

The theory of belief functions, sometimes referred to as evidence theory or Dempster-Shafer theory, was first introduced by Arthur P. Dempster in the context of statistical inference, to be later developed by Glenn Shafer as a general framework for modelling epistemic uncertainty. Belief theory and the closely related random set theory form a natural framework for modelling situations in which data are missing or scarce: think of extremely rare events such as volcanic eruptions or power plant meltdowns, problems subject to huge uncertainties due to the number and complexity of the factors involved (e.g. climate change), but also the all-important issue with generalisation from small training sets in machine learning.

This short talk abstracted from an upcoming half-day tutorial at IJCAI 2016 is designed to introduce to non-experts the principles and rationale of random sets and belief function theory, review its rationale in the context of frequentist and Bayesian interpretations of probability but also in relationship with the other main approaches to non-additive probability, survey the key elements of the methodology and the most recent developments, discuss current trends in both its theory and applications. Finally, a research program for the future is outlined, which include a robustification of Vapnik' statistical learning theory for an Artificial Intelligence 'in the wild'.

By focusing on Eliade’s early life and writings (1921–1936), and specifically on his work on Yoga, I address in this paper a question that is frequently asked about the problematic relation between personal belief and scholarship in fields such as the history of religions. Eliade’s interest in Yoga showcases his own personal spiritual quest, of course. But it is also informed by his earlier readings of, and reflections on, two polar aspects of the religious experience: the magical and the mystical. For Eliade, these two categories provide not only a key to unlocking the conceptual and pragmatic dimensions of Yoga, but also the impetus for charting a systematic theory of religion. The last major expression of these early ideas can be seen in Yoga: Essai sur les origines de la mystique indienne (1936). Eliade’s thoughts on Yoga and religion then took different paths, reaching their full expression in works published in Paris after 1945. In his later books on Yoga – Techniques du Yoga (1948), Le Yoga: Immortalité et liberté (1954) – the twofold schema of the magical and the mystical no longer plays a key hermeneutical role. Nonetheless, understanding this early phase of his work is crucial if we are to provide a comprehensive account of the development of his thought on Yoga. This early phase in his scholarly career and experiential transformation, marked by his trip to India, is also very important in terms of understanding the further developments of his ideas on history of religions, as well as to evince the ground on which Eliade built his own – never confessed – private religious belief.

Computer vision is an ever growing discipline whose ambitious goal is to enable machines with the intelligent
visual skills humans and animals are provided by Nature, allowing them to interact effortlessly
with complex, dynamic environments. Designing automated visual recognition and sensing systems
typically involves tackling a number of challenging tasks, and requires an impressive variety of sophisticated
mathematical tools. In most cases, the knowledge a machine has of its surroundings is at best
incomplete – missing data is a common problem, and visual cues are affected by imprecision. The need
for a coherent mathematical ‘language’ for the description of uncertain models and measurements then
naturally arises from the solution of computer vision problems.

The theory of evidence (sometimes referred to as ‘evidential reasoning’, ‘belief theory’ or ‘Dempster-
Shafer theory’) is, perhaps, one of the most successful approaches to uncertainty modelling, as arguably
the most straightforward and intuitive approaches to a generalized probability theory. Emerging in the
last Sixties from a profound criticism of the more classical Bayesian theory of inference and modelling
of uncertainty, it stimulated in the last decades an extensive discussion of the epistemic nature of both
subjective ‘degrees of beliefs’ and frequentist ‘chances’ or relative frequencies. More recently, a renewed
interest in belief functions, the mathematical generalization of probabilities which are the object of study
of the theory of evidence, has seen a blossoming of applications to a variety of fields of applied science.

In this Book we are going to show how, indeed, the fruitful interaction of computer vision and evidential
reasoning is able stimulate a number of advances in both fields. From a methodological point of
view, novel theoretical advances concerning the geometric and algebraic properties of belief functions as
mathematical objects will be illustrated in some detail in Part II, with a focus on a perspective ‘geometric
approach’ to uncertainty and an algebraic solution of the issue of conflicting evidence. In Part III we will
illustrate how these new perspectives on the theory of belief functions arise from important computer vision
problems, such as articulated object tracking, data association and object pose estimation, to which
in turn the evidential formalism can give interesting new solutions. Finally, some initial steps towards
a generalization of the notion of total probability to belief functions will be taken, in the perspective of
endowing the theory of evidence with a complete battery of estimation and inference tools to the benefit
of scientists and practitioners.

Although born within the remit of mathematical statistics, the theory of belief functions has later evolved towards subjective interpretations which have distanced it from its mother field, and have drawn it nearer to artificial intelligence.
The purpose of this talk, in its first part, is to understanding belief theory in the context of mathematical probability and its main interpretations, Bayesian and frequentist statistics, contrasting these three methodologies according to their treatment of uncertain data.
In the second part we recall the existing statistical views of belief function theory, due to the work by Dempster, Almond, Hummel and Landy, Zhang and Liu, Walley and Fine, among others.
Finally, we outline a research programme for the development of a fully-fledged theory of statistical inference with random sets. In particular, we discuss the notion of generalised lower and upper likelihoods, the formulation of a framework for logistic regression with belief functions, the generalisation of the classical total probability theorem to belief functions, the formulation of parametric models based of random sets, and the development of a theory of random variables and processes in which the underlying probability space is replaced by a random set space.

The authors present a remarkable site with a remarkable interpretation: a structured platform of dugong bones, containing skulls laid in parallel and ribs in sets, together with artefacts of the Neolithic period. They propose that the bones have been symbolically arranged and the mound as a whole had a ritual purpose -an interpretation endorsed by analogy with dugong platforms noted in the Torres Strait in recent times.

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned.

In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence.

The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.

In the last few years, evidence theory, also known as Dempster-Shafer theory or belief functions theory, have received growing attention in many fields such as artificial intelligence, computer vision, telecommunications and networks, robotics, and finance. This is due to the fact that imperfect information permeates the real-world applications , and as a result, it must be incorporated into any information system that aims to provide a complete and accurate model of the real world. Although, it is in an early stage of development relative to classical probability theory , evidence theory has proved to be particularly useful to represent and reason with imperfect information in a wide range of real-world applications. In such cases, evidence theory provides a flexible framework for handling and mining uncertainty and imprecision as well as combining evidence obtained from multiple sources and modeling the conflict between them. The purpose of this paper is threefold. First, it introduces the basics of the belief functions theory with emphasis on the transferable belief model. Second , it provides a practical case study to show how the belief functions theory was used in a real network application, thereby providing guidelines for how the evidence theory may be used in telecommunications and networks. Lastly, it surveys and discusses a number of examples of applications of the evidence theory in telecommunications and network technologies.

In the last few years, evidence theory, also known Q1 6 as Dempster-Shafer theory or belief functions theory, have 7 received growing attention in many fields such as artifi-8 cial intelligence, computer vision, telecommunications and 9 networks, robotics, and finance. This is due to the fact 10 that imperfect information permeates the real-world appli-11 cations, and as a result, it must be incorporated into any 12 information system that aims to provide a complete and 13 accurate model of the real world. Although, it is in an early 14 stage of development relative to classical probability the-15 ory, evidence theory has proved to be particularly useful 16 to represent and reason with imperfect information in a 17 wide range of real-world applications. In such cases, evi-18 dence theory provides a flexible framework for handling and 19 mining uncertainty and imprecision as well as combining 20 evidence obtained from multiple sources and modeling the 21 conflict between them. The purpose of this paper is three-22 fold. First, it introduces the basics of the belief functions 23 theory with emphasis on the transferable belief model. Sec-24 ond, it provides a practical case study to show how the belief 25 functions theory was used in a real network application, 26 thereby providing guidelines for how the evidence theory 27

Decision trees classifiers are popular classification methods. In this paper, we extend to multi-class problems a decision tree method based on belief functions previously described for 2-class problems only. We propose two ways to achieve this extension: combining multiple 2-class trees together and directly extending the estimation of belief functions within the tree to the multi-class setting. We provide experiment results and compare them to classical decision trees.

Probability theory is far from being the most general mathematical theory of uncertainty. A number of arguments point at its inability to describe second-order (‘Knightian’) uncertainty. In response, a wide array of theories of uncertainty have been proposed, many of them generalisations of classical probability. As we show here, such frameworks can be organised into clusters sharing a common rationale, exhibit complex links, and are characterised by different levels of generality. Our goal is a critical appraisal of the current landscape in uncertainty theory.

Evidence theory, also called belief function theory, provides an efficient tool to represent and combine uncertain information for pattern classification. Evidence combination can be interpreted, in some applications, as classifier fusion. The sources of evidence corresponding to multiple classifiers usually exhibit different classification qualities, and they are often discounted using different weights before combination. In order to achieve the best possible fusion performance, a new credal belief redistribution (CBR) method is proposed to revise such evidence. The rationale of CBR consists of transferring belief from one class not just to other classes, but also to the associated disjunctions of classes (i.e., meta-classes). As classification accuracy for different objects in a given classifier can also vary, the evidence is revised according to prior knowledge mined from its training neighbors. If the selected neighbors are relatively close to the evidence, a large amount of
belief will be discounted for redistribution. Otherwise, only a small
fraction of belief will enter the redistribution procedure. An imprecision
matrix estimated based on these neighbors is employed to specifically redistribute the discounted beliefs. This matrix expresses the likelihood of misclassification (i.e., the probability of a test pattern belonging to a class different from the one assigned to it by the classifier). InCBR, the discounted beliefs are divided into two parts. One part is transferred between singleton classes, whereas the other is cautiously committed to the associated meta-classes. By doing this, one can efficiently reduce the chance of misclassification by modeling partial imprecision. The multiple revised pieces of evidence are finally combined by the Dempster–Shafer rule to reduce uncertainty and further improve classification accuracy. The effectiveness of CBR is extensively  validated on several real datasets from the UCI repository and critically compared with that of other related fusion methods.

In example-based human pose estimation, the configuration of an evolving object is sought given visual evidence, having to rely uniquely on a set of sample images. We assume here that, at each time instant of a training session, a number of feature measurements is extracted from the available images, while ground truth is provided in the form of the true object pose. In this scenario, a sensible approach consists in learning maps from features to poses, using the information provided by the training set. In particular, multi-valued mappings linking feature values to set of training poses can be constructed. To this purpose we propose a Belief Modeling Regression (BMR) approach in which a probability measure on any individual feature space maps to a convex set of probabilities on the set of training poses, in a form of a belief function. Given a test image, its feature measurements translate into a collection of belief functions on the set of training poses which, when combined, yield there an entire family of probability distributions. From the latter either a single central pose estimate or a set of extremal ones can be computed, together with a measure of how reliable the estimate is. Contrarily to other competing models, in BMR the sparsity of the training samples can be taken into account to model the level of uncertainty associated with these estimates. We illustrate BMR's performance in an application to human pose recovery, showing how it outperforms our implementation of both Relevant Vector Machine and Gaussian Process Regression. Finally, we discuss motivation and advantages of the proposed approach with respect to its most direct competitors. Index Terms Example-based pose estimation, feature-pose maps, theory of evidence, belief functions.

Conditioning is crucial in applied science when inference involving time series is involved. Belief
calculus is an effective way of handling such inference in the presence of uncertainty, but different approaches
to conditioning in that framework have been proposed in the past, leaving the matter unsettled.
We propose here an approach to the conditioning of belief functions based on geometrically projecting
them onto the simplex associated with the conditioning event in the space of all belief functions. We show
here that such a geometric approach to conditioning often produces simple results with straightforward
interpretations in terms of degrees of belief. The question of whether classical approaches, such as for
instance Dempster’s conditioning, can also be reduced to some form of distance minimization remains
open: the study of families of combination rules generated by (geometric) conditioning rules appears to
be the natural prosecution of the presented research.

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters.

Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

The theory of belief functions, sometimes referred to as evidence theory or Dempster-Shafer theory, was first introduced by Arthur P. Dempster in the context of statistical inference, to be later developed by Glenn Shafer as a general framework for modelling epistemic uncertainty. The methodology is now well established as a general framework for reasoning with uncertainty, with well-understood connections to related frameworks such as probability, possibility, random set and imprecise probability theories. Importantly, in recent years the number of papers published on the theory and application of belief functions has been booming (reaching over 800 in 2014 alone), displaying strong growth in particular in the East Asian community and among practitioners working on multi-criteria decision making, earth sciences, and sensor fusion. Belief functions are a natural tool to cope with heavy uncertainty, lack of evidence and missing data, and extremely rare events.

An early debate on the rationale of belief functions gave a strong contribution to the growth and success of the UAI community and series of conference in the Eighties and Nineties, thanks to the contribution of scientists of the caliber of Glenn Shafer, Judea Pearl, Philippe Smets and Prakash Shenoy, among others. Ever since the UAI and BELIEF community have somewhat diverged, and the proposers’ effort has been recently directed towards going back to a closer relationships and exchange of ideas between the two communities. This was one of the aims of the recent BELIEF 2014 International Conference of which the proposers were General Chair and member of the Steering Committee, respectively. A number of books are being published on the subject as we speak, and the impact of the belief function approach to uncertainty is growing.

The tutorial aims at bridging the gap between researchers in the field and the wider AI and Uncertainty Theory community, with the longer term goal of a more fruitful collaboration and dissemination of ideas.

La résolution des requêtes Top-k est un problème important et difficile dans les systèmes large échelle. Cette résolution se fait par agrégation afin d'optimiser la recherche et de manipuler moins de quantités de données. Dans cet article, nous présentons un nouvel algorithme dans le but de résoudre ce type de problème efficacement. Notre algorithme est composé de trois phases et ne définit qu'un seul seuil, à la différence des autres algorithmes déjà existants. Les simulations sur des collections de données réelles et la comparaison avec des algorithmes très connus montrent que notre solution est très compétitive et plus rapide en temps d'exécution.

This special issue presents articles submitted in response to calls for papers in the domains of public health and of comparisons of fusion methods in real-world applications. The fruits of the two searches were combined for presentation under the new rubric, in one issue. We aim to increase awareness of and involvement in the discipline of information fusion among researchers who are unfamiliar with the discipline and might benefit from new approaches. We also hope to refocus the attention of those
with experience and expertise in information fusion on new application areas, in the hopes that new partnerships and collaborations might result across disciplines..

Random set theory, originally born within the remit of mathematical statistics, lies nowadays at the interface of statistics and AI. Arguably more mathematically complex than standard probability, the field is now facing open issues such as the formulation of generalised laws of probability, the generalisation of the notion of random variable to random set spaces, the extension of the notion of random process, and so on. Frequentist inference with random sets can be envisaged to better describe common situations such as lack of data and set-valued observations. To this aim, parameterised families of random sets (and Gaussian random sets in particular) are a crucial area of investigation. In particular, we will present some recent work on the generalisation of the notion of likelihood, as the basis for a generalised logistic regression framework capable to better estimate rare events; a random set-version of maximum-entropy classifiers; and a recent generalisation of the law of total probability to belief functions. In a longer-term perspective, random set theory can be instrumental to new robust foundations for statistical machine learning allowing the formulation of models and algorithms able to deal with mission-critical applications ‘in the wild’, in a mutual beneficial exchange between statistics and artificial intelligence.

This paper extends the decision tree technique to an uncertain environment where the uncertainty is represented by belief functions as interpreted in the Transferable Belief Model (TBM). This so-called belief decision tree is a new classification method adapted to uncertain data. We will be concerned with the construction of the belief decision tree from a training set where the knowledge about the instances' classes is represented by belief functions, and its use for the classification of new instances where the knowledge about the attributes' values is represented by belief functions. Keywords: Belief Functions, Decision Tree, Belief Decision Tree, Classification, Transferable Belief Model. 1 Introduction Several learning methods have been developed to ensure classification. Among these, the decision tree method may be one of the most commonly used in supervised learning approaches. Indeed decision trees are characterized by their capability to break down a complex decis...

In this Book we argue that the fruitful interaction of computer vision and belief calculus is capable of stimulating significant advances in both fields. From a methodological point of view, novel theoretical results concerning the geometric and algebraic properties of belief functions as mathematical objects are illustrated and discussed in Part II, with a focus on both a perspective 'geometric approach' to uncertainty and an algebraic solution to the issue of conflicting evidence. In Part III we show how these theoretical developments arise from important computer vision problems (such as articulated object tracking, data association and object pose estimation) to which, in turn, the evidential formalism is able to provide interesting new solutions. Finally, some initial steps towards a generalization of the notion of total probability to belief functions are taken, in the perspective of endowing the theory of evidence with a complete battery of estimation and inference tools to the benefit of all scientists and practitioners.

The notion of belief likelihood function of repeated trials is introduced, whenever the uncertainty for individual trials is encoded by a belief measure (a finite random set). This gen-eralises the traditional likelihood function, and provides a natural setting for belief inference from statistical data. Factorisation results are proven for the case in which conjunctive or disjunctive combination are employed, leading to analytical expressions for the lower and upper likelihoods of 'sharp' samples in the case of Bernoulli trials, and to the formulation of a generalised logistic regression framework.

This special issue of the International Journal of Approximate Reasoning (IJAR) collects a number of significant papers published at the 3rd International Conference on Belief Functions (BELIEF 2014).
The series of biennial BELIEF conferences, organized by the Belief Functions and Applications Society (BFAS), is dedicated to the confrontation of ideas, the reporting of recent achievements, and the presentation of the wide range of applications of this theory. The series started in Brest, France, in 2010, while the second edition was held in Compiègne, France, in May 2012. The upcoming BELIEF 2016 will take place in Prague in September 2016.

In this work we consider the case of the ranking aggregation problem that includes the true ranking in its formulation. The goal is to find an estimation of an unknown true ranking given a set of rankings provided by different quality experts. This is the case when bioinformatic experts provide ranked items involved in an unknown biological phenomenon regulated by its own physical reality. We devise an innovative solution called Belief Ranking Estimator (BRE), based on the belief function framework that permits to represent beliefs on the correctness of each item rank as well as uncertainty on the quality of the rankings from the subjective point of view of the expert. Moreover, weights computed using a true-ranking estimator are applied to the original belief basic assignment in order to take into account the quality of the input rankings. The results of an empirical comparison of BRE with weighting schema against competitor methods for ranking aggregation show that our method improves significantly the performance when the quality of the ranking is heterogeneous.

In this paper we discuss the semantics and properties of the relative belief transform, a probability transformation of belief functions closely related to the classical plausibility transform. We discuss its rationale in both the probability-bound and Shafer's interpretations of belief functions. Even though the resulting probability (as it is the case for the plausibility transform) is not consistent with the original belief function, an interesting rationale in terms of optimal strategies in a non-cooperative game can be given in the probability-bound interpretation to both relative belief and plausibility of singletons. On the other hand, we prove that relative belief commutes with Dempster's orthogonal sum, meets a number of properties which are the duals of those met by the relative plausibility of singletons, and commutes with convex closure in a similar way to Dempster's rule. This supports the argument that relative plausibility and belief transform are indeed naturally associated with the D-S framework, and highlights a classification of probability transformations in two families, according to the operator they relate to. Finally, we point out that relative belief is only a member of a class of \relative mass" mappings, which can be interpreted as low-cost proxies for both plausibility and pignistic transforms.

In this paper we propose a generalised maximum-entropy classification framework, in which the empirical expectation of the feature functions is bounded by the lower and upper expectations associated with the lower and upper probabilities associated with a belief measure. This generalised setting permits a more cautious appreciation of the information content of a training set. We analytically derive the Karush-Kuhn-Tucker conditions for the generalised max-entropy classifier in the case in which a Shannon-like entropy is adopted.

In this paper we extend our geometric approach to the theory of evidence in order to include other important classes of finite fuzzy measures. In particular we describe the geometric counterparts of possibility measures or fuzzy sets, represented as consonant belief functions. The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of simplicial complex.

In example-based pose estimation, the configuration or “pose” of an evolving object is sought given visual evidence,
having to rely uniquely on a set of examples. In a training stage, a number of features are extracted from the available images and ground truth poses are acquired. In this scenario, a sensible approach consists in learning maps from features to poses, using the training data. In particular, multi-valued mappings linking feature values to a set of training poses can be easily constructed: in this paper, we propose to use these mappings for pose estimation. In the proposed method, a probability measure on any feature space is naturally mapped to a convex set of probabilities on the set of training poses, in a form of a “belief function”.

Given a test image, its features translate into a collection of
belief functions on the set of training poses, which when combined yield there an entire family of probability distributions. From the latter, both a single, central pose estimate and a set of extremal
estimates can be computed, together with a measure of how
reliable the estimate is. We call this technique “Belief Modeling
Regression”. We demonstrate its effectiveness by comparing it
to popular mapping techniques such as Gaussian Process and
Relevance Vector Regression under a varied set of experiments.

"""The present work presents a general theoretical framework for the study of operators which merge partial probabilistic evidence from different sources which are individually coherent, but may be collectively incoherent. We consider a number of principles for such an operator to satisfy including a set of principles derived from those of Konieczny and Pino Perez [11] which were formulated for the different context of propositional merging.

Finally we investigate two specific such merging operators derived from the Kullback-Leibler notion of informational distance: the social entropy operator, and its dual, the linear entropy operator. The first of these is strongly related to both the multi-agent normalised  geometric mean pooling operator and the single agent maximum entropy inference process,
ME.  By contrast the linear entropy operator is similarly related to both the arithmetic mean pooling operator and the limit centre of mass inference process, CM-infinity."""

In this paper, we analyze Shafer’s belief functions (BFs) as geometric entities, focusing in particular on the geometric behavior of Dempster’s rule of combination in the belief space, i.e., the set of all the admissible BFs defined over a given finite domain Theta. The study of the orthogonal sums of affine subspaces allows us to unveil a convex decomposition of Dempster’s rule of combination in terms of Bayes’ rule of conditioning and prove that under specific conditions orthogonal sum and affine closure commute. A direct consequence of these results is the simplicial shape of the conditional subspaces , i.e., the sets of all the possible combinations of a given BF s. We show how Dempster’s rule exhibits a rather elegant behavior when applied to BFs assigning the same mass to a fixed subset (constant mass loci). The resulting affine spaces have a common intersection that is characteristic of the conditional subspace, called focus. The affine geometry of these foci eventually suggests an interesting geometric construction of the orthogonal sum of two BFs.

—When several networks (e.g., Wi-Fi, UMTS, and LTE) cover the same region, the mobile terminals that are equipped with multiple network interfaces provide the possibility for mobile end-users to select their believed best network. This is known as the network selection problem, which is a decision making problem with multiple criteria (network conditions, service requirements, terminal characteristics, and user needs). Many network selection solutions using different mathematical theories have been proposed in the literature to allow the best connectivity for applications, users, and terminals. Unfortunately, most approaches for the network selection do not make effective selection decisions, since they are vulnerable to the uncertainty and imprecision related to network state information. In this paper, we investigate the belief functions theory to devise an efficient lightweight uncertainty-aware network interface selection scheme. We provide analytical studies and simulation experiments to demonstrate the efficiency of the proposed solution.

Nous présentons dans cet article, en premier lieu, une étude comparative de trois théories mathématiques permettant de
réaliser la fusion des sources d’information. Cette étude vise à dégager les caractéristiques inhérentes aux théories des
possibilités, des fonctions de croyance (TFC) et du raisonnement plausible et paradoxal (DSmT) afin d’établir une stratégie
de choix qui nous permet d’adopter la théorie la plus adéquate pour résoudre un problème de fusion en tenant compte
des informations acquises et des imperfections qui les accompagnent. En deuxième lieu, nous présentons également une
nouvelle méthode pour déterminer les éléments focaux paradoxaux pertinents à l’aide des histogrammes de niveaux de gris
et qui permet aussi de réduire la cardinalité de l’hyper-puissance. Ainsi, nous avons élaboré une nouvelle technique pour
ordonner et coder les éléments focaux généralisés. Cette méthode est exploitée, en particulier, pour calculer la cardinalité
de Dezert et Smarandache.

A category of learning problems in which the class membership of training patterns is assessed by an expert and encoded in the form of a possibility distribution is considered. Each example i thus consists in a feature vector x i and a possibilistic label (u i 1 ; : : : ; u i c ), where u i k denotes the possibility of that example belonging to class k. This problem is tackled in the framework of Evidence Theory. The evidential distance-based classiÿer previously introduced by one of the authors is extended to handle possibilistic labeling of training data. Two approaches are proposed, based either on the transformation of each possibility distribution into a consonant belief function, or on the use of generalized belief structures with fuzzy focal elements. In each case, a belief function modeling the expert's beliefs concerning the class membership of each new pattern is obtained. This information may then be either interpreted by a human operator to support decision-making, or automatically processed to yield a ÿnal class assignment through the computation of pignistic probabilities. Experiments with synthetic and real data demonstrate the ability of both classiÿcation schemes to make e ective use of possibilistic labels as training information.

In this paper we extend our geometric approach to the theory of evidence
in order to include other important classes of nite fuzzy measures.
In particular we describe the geometric counterparts of possibility measures or fuzzy sets, represented as consonant belief functions.
The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions
in terms of the notion of simplicial complex.

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.

The theory of belief functions, sometimes referred to as evidence theory or Dempster-Shafer theory, was first introduced by Arthur P. Dempster in the context of statistical inference, to be later developed by Glenn Shafer as a general framework for modelling epistemic uncertainty. Belief theory and the closely related random set theory form natural frameworks for modelling situations in which data are missing or scarce: think of extremely rare events such as volcanic eruptions or power plant meltdowns, problems subject to huge uncertainties due to the number and complexity of the factors involved (e.g. climate change), but also the all-important issue with generalisation from small training sets in machine learning. This tutorial is designed to introduce the principles and rationale of random sets and belief function theory to mainstream statisticians, mathematicians and working scientists, survey the key elements of the methodology and the most recent developments, make practitioners aware of the set of tools that have been developed for reasoning in the belief function framework on real-world problems. Attendees will acquire first-hand knowledge of how to apply these tools to significant problems in major application fields such as computer vision, climate change, and others. A research programme for the future of random set theory and high impact applications is eventually outlined.

The notion of belief likelihood function of repeated trials is introduced, whenever the uncertainty for individual trials is encoded by a belief measure (a finite random set). This generalises the traditional likelihood function, and provides a natural setting for belief inference from statistical data. Factorisation results are proven for the case in which conjunctive or disjunctive combination are employed, leading to analytical expressions for the lower and upper likelihoods of `sharp' samples in the case of Bernoulli trials, and to the formulation of a generalised logistic regression framework.

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