Belief Functions

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.

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.

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, Dempter'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)dependency, and only consider such assumptions as possible additional information.

In this paper, we analyze from a geometric perspective the meaningful relations taking place between belief and probability functions in the framework of the geometric approach to the theory of evidence. Starting from the case of binary domains, we identify and study three major geometric entities relating a generic belief function (b.f.) to the set of probabilities P: 1) the dual line connecting belief and plausibility functions; 2) the orthogonal complement of P; and 3) the simplex of consistent probabilities. Each of them is in turn associated with a different probability measure that depends on the original b.f. We focus in particular on the geometry and properties of the orthogonal projection of a b.f. onto P and its intersection probability, provide their interpretations in terms of degrees of belief, and discuss their behavior with respect to affine combination.

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 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.

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 (MCS), 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. Extensions and properties of the basic fusion rule are also studied.

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.

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.

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.

In this paper we extend the geometric approach to the theory of evidence in order
to include the class of necessity measures, represented on a finite domain of “frame”
by consonant belief functions (b.f.s). The correspondence between chains of subsets
and convex sets of b.f.s 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 particular we focus on the set of outer consonant
approximations of a belief function, showing that for each maximal chain of subsets
these approximations form a polytope. The maximal such approximation with
respect to the weak inclusion relation between b.f.s is one of the vertices of this
polytope, and is generated by a permutation of the elements of the frame.

The paper builds a belief hierarchy as a common framework to all uncertainty measures for which an actor is ambiguous about his uncertain beliefs. The belief hierarchy is moreover interpreted by distinguishing physical and psychical worlds, associated to objective and subjective probabilities. Various principles are introduced in order to transform a belief hierarchy, which are applied to establish some links between different contexts of belief change. The numerous belief change rules proposed in the literature can then be associated to specific belief hierarchies and to specific contexts of belief change and epistemic justification of these rules given. The papers brings new and original analysis of the relation between subjective and objective beliefs and on the possibility of learning some generic probabilistic belief despite its resiliency.

In this paper we propose a credal representation of the interval probability associated with a belief function (b.f.), and show how it relates to several classical Bayesian transformations of b.f.s through the notion of “focus” of a pair of simplices.
While a belief function corresponds to a polytope of probabilities consistent with it, the related interval probability is geometrically represented by a pair of upper and lower simplices. Starting from the interpretation of the pignistic function as the center of mass of the credal set of consistent probabilities, we prove that relative belief of singletons, relative plausibility of singletons and intersection probability can all be described as foci of different pairs of simplices in the region of all probability measures. The formulation of frameworks similar to the Transferable Belief Model for such Bayesian transformations appears then at hand.

In this paper the geometric structure of the space S_Theta of the belief functions defined over a discrete set Theta (belief space) is analyzed. Using the Moebius inversion lemma we prove the recursive bundle structure of the belief space and show how an arbitrary belief function can be uniquely represented as a convex combination of certain elements of the fibers, giving $\mathcal{S}$ the form of a simplex. The commutativity of orthogonal sum and convex closure operator is proved and used to depict the geometric structure of conditional subspaces, i.e. sets of belief functions conditioned by a given function s.
Future applications of this geometric methods to classical problems like probabilistic approximation and canonical decomposition are outlined.

In this paper we adopt the geometric approach to the theory of evidence to
study the geometric counterparts of the plausibility functions, or upper probabilities.
The computation of the coordinate change between the two natural
reference frames in the belief space allows us to introduce the dual notion
of basic plausibility assignment and understand its relation with the classical
basic probability assignment. The convex shape of the plausibility space P is
recovered in analogy to what was done for the belief space, and the pointwise geometric relation between a belief function and the corresponding plausibility vector is discussed. The orthogonal projection of an arbitrary belief
function s onto the probabilistic subspace is computed and compared with
other significant entities, such as the relative plausibility and mean probability
vectors.

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.

The study of finite non-additive measures or “belief functions” has been recently posed in connection with combinatorics and convex geometry. As a matter of fact, as belief functions are completely specified by the associated belief values on the events of the frame on which they are defined, they can be represented as points of a Cartesian space. The space of all belief functions B or “belief space” is a simplex whose vertices are BF focused on single events. In this paper we present an alternative description of the space of belief functions in terms of differential geometric notions. The belief space possesses indeed a recursive bundle structure inherently related to the mass assignment mechanism, in which basic probability is recursively assigned to events of increasing size. A formal proof of the decomposition of B together with a characterization of bases and fibers as simplices are provided.

In this paper we prove that a recent Bayesian approximation of belief functions, the relative belief of singletons, meets a number of properties with respect to Dempster's rule of combination which mirrors those satisfied by the relative plausibility of singletons. In particular, its operator commutes with Dempster's sum of plausibility functions, while perfectly representing a plausibility function when combined through Dempster's rule. This suggests a classification of all Bayesian approximations into two families according to the operator they relate to.

In this paper we investigate the properties of the relative plausibility function, the probability built by normalizing the plausibilities of singletons associated with a belief function. On one side, we stress how this probability is a perfect representative of the original belief function when combined with any arbitrary probability through Dempster's rule. This leads to conjecture that this function should also be the solution of the probabilistic approximation problem, formulated naturally in terms of Dempster's rule. On the other side, the geometric properties of relative plausibilities are studied in the context of the geometric approach to the theory of evidence, yielding a description of the representation property which suggests a sketch for the general proof of our conjecture.

"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. Two different such simplices can be defined, as each belief function can be represented as either the vector of its basic probability values or the vector of its belief values. 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."

In this paper we introduce three alternative combinatorial formulations of the theory of evidence (ToE), by proving that both plausibility and commonality functions share the structure of \sum function" with belief functions. We compute their Moebius inverses, which we call basic plausibility and commonality assignments. In the framework of the geometric approach to uncertainty measures the equivalence of the associated formulations of the ToE is mirrored by the geometric congruence of the related simplices. We can then describe the point-wise geometry of these sum functions in terms of rigid transformations mapping them onto each other. Combination rules can be applied to plausibility and commonality functions through their Moebius inverses, leading to interesting applications of such inverses to the probabilistic transformation problem.

The problem of conflict measurement between information sources knows a regain of interest. In most works related to this issue, Dempter's rule plays a central role. In this paper, we propose to revisit conflict from a different perspective. We do not make a priori assumption about dependencies and start from the definition of conflicting sets, studying its possible extensions to the framework of belief functions.

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.

Grouping 3D-objects into (semantically) meaningful categories is a challenging and important problem in 3D-mining and shape processing. Here, we present a novel approach to categorize 3D-objects. The method described in this paper, is a belief function based approach and consists of two stages. The training stage, where 3D-objects in the same category are processed and a set of representative parts is constructed, and the labeling stage, where unknown objects are categorized. The experimental results obtained on the Tosca-Sumner and the Shrec07 datasets show that the system efficiently performs in categorizing 3D-models.

In this paper we discuss the problem of approximating a belief function (b.f.) with a necessity measure or \consonant belief function" (co.b.f.) from a geometric point of view. We focus in particular on outer consonant approximations, i.e. co.b.f.s less committed than the original b.f. in terms of degrees of belief. We show that for each maximal chain of focal elements the set of outer consonant approximation is a polytope.
We describe the vertices of such polytope, and characterize the geometry
of maximal outer approximations.

In this paper we build on previous work on the geometry of Dempster's rule to investigate the geometric behaviour of various other combination rules, including Yager's, Dubois', and disjunctive combination , starting from the case of binary frames of discernment. Believabil-ity measures for unnormalised belief functions are also considered. A research programme to complete this analysis is outlined.

Keywords: Geometry · Yager's and Dubois' combination · conjunctive and disjunctive combination · unnormalised belief functions.

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..

As Dempster-Shafer theory spreads in different applications fields involving complex systems, the need for algorithms randomly generating mass functions arises. As such random generation is often perceived as secondary, most proposed algorithms use procedures whose sample statistical properties are difficult to characterize. Thus, although they produce randomly generated mass functions, it is difficult to control the sample statistical laws. In this paper, we briefly review classical algorithms, explaining why their statistical properties are hard to characterize, and then provide simple procedures to perform efficient and controlled random generation. Thomas Burger: CNRS (FR3425), CEA (iRTSV/BGE), INSERM (

"""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."""

Object tracking consists of reconstructing the configuration of an articulated body from a sequence of images provided by one or more cameras. In this paper we present a general method for pose estimation based on the evidential reasoning. The proposed framework integrates different levels of description of the object to improve robustness and precision, overcoming the limitations of approaches using single-feature representations. Several image descriptions extracted from a single-camera view are fused together using the Dempster-Shafer ”theory of evidence”. Feature data are expressed as belief functions over the set of their possibile values. There is no need of any a-priori assumptions about the model of the object. Learned refinement maps between feature spaces and the parameter space Q describing the configuration of the object characterize the relationships among distinct representations of the pose and play the role of the model. During training the object follows a sample trajectory in Q. Each feature space is reduced to a discrete frame of discernment (FOD) and refinements are built by mapping these FODs into subsets of the sample trajectory. During tracking new sensor data are converted to belief functions which are projected and combined in the approximate state space. Resulting degrees of belief indicate the best pose estimate at the current time step. The choice of a sufficiently dense (in a topological sense) sample trajectory is a critical problem.
Experimental results concerning a simple tracking system are shown.

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.

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 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.

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.