The geometry of relative plausibilities

2006

In this work we extend the geometric approach to the theory of evidence in order to study the geometric behavior of the two quantities inherently associated with a belief function. i.e. the plausibility and commonality functions. After introducing the analogous of the basic probability assignment for plausibilities and commonalities, we exploit it to understand the simplicial form of both plausibility and commonality spaces. Given the intuition provided by the binary case we prove the congruence of belief, plausibility, and commonality spaces for both standard and unnormalized belief functions, and describe the point-wise geometry of these sum functions in terms of the rigid transformation mapping them onto each other. This leads us to conjecture that the D-S formalism may be in fact a geometric calculus in the line of geometric probability, and opens the way to a wider application of discrete mathematics to subjective probability.

2021

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 epistemic uncertainty – unfortunately, different approaches to conditioning in the belief function framework have been proposed in the past, leaving the matter somewhat unsettled. Inspired by the geometric approach to uncertainty, in this paper we propose 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. This raises the question of whether classical approaches, such as for instance Dempster’s conditioning, can also be reduced to some form of distance minimisation in a suitable space. The study of families of combination rul...

2011

In this paper we study the problem of conditioning a belief function (b.f.) b with respect to an event A by geometrically projecting such belief function onto the simplex associated with A in the space of all belief functions. Deflning geometric conditional b.f.s by minimizing Lp distances between b and the conditioning simplex in such \belief" space (rather than in the \mass" space) produces complex results with less natural interpretations in terms of degrees of belief. The question of weather classical approaches, such as Dempster’s conditioning, can be themselves reduced to some form of distance minimization remains open: the generation of families of combination rules generated by (geometrical) conditioning appears to be the natural prosecution of this line of research.

2005

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.

ANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCE, 2010

Probability transformation of belief functions can be classified into different families, according to the operator they commute with. In particular, as they commute with Dempster's rule, relative plausibility and belief transforms form one such "epistemic" family, and possess natural rationales within Shafer's formulation of the theory of evidence. However, the relative belief transform only exists when some mass is assigned to singletons. We show here that relative belief is only a member of a class of "relative mass" mappings, which can be interpreted as lowcost proxies for both plausibility and pignistic transforms.

In this paper we study the problem of conditioning a belief function (b.f.) b with respect to an event A by geometrically projecting such belief function onto the simplex associated with A in the space of all belief functions. Defining geometric conditional b.f.s by minimizing L p distances between b and the conditioning simplex in such "belief" space (rather than in the "mass" space) produces complex results with less natural interpretations in terms of degrees of belief. The question of weather classical approaches, such as Dempster's conditioning, can be themselves reduced to some form of distance minimization remains open: the generation of families of combination rules generated by (geometrical) conditioning appears to be the natural prosecution of this line of research.

Proceedings of ISIPTA, 2003

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.

Arxiv preprint, arXiv:1810.10341, 2018

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.

Belief Functions: Theory and Applications, 2018

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. Believability measures for unnormalised belief functions are also considered. A research programme to complete this analysis is outlined.

Logics in Artificial Intelligence, 2008

2014

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.

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.

Proceedings of BELIEF 2010

"Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an e ffective way of handling such inference in the presence of uncertainty, but di fferent 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 di fferent 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."

2000

We discuss the justifications of Bayesianism by Cox and Jaynes, and relate them to a recent critique by Halpern(JAIR, vol 10(1999), pp 67–85). We show that a problem with Halperns example is that a finite and natural refinement of the model leads to inconsistencies, and that the same is the case with every model in which rescalability to probability cannot be done. We also discuss other problems with the justifications and assumptions usually made on the function F describing plausibility of conjunction. We note that the commonly postulated monotonicity condition should be strengthened to strict monotonicity before Cox justification becomes convincing. On the other hand, we note that the commonly assumed regularity requirements on F (like continuity) or its domain (like denseness) are unnecessary.

submitted to the International Journal of …, 2007