Geometry of relative plausibility and relative belief of singletons

Cuzzolin, Fabio

doi:10.1007/s10472-010-9186-x

Cited by 10 publications

(11 citation statements)

References 27 publications

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“…is the barycenter of the credal set P[b] associated with b. Although other transforms have been proposed [45], [48], [46], [47], empirically their performances in the human pose estimation experiments presented here have been proven comparable. C. Assessing evidential models 1) Robustness: as a consequence of computing the belief estimate by conjunctive combination, a non-zero mass may be assigned to the empty set.…”

Section: )mentioning

confidence: 92%

A Belief-Theoretical Approach to Example-Based Pose Estimation

Gong

Cuzzolin

2018

IEEE Trans. Fuzzy Syst.

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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.

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Section: )mentioning

confidence: 92%

A Belief-Theoretical Approach to Example-Based Pose Estimation

Gong

Cuzzolin

2018

IEEE Trans. Fuzzy Syst.

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“…The geometry of the relationship between measures of different kinds has also been extensively studied [50,30,17,31], with particular attention to the problem of transforming a belief function into a classical probability measure [9,112,99] (see Section 3). One can distinguish between an affine family of probability transformations [20] (those which commute with affine combination in the belief space), and an epistemic family of transforms [19], formed by the relative belief and relative plausibility of singletons [28,27,37,46,33], which possess dual properties with respect to Dempster's sum [24]. The problem of finding the possibility measure which best approximates a given belief function [2] can also be approached in geometric terms [29,47,39,40].…”

Section: The Geometry Of Uncertainty Measuresmentioning

confidence: 99%

“…plausibility of singletons and belief of singletons, respectively [27,24,33,37], for sake of consistency of nomenclature. However, P l is traditionally referred to as the contour function.…”

Section: Justification For the Namementioning

confidence: 99%

The intersection probability: betting with probability intervals

Cuzzolin¹

2022

Preprint

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Probability intervals are an attractive tool for reasoning under uncertainty. Unlike belief functions, though, they lack a natural probability transformation to be used for decision making in a utility theory framework. In this paper we propose the use of the intersection probability, a transform derived originally for belief functions in the framework of the geometric approach to uncertainty, as the most natural such transformation. We recall its rationale and definition, compare it with other candidate representives of systems of probability intervals, discuss its credal rationale as focus of a pair of simplices in the probability simplex, and outline a possible decision making framework for probability intervals, analogous to the Transferable Belief Model for belief functions.

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“…The geometry of the relationship between measures of different kinds has also been extensively studied [51,31,18,32], with particular attention to the problem of transforming a belief function into a classical probability measure [11,142,129]. One can distinguish between an 'affine' family of probability transformations [21] (those which commute with affine combination in the belief space), and an 'epistemic' family of transforms [20], formed by the relative belief and relative plausibility of singletons [28,29,38,47,34], which possess dual properties with respect to Dempster's sum [25]. Semantics for the main probability transforms can be provided in terms of credal sets, i.e., convex sets of probabilities [32].…”

Section: Geometry Approach To Uncertaintymentioning

confidence: 99%

A geometric approach to conditioning belief functions

Cuzzolin

2021

Preprint

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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 rules generated by (geometric) conditioning rules appears to be the natural prosecution of the presented research.

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Geometry of relative plausibility and relative belief of singletons

Cited by 10 publications

References 27 publications

A Belief-Theoretical Approach to Example-Based Pose Estimation

A Belief-Theoretical Approach to Example-Based Pose Estimation

The intersection probability: betting with probability intervals

A geometric approach to conditioning belief functions

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