Dempster-Shafer Analysis

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.

This paper presents a basic tutorial on epistemic uncertainty quantification methods. Epistemic uncertainty, characterizing lack-of-knowledge, is often prevalent in engineering applications. However, the methods we have for analyzing and propagating epistemic uncertainty are not as nearly widely used or well-understood as methods to propagate aleatory uncertainty (e.g. inherent variability characterized by probability distributions). We examine three methods used in propagating

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'.

Existing research in association mining has focused mainly on how to expedite the search for frequently co-occurring groups of items in “shopping cart” type of transactions; less attention has been paid to methods that exploit these “frequent itemsets” for prediction purposes. This paper contributes to the latter task by proposing a technique that uses partial information about the contents of a shopping cart for the prediction of what else the customer is likely to buy. Using the recently proposed data structure of itemset trees (IT-trees), we obtain, in a computationally efficient manner, all rules whose antecedents contain at least one item from the incomplete shopping cart. Then, we combine these rules by uncertainty processing techniques, including the classical Bayesian decision theory and a new algorithm based on the Dempster-Shafer (DS) theory of evidence combination.

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.

Nowadays ontologies present a growing interest in Data Fusion applications. As a matter of fact, the ontologies are seen as a semantic tool for describing and reasoning about sensor data, objects, relations and general domain theories. In addition, uncertainty is perhaps one of the most important characteristics of the data and information handled by Data Fusion. However, the fundamental nature of ontologies implies that ontologies describe only asserted and veracious facts of the world. Different probabilistic, fuzzy and evidential approaches already exist to fill this gap; this paper recaps the most popular tools. However none of the tools meets exactly our purposes. Therefore, we constructed a Dempster-Shafer ontology that can be imported into any specific domain ontology and that enables us to instantiate it in an uncertain manner. We also developed a Java application that enables reasoning about these uncertain ontological instances.

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.

Cat is one of the favourite pets in Indonesia, but there are some people who have abandoned their cats. Some of the reason is because their cats suffered from skin diseases. Skin diseases in Indonesia are commonly due to Ear Mites, Flea, Ringworm and Scabies. Along with the development of technology, Artificial Intelligence (AI) is needed to assist in the diagnosis process of skin diseases in cats like by using an expert system. To determine the best method that is used to build this system the comparative analysis method is carried out. The methods that can be used are Certainty Factor and Dempster-Shafer methods. Based on the analysis, earned accuracy value is 86.67% and error system is 13.33% in both methods. Despite having the results of the same accuracy and error, the average results of the two methods of diagnosis have a considerable margin of 10.9%. From the confusion matrix test results obtained accuracy value of 98.33%, precision value of 100%, and recall value of 93.33%.

In this talk, I further develop the idea of using Dempster-Shafer theory as a basis of Machine Learning. In particular, I propose one way of representing Machine Learning Computations as Dempster-Shafer sets. Finally, I extend the representation to build a classifier.

Context-sensing for context-aware HCI challenges the traditional sensor fusion methods with dynamic sensor configuration and measurement requirements commensurate with human perception. The Dempster-Shafer theory of evidence has uncertainty management and ...

This paper addresses classification problems in which the class membership of training data is only partially known. Each learning sample is assumed to consist in a feature vector and an imprecise and/or uncertain “soft” label m i defined as a Dempster-Shafer basic belief assignment over the set of classes. This framework thus generalizes many kinds of learning problems including supervised, unsupervised and semi-supervised learning. Here, it is assumed that the feature vectors are generated from a mixture model. Using the General Bayesian Theorem, we derive a criterion generalizing the likelihood function. A variant of the EM algorithm dedicated to the optimization of this criterion is proposed, allowing us to compute estimates of model parameters. Experimental results demonstrate the ability of this approach to exploit partial information about class labels.