Belief functions: past, present and future (original) (raw)
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Belief functions (Random sets) for the working scientist - A IJCAI 2016 Tutorial
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
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.
Reasoning with random sets: An agenda for the future
arXiv:2401.09435, 2023
In this paper, we discuss a potential agenda for future work in the theory of random sets and belief functions, touching upon a number of focal issues: the development of a fully-fledged theory of statistical reasoning with random sets, including the generalisation of logistic regression and of the classical laws of probability; the further development of the geometric approach to uncertainty, to include general random sets, a wider range of uncertainty measures and alternative geometric representations; the application of this new theory to high-impact areas such as climate change, machine learning and statistical learning theory.
Belief functions: Theory and applications (BELIEF 2014)
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.
International Journal of Finance, Entrepreneurship & Sustainability
The main purpose of this article is to introduce the Dempster-Shafer (DS) Theory of Belief Functions. The DS Theory is founded on the mathematical theory of probability and is a broader framework than the probability theory. It reduces to probability theoryunder a special condition. In addition, the article illustrates problems in representing pure positive, pure negative evidence, and ambiguity under probability theory and shows how this problem is resolved under DS Theory. Next, the article describes and illustrates Dempster’s rule of combination to combine two or more items of evidence. Also, the article introduces Evidential Reasoning approach and its applications to various disciplines such as accounting, auditing, information systems, and information quality. Examples are provided where DS Theory is being used for developing AI and Expert systems within the business disciplines and outside of the business 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.