The total belief theorem

In this paper, motivated by the treatment of conditional constraints in the data association problem, we state and prove the generalisation of the law of total probability to belief functions, as finite random sets. Our results apply to the case in which Dempster’s conditioning is employed. We show that the solution to the resulting total belief problem is in general not unique, whereas it is unique when the a-priori belief function is Bayesian. Examples and case studies underpin the theoretical contributions. Finally, our results are compared to previous related work on the generalisation of Jeffrey’s rule by Spies and Smets

Total belief theorem and conditional belief functions

In this paper new theoretical results for reasoning with belief functions are obtained and discussed. After a judicious decomposition of the set of focal elements of a belief function, we establish the Total Belief Theorem (TBT) which is the direct generalization of the Total Probability Theorem when working in the framework of belief functions. The TBT is also generalized for dealing with different frames of discernments thanks to Cartesian product space. From TBT, we can derive and define formally the expressions of conditional belief functions which are consistent with the bounds of imprecise conditional probability. This work provides a direct establishment and solid justification of Fagin-Halpern belief conditioning formulas. The well-known Bayes' Theorem of Probability Theory is then generalized in the framework of belief functions and we illustrate it with an example at the end of this paper

Une formulation simplifiée du théorème de Bayes généralisé

International audienceIn this paper we present a simple formulation of the Generalized Bayes' Theorem (GBT) which extends Bayes' theorem in the framework of belief functions. We also present the condition under which this new formulation is valid. We illustrate our theoretical results with simple examples

Reasoning with random sets: An agenda for the future

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.Comment: 94 pages, 17 figure

Advances and Applications of DSmT for Information Fusion. Collected Works, Volume 5

This fifth volume on Advances and Applications of DSmT for Information Fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics, and is available in open-access. The collected contributions of this volume have either been published or presented after disseminating the fourth volume in 2015 in international conferences, seminars, workshops and journals, or they are new. The contributions of each part of this volume are chronologically ordered. First Part of this book presents some theoretical advances on DSmT, dealing mainly with modified Proportional Conflict Redistribution Rules (PCR) of combination with degree of intersection, coarsening techniques, interval calculus for PCR thanks to set inversion via interval analysis (SIVIA), rough set classifiers, canonical decomposition of dichotomous belief functions, fast PCR fusion, fast inter-criteria analysis with PCR, and improved PCR5 and PCR6 rules preserving the (quasi-)neutrality of (quasi-)vacuous belief assignment in the fusion of sources of evidence with their Matlab codes. Because more applications of DSmT have emerged in the past years since the apparition of the fourth book of DSmT in 2015, the second part of this volume is about selected applications of DSmT mainly in building change detection, object recognition, quality of data association in tracking, perception in robotics, risk assessment for torrent protection and multi-criteria decision-making, multi-modal image fusion, coarsening techniques, recommender system, levee characterization and assessment, human heading perception, trust assessment, robotics, biometrics, failure detection, GPS systems, inter-criteria analysis, group decision, human activity recognition, storm prediction, data association for autonomous vehicles, identification of maritime vessels, fusion of support vector machines (SVM), Silx-Furtif RUST code library for information fusion including PCR rules, and network for ship classification. Finally, the third part presents interesting contributions related to belief functions in general published or presented along the years since 2015. These contributions are related with decision-making under uncertainty, belief approximations, probability transformations, new distances between belief functions, non-classical multi-criteria decision-making problems with belief functions, generalization of Bayes theorem, image processing, data association, entropy and cross-entropy measures, fuzzy evidence numbers, negator of belief mass, human activity recognition, information fusion for breast cancer therapy, imbalanced data classification, and hybrid techniques mixing deep learning with belief functions as well

Théorème des croyances totales et théorème de Bayes généralisé

International audienceThis paper presents two new theoretical contributions for reasoning under uncertainty: 1) the Total Belief Theorem (TBT) which is a direct generalization of the Total Probability Theorem, and 2) the Generalized Bayes' Theorem drawn from TBT. A constructive justification of Fagin-Halpern belief conditioning formulas proposed in the nineties is also given. We also show how our new approach and formulas work through simple illustrative examples