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## Conjunctive rules in the theory of belief functions and their justification through decisions models.

P. 137-145.

Andrey G. Bronevich, Rozenberg I.

In the paper we argue that aggregation rules in the theory of belief functions should be in accordance with underlying decision models, i.e. aggregation produced in conjunctive manner has to produce the order embedded to the union of partial orders constructed in each source of information; and if we take models based on imprecise probabilities, then such aggregation exists if the intersection of underlying credal sets is not empty. In the opposite case there is contradiction in information and the justifiable functional to measure it is the functional giving the smallest contradiction by applying all possible conjunctive rules. We give also the axiomatics of this contradiction measure.

### In book

Vol. 9861: Belief Functions: Theory and Applications. , Springer, 2016

Bronevich A., Розенберг И. Н., International Journal of Approximate Reasoning 2015 Vol. 56 P. 122-136

In the paper we investigate the criteria of choosing generalized Dempster–Shafer rules for aggregating sources whose information is represented by belief functions. The approach is based on measuring various types of uncertainty in information and we use for this purpose in particular linear imprecision indices. Some results concerning properties of such rules are also presented. ...

Added: March 5, 2015

Bronevich A., Lepskiy A., Information Fusion 2017 P. 1-12

This paper is devoted to the study of information aggregation from different sourced in the frame of belief function theory. The paper has two main aims. In the first part of this paper we study the so called conjunctive rules of aggregation that generalize the classical Dempster rule. The aim is to research the relation ...

Added: December 13, 2015

Karabekyan D., Журнал Новой экономической ассоциации 2022 № 5(57) С. 24-37

Some distortions are possible in the process of preference aggregation. For example, one voter who is pivotal for some preference profi le may not read instructions properly and accidently submit wrong preference. We study how different voting rules react to these distortions for three, four and fi ve alternatives with computer modelling. One of the ...

Added: January 17, 2023

Measures of conflict, basic axioms and their application to the clusterization of a body of evidence

Andrey G. Bronevich, Alexander E. Lepskiy, Fuzzy Sets and Systems 2022 Vol. 446 P. 277-300

There are several approaches for evaluating conflict within belief functions. In this paper, we develop one of them based on axioms and show its connections to the decomposition approach. We describe a class of conflict measures satisfying this system of axioms and show that measuring conflict can be realized through the clusterization of a body ...

Added: August 27, 2022

Bronevich A., Rozenberg I., , in : Belief Functions: Theory and Applications 5th International Conference, BELIEF 2018, Compiègne, France, September 17-21, 2018, Proceedings. Vol. 11069.: Springer, 2018. P. 31-38.

The aim of this paper is to show that the Kantorovich problem, well known in models of economics and very intensively studied in probability theory in recent years, can be viewed as the basis of some constructions in the theory of belief functions. We demonstrate this by analyzing specialization relation for finitely defined belief functions ...

Added: October 8, 2018

Andrey G. Bronevich, Rozenberg I., , in : Advances in Intelligent Systems and Computing. Vol. 456: Soft Methods for Data Science.: Springer, 2017. P. 87-94.

In the paper we continue investigations started in the paper presented at ISIPTA’15, where the notions of lower and upper generalized credal sets has been introduced. Generalized credal sets are models of imprecise probabilities, where it is possible to describe contradiction in information, when the avoiding sure loss condition is not satisfied. The paper contains ...

Added: October 17, 2016

Bronevich A., Розенберг И. Н., , in : Belief Functions: Theory and Applications. Vol. 8764.: Dordrecht, L., Heidelberg, NY : Springer, 2014. P. 21-28.

In the paper we investigate the criteria of choosing generalized Dempster-Shafer rules for aggregating sources of information presented by belief functions. The approach is based on measuring various types of uncertainty in information and we use for this linear imprecision indices. Some results concerning properties of such rules are also presented. ...

Added: March 5, 2015

Springer, 2016

The theory of belief functions, also referred to as evidence theory or Dempster-Shafer theory, is a well-established general framework for reasoning with uncertainty. It has well-understood connections to other frameworks, such as probability, possibility, and imprecise probability theories. First introduced by Arthur P. Dempster in the context of statistical inference, the theory was later developed ...

Added: September 28, 2016

Lepskiy A., Meshcheryakova N., , in : Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2020. * 2. Vol. 1238.: Cham : Springer, 2020. P. 283-296.

We apply Dempster-Shafer theory in order to reveal important elements in undirected weighted networks. We estimate cooperation of each node with different groups of vertices that surround it via construction of belief functions. The obtained intensities of cooperation are further redistributed over all elements of a particular group of nodes that results in pignistic probabilities ...

Added: July 8, 2020

Bronevich A., Rozenberg I., , in : Proceedings of the 9th International Symposium on Imprecise Probability: Theories and Applications. : Rome : Aracne editrice, 2015. P. 67-76.

In the paper we consider the generalization of the conjunctive rule in the theory of imprecise probabilities. Let us remind that the conjunction rule, produced on credal sets,gives their intersection and it is not defined if this intersection is empty. In the last case the sources of information are called contradictory1. Meanwhile, in the Dempster-Shafer ...

Added: February 29, 2016

Bronevich A., Rozenberg I., International Journal of Approximate Reasoning 2019 Vol. 112 P. 119-139

In the paper, we formalize the notion of contradiction between belief functions: we argue that belief functions are not contradictory if they provide non-contradictory models for decision-making. To elaborate on this idea, we take the decision rule from imprecise probabilities and show that sources of information described by belief functions are not contradictory iff the ...

Added: September 25, 2019

Bronevich A., Lepskiy A., , in : Proceedings of the 11th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2019). Vol. 1.: P. : Atlantis Press, 2019. P. 328-333.

In real applications, sometimes it is necessary to evaluate inner or external conflict of pieces of evidence. However, these numerical values cannot give us explanations why this conflict occurs. Thus, we need deeper analysis of available information. In the paper, we propose the clusterization of a given evidence on pieces of evidence in a way ...

Added: September 25, 2019

Jürgen Eichberger, Pasichnichenko I., Journal of Economic Theory 2021 Vol. 198 Article 105369

In this paper, we study choice under uncertainty with belief functions. Belief functions can capture par- tial information by describing what is objectively known about the probabilities of events. State-contingent acts together with a belief function over states induce belief functions over outcomes. We assume that de- cision makers have preferences over belief functions that ...

Added: October 31, 2021

N. L. Polyakov, Shamolin M. V., Journal of Mathematical Sciences 2023 Vol. 272 No. 5 P. 667-671

In the paper, combinatorial theorems related to the theory of social choice are obtained. These theorems describe general conditions under which the problem on the preserving the preference set D by anarbitrary aggregation rule f and the problem on the compatibility of the preference set D with a pair (f,C) can be reduced to similar ...

Added: May 19, 2023

Bronevich A., Rozenberg I., International Journal of General Systems 2018 Vol. 27 No. 1 P. 67-96

To model conflict, non-specificity and contradiction in information,
upper and lower generalized credal sets are introduced. Any upper
generalized credal set is a convex subset of plausibility measures
interpreted as lower probabilities whose bodies of evidence consist of
singletons and a certain event. Analogously, contradiction is modelled
in the theory of evidence by a belief function that is greater than
zero ...

Added: February 2, 2018

Lepskiy A., , in : Advances in Intelligent Systems and Computing. Vol. 456: Soft Methods for Data Science.: Springer, 2017. P. 311-318.

The qualitative characteristics of the combining evidence with the help of Dempster’s rule with discounting is studied in this paper in the framework of Dempster-Shafer theory. The discount coefficient (discounting rate) characterizes the reliability of information source. The conflict between evidence and change of ignorance after applying combining rule are considered in this paper as ...

Added: September 28, 2016

Springer Nature Switzerland AG, 2019

Added: September 25, 2019

Bronevich A., Penikas H. I., Lepskiy A. et al., / Высшая школа экономики. Серия WP7 "Математические методы анализа решений в экономике, бизнесе и политике". 2015. № 10.

There is one of the investment strategies on a stock market called a choice of securities including
its buying and sale based on financial analysts’ recommendations. The distinguishing characteristic of
these recommendations is that in each of it for the same securities there can be different recommendations
(«Buy», «Sell», «Hold») as well as for one type of recommendations ...

Added: December 2, 2015

Dominiak A., Eichberger J. T., Games and Economic Behavior 2021 Vol. 128 P. 125-159

We propose a new solution concept, called Context-Dependent Equilibrium Under Ambi- guity (CD-EUA), for strategic games where players’ beliefs may be influenced by exogenous context-related information. Players’ beliefs about the strategic behavior of their opponents are represented by belief functions. The notion of belief functions allows us to combine ex- ogenous context information in the spirit ...

Added: October 31, 2021

Lepskiy A., Smolev V., , in : Atlantis Studies in Uncertainty Modelling, Proceedings of the 2019 Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (EUSFLAT 2019). Vol. 1.: Atlantis Press, 2019. P. 321-327.

Added: September 29, 2019