Upper and lower entropies of belief functions using compatible probability functions

Author(s):  
C. W. R. Chau ◽  
P. Lingras ◽  
S. K. M. Wong
Author(s):  
Chunlai Zhou ◽  
Biao Qin ◽  
Xiaoyong Du

In reasoning under uncertainty in AI, there are (at least) two useful and different ways of understanding beliefs: the first is as absolute belief or degree of belief in propositions and the second is as belief update or measure of change in belief. Pignistic and plausibility transformations are two well-known probability transformations that map belief functions to probability functions in the Dempster-Shafer theory of evidence. In this paper, we establish the link between pignistic and plausibility transformations by devising a belief-update framework for belief functions where plausibility transformation works on belief update while pignistic transformation operates on absolute belief. In this framework, we define a new belief-update operator connecting the two transformations, and interpret the framework in a belief-function model of parametric statistical inference. As a metaphor, these two transformations projecting the belief-update framework for belief functions to that for probabilities are likened to the fire projecting reality into shadows on the wall in Plato's cave.


Author(s):  
Philippe Smets

This note is a very short presentation of the transferable belief model (TBM), a model for the representation of quantified beliefs based on belief functions. Details must be found in the recent literature. The TBM covers the same domain as the subjective probabilities except probability functions are replaced by belief functions which are much more general. The model is much more flexible than the Bayesian one and allows the representation of states of beliefs not adequately represented with probability functions. The theory of belief functions is often called the Dempster-Shafer’s theory, but this term is unfortunately confusing.


Synthese ◽  
2020 ◽  
Author(s):  
David Atkinson ◽  
Jeanne Peijnenburg

Abstract In a recent paper Ronald Meester and Timber Kerkvliet argue by example that infinite epistemic regresses have different solutions depending on whether they are analyzed with probability functions or with belief functions. Meester and Kerkvliet give two examples, each of which aims to show that an analysis based on belief functions yields a different numerical outcome for the agent’s degree of rational belief than one based on probability functions. In the present paper we however show that the outcomes are the same. The only way in which probability functions and belief functions can yield different solutions for the agent’s degree of belief is if they are applied to different examples, i.e. to different situations in which the agent finds himself.


Author(s):  
THOMAS WEILER

This paper addresses the approximation of belief functions by probability functions where the approximation is based on minimizing the Euclidean distance. First of all, we simplify this optimization problem so it becomes equivalent to a standard problem in linear algebra. For the simplified optimization problem, we provide the analytic solution. Furthermore, we show that for Dempster-Shafer belief the simplified optimization problem is equivalent to the original one. In terms of semantics, we compare the approximation of belief functions to various alternative approaches, e.g. pignistic transformation for Dempster-Shafer belief and Shapley value for fuzzy belief functions. For the later one, we give an example where the approximation method has some obvious statistical advantages. Additionally, for the approximation of additive belief functions, we can provide a semantical justification.


Author(s):  
Jens Beckert ◽  
Richard Bronk

This chapter provides a theoretical framework for considering how imaginaries and narratives interact with calculative devices to structure expectations and beliefs in the economy. It analyses the nature of uncertainty in innovative market economies and examines how economic actors use imaginaries, narratives, models, and calculative practices to coordinate and legitimize action, determine value, and establish sufficient conviction to act despite the uncertainty they face. Placing the themes of the volume in the context of broader trends in economics and sociology, the chapter argues that, in conditions of widespread radical uncertainty, there is no uniquely rational set of expectations, and there are no optimal strategies or objective probability functions; instead, expectations are often structured by contingent narratives or socially constructed imaginaries. Moreover, since expectations are not anchored in a pre-existing future reality but have an important role in creating the future, they become legitimate objects of political debate and crucial instruments of power in markets and societies.


Author(s):  
Jianping Fan ◽  
Jing Wang ◽  
Meiqin Wu

The two-dimensional belief function (TDBF = (mA, mB)) uses a pair of ordered basic probability distribution functions to describe and process uncertain information. Among them, mB includes support degree, non-support degree and reliability unmeasured degree of mA. So it is more abundant and reasonable than the traditional discount coefficient and expresses the evaluation value of experts. However, only considering that the expert’s assessment is single and one-sided, we also need to consider the influence between the belief function itself. The difference in belief function can measure the difference between two belief functions, based on which the supporting degree, non-supporting degree and unmeasured degree of reliability of the evidence are calculated. Based on the divergence measure of belief function, this paper proposes an extended two-dimensional belief function, which can solve some evidence conflict problems and is more objective and better solve a class of problems that TDBF cannot handle. Finally, numerical examples illustrate its effectiveness and rationality.


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