scholarly journals An Algebraic Study of the Notion of Independence of Frames

2014 ◽  
pp. 239-267
Author(s):  
Fabio Cuzzolin
Keyword(s):  
Belief Function ◽  
Point Of View ◽  
Boolean Algebras ◽  
Mathematical Representation ◽  
Belief Functions ◽  
Modular Lattices ◽  
Algebraic Point ◽  
Theory Of Belief Functions ◽  
Standard Linear ◽  
Theory Of Evidence

The theory of belief functions or “theory of evidence” allows the mathematical representation of uncertain pieces of evidence on which decisions can be based. Frequently, different pieces of evidence belong to distinct, albeit related, domains or “frames”: for instance, audio and video clues can be combined to infer the identity of a person from a video. Evidence encoded by different belief functions on separate frames can be merged on a common frame, a combination which is guaranteed to exist if and only if the frames are “independent” in the sense of Boolean algebras. In all other cases the evidence conflicts. Independence of frames and belief function combinability are then strictly related. In this chapter, the authors discuss the notion of independence of frames in the theory of evidence from an algebraic point of view, starting from an analogy with standard linear independence. The final goal is to search for a solution of the problem of conflicting belief function via a generalization of the classical Gram-Schmidt algorithm for vector orthogonalization. Families of frames can be given several algebraic interpretations in terms of semi-modular lattices, matroids, and geometric lattices. Each of those structures is endowed with a particular (extended) independence relation, which we prove to be distinct albeit related to independence of frames.

Belief Function Approach to Evidential Reasoning in Causal Maps

2005 ◽  
pp. 109-141
Author(s):  
Rajendra P. Srivastava ◽  
Mari W. Buche ◽  
Tom L. Roberts
Keyword(s):  
Job Satisfaction ◽  
Information Technology ◽  
Belief Function ◽  
Evidential Reasoning ◽  
Belief Functions ◽  
Causal Maps ◽  
Dempster Shafer Theory ◽  
Evidential Reasoning Approach ◽  
Theory Of Belief Functions ◽  
Shafer Theory

The purpose of this chapter is to demonstrate the use of the evidential reasoning approach under the Dempster-Shafer (D-S) theory of belief functions to analyze revealed causal maps (RCM). The participants from information technology (IT) organizations provided the concepts to describe the target phenomenon of Job Satisfaction. They also identified the associations between the concepts. This chapter discusses the steps necessary to transform a causal map into an evidential diagram. The evidential diagram can then be analyzed using belief functions technique with survey data, thereby extending the research from a discovery and explanation stage to testing and prediction. An example is provided to demonstrate these steps. This chapter also provides the basics of Dempster-Shafer theory of belief functions and a step-by-step description of the propagation process of beliefs in tree-like evidential diagrams.

Dempster Combination Rule for Signed Belief Functions

1998 ◽  
Vol 06 (01) ◽  
pp. 79-102 ◽  
Author(s):  
Ivan Kramosil
Keyword(s):  
Large Class ◽  
Probability Space ◽  
Binary Operation ◽  
Random Variables ◽  
Belief Function ◽  
Measurable Space ◽  
Point Of View ◽  
Belief Functions ◽  
Combination Rule ◽  
Real Numbers

A possibility to define a binary operation over the space of pairs of belief functions, inverse or dual to the well-known Dempster combination rule in the same sense in which substraction is dual with respect to the addition operation in the space of real numbers, can be taken as an important problem for the purely algebraic as well as from the application point of view. Or, it offers a way how to eliminate the modification of a belief function obtained when combining this original belief function with other pieces of information, later proved not to be reliable. In the space of classical belief functions definable by set-valued (generalized) random variables defined on a probability space, the invertibility problem for belief functions, resulting from the above mentioned problem of "dual" combination rule, can be proved to be unsolvable up to trivial cases. However, when generalizing the notion of belief functions in such a way that probability space is replaced by more general measurable space with signed measure, inverse belief functions can be defined for a large class of belief functions generalized in the corresponding way. "Dual" combination rule is then defined by the application of the Dempster rule to the inverse belief functions.

scholarly journals Plato's Cave in the Dempster-Shafer land--the Link between Pignistic and Plausibility Transformations

2017 ◽  
Author(s):  
Chunlai Zhou ◽  
Biao Qin ◽  
Xiaoyong Du
Keyword(s):  
Statistical Inference ◽  
Belief Function ◽  
Belief Functions ◽  
Reasoning Under Uncertainty ◽  
Function Model ◽  
Probability Functions ◽  
Degree Of Belief ◽  
Dempster Shafer Theory ◽  
Shafer Theory ◽  
Theory Of Evidence

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.

scholarly journals Decision making with consonant belief functions: Discrepancy resulting with the probability transformation method used

2014 ◽  
Vol 24 (3) ◽  
pp. 359-370 ◽  
Author(s):  
Esma Cinicioglu
Keyword(s):  
Decision Making ◽  
Probability Distributions ◽  
Belief Function ◽  
Transformation Method ◽  
Research Society ◽  
Belief Functions ◽  
Belief Function Theory ◽  
Probability Transformation ◽  
Theory Of Belief Functions ◽  
Transformation Methods

Dempster?Shafer belief function theory can address a wider class of uncertainty than the standard probability theory does, and this fact appeals the researchers in operations research society for potential application areas. However, the lack of a decision theory of belief functions gives rise to the need to use the probability transformation methods for decision making. For representation of statistical evidence, the class of consonant belief functions is used which is not closed under Dempster?s rule of combination but is closed under Walley?s rule of combination. In this research, it is shown that the outcomes obtained using both Dempster?s and Walley?s rules do result in different probability distributions when pignistic transformation is used. However, when plausibility transformation is used, they do result in the same probability distribution. This result shows that the choice of the combination rule and probability transformation method may have a significant effect on decision making since it may change the choice of the decision alternative selected. This result is illustrated via an example of missile type identification.

Extended two-dimensional belief function based on divergence measurement

2020 ◽  
pp. 1-8
Author(s):  
Jianping Fan ◽  
Jing Wang ◽  
Meiqin Wu
Keyword(s):  
Belief Function ◽  
Distribution Functions ◽  
Divergence Measure ◽  
Uncertain Information ◽  
Belief Functions ◽  
Two Dimensional ◽  
Probability Distribution Functions ◽  
Basic Probability ◽  
The Difference ◽  
Supporting Degree

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.

scholarly journals CONFLICTS AND FAIR TESTING

2006 ◽  
Vol 17 (04) ◽  
pp. 797-813 ◽  
Author(s):  
ROBI MALIK ◽  
DAVID STREADER ◽  
STEVE REEVES
Keyword(s):  
Discrete Event Systems ◽  
Process Algebra ◽  
Discrete Event ◽  
Denotational Semantics ◽  
Point Of View ◽  
Algebraic Point ◽  
Event Systems ◽  
Common Task ◽  
Testing Theory ◽  
Process Algebraic

This paper studies conflicts from a process-algebraic point of view and shows how they are related to the testing theory of fair testing. Conflicts have been introduced in the context of discrete event systems, where two concurrent systems are said to be in conflict if they can get trapped in a situation where they are waiting or running endlessly, forever unable to complete their common task. In order to analyse complex discrete event systems, conflict-preserving notions of refinement and equivalence are needed. This paper characterises an appropriate refinement, called the conflict preorder, and provides a denotational semantics for it. Its relationship to other known process preorders is explored, and it is shown to generalise the fair testing preorder in process-algebra for reasoning about conflicts in discrete event systems.

scholarly journals Localizability of unicycle mobiles robots: An algebraic point of view

2012 ◽  
Author(s):  
Hugues Sert ◽  
Wilfrid Perruquetti ◽  
Annemarie Kokosy ◽  
Xin Jin ◽  
Jorge Palos
Keyword(s):  
Point Of View ◽  
Algebraic Point

DEMPSTER BELIEF FUNCTIONS ARE BASED ON THE PRINCIPLE OF COMPLETE IGNORANCE

2000 ◽  
Vol 08 (03) ◽  
pp. 271-284 ◽  
Author(s):  
PETER P. WAKKER
Keyword(s):  
Bayesian Approach ◽  
Belief Function ◽  
Belief Functions ◽  
Strict Adherence ◽  
Second Stage ◽  
Subjective Information ◽  
State Of Nature ◽  
Complete Ignorance ◽  
True State ◽  
The Bayesian Approach

This paper shows that a "principle of complete ignorance" plays a central role in decisions based on Dempster belief functions. Such belief functions occur when, in a first stage, a random message is received and then, in a second stage, a true state of nature obtains. The uncertainty about the random message in the first stage is assumed to be probabilized, in agreement with the Bayesian principles. For the uncertainty in the second stage no probabilities are given. The Bayesian and belief function approaches part ways in the processing of the uncertainty in the second stage. The Bayesian approach requires that this uncertainty also be probabilized, which may require a resort to subjective information. Belief functions follow the principle of complete ignorance in the second stage, which permits strict adherence to objective inputs.

MAGNETIC GROUP ON THE PSEUDOSPHERE

1992 ◽  
Vol 06 (21) ◽  
pp. 3525-3537 ◽  
Author(s):  
V. BARONE ◽  
V. PENNA ◽  
P. SODANO
Keyword(s):  
Magnetic Field ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Constant Magnetic Field ◽  
Coherent States ◽  
Compact Operators ◽  
Point Of View ◽  
Algebraic Point ◽  
Planar Limit ◽  
Discrete Part

The quantum mechanics of a particle moving on a pseudosphere under the action of a constant magnetic field is studied from an algebraic point of view. The magnetic group on the pseudosphere is SU(1, 1). The Hilbert space for the discrete part of the spectrum is investigated. The eigenstates of the non-compact operators (the hyperbolic magnetic translators) are constructed and shown to be expressible as continuous superpositions of coherent states. The planar limit of both the algebra and the eigenstates is analyzed. Some possible applications are briefly outlined.

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