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.

DEMPSTER COMBINATION RULE WITH BOOLEAN-LIKE PROCESSED BELIEF FUNCTIONS

2001 ◽  
Vol 09 (01) ◽  
pp. 105-121 ◽  
Author(s):  
IVAN KRAMOSIL
Keyword(s):  
Empirical Data ◽  
Random Variables ◽  
Unit Interval ◽  
Probability Measures ◽  
Belief Functions ◽  
Statistical Independence ◽  
Combination Rule ◽  
Real Numbers

It is a well-known fact that the Dempster combination rule for combination of uncertainty degrees coming from two or more sources is legitimate only if the combined empirical data, charged with uncertainty and taken as random variables, are statistically (stochastically) independent. We shall prove, however, that for a particular but large enough class of probability measures, an analogy of Dempster combination rule, preserving its extensional character but using some nonstandard and boolean-like structures over the unit interval of real numbers, can be obtained without the assumption of statistical independence of input empirical data charged with uncertainty.

On the Equivalence of Modes of Convergence(1)

1973 ◽  
Vol 16 (4) ◽  
pp. 571-575 ◽  
Author(s):  
R. J. Tomkins
Keyword(s):  
Probability Space ◽  
Random Variables ◽  
Real Numbers ◽  
Probability Zero

Let (Ω,ℱ, P) be a probability space. Let R denote the set of real numbers and the set of all random variables defined on Ω. Throughout this work, random variables which differ only on a set of probability zero will be considered identical. EX represents, as usual, the expectation of .

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.

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

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 On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

Stochastic comparisons of order statistics under multivariate imperfect repair

1996 ◽  
Vol 33 (1) ◽  
pp. 156-163 ◽  
Author(s):  
Taizhong Hu
Keyword(s):  
Order Statistics ◽  
Probability Space ◽  
Random Variables ◽  
Stochastic Comparisons ◽  
Special Model ◽  
Imperfect Repair ◽  
Monotone Coupling ◽  
Common Probability

A monotone coupling of order statistics from two sets of independent non-negative random variables Xi, i = 1, ···, n, and Yi, i = 1, ···, n, means that there exist random variables X′i, Y′i, i = 1, ···, n, on a common probability space such that , and a.s. j = 1, ···, n, where X(1) ≦ X(2) ≦ ·· ·≦ X(n) are the order statistics of Xi, i = 1, ···, n (with the corresponding notations for the X′, Y, Y′ sample). In this paper, we study the monotone coupling of order statistics of lifetimes in two multi-unit systems under multivariate imperfect repair. Similar results for a special model due to Ross are also given.

scholarly journals Complete convergence and complete moment convergence for negatively dependent random variables under sub-linear expectations

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1093-1104
Author(s):  
Qunying Wu ◽  
Yuanying Jiang
Keyword(s):  
Probability Space ◽  
Complete Convergence ◽  
Random Variables ◽  
Complete Moment Convergence ◽  
Convergence Theorems ◽  
Dependent Random Variables ◽  
Moment Convergence ◽  
Negatively Dependent Random Variables

This paper we study and establish the complete convergence and complete moment convergence theorems under a sub-linear expectation space. As applications, the complete convergence and complete moment convergence for negatively dependent random variables with CV (exp (ln? |X|)) < ?, ? > 1 have been generalized to the sub-linear expectation space context. We extend some complete convergence and complete moment convergence theorems for the traditional probability space to the sub-linear expectation space. Our results generalize corresponding results obtained by Gut and Stadtm?ller (2011), Qiu and Chen (2014) and Wu and Jiang (2016). There is no report on the complete moment convergence under sub-linear expectation, and we provide the method to study this subject.

Sign in / Sign up

Export Citation Format

Share Document