Dempster's Rule Of Combination Research Articles

Evidential segmentation scheme of multi-echo MR images for the detection of brain tumors using neighborhood information

In this paper we propose and study an evidential segmentation scheme of multi-echo MR images for the detection of brain tumors. We show that the modeling by means of evidence theory is well suited to the processing of redundant and complementary data as the MR images. Moreover neighborhood relationship between voxels is taken into account via Dempster's combination rule. We show that using this information improves the classification results previously obtained and leads to a real region-based segmentation. Moreover, the combination of spatial information allows to compute a measure of conflict, which reflects the spatial organization of the data: the conflict is higher at the boundaries between different structures. Thus, it provides a new source of evidence that the specialist can aggregate with the segmentation results to soften its own decision.

Evidence Theory and VPRS model

The Rough Set Theory (RST) was proposed by Pawlak [4] as a new mathematical approach to deal with uncertain knowledge in expert systems. In 1991 Ziarko [11] proposed the Variable Precision Rough Set Model (VPRSM) as a certain extension of the rough set theory. VPRSM approach makes it possible to use a certain level of misclassification.The aim of this paper is to introduce belief and plausibility functions defined by the β-approximation regions. On the basis of the β-approximation regions, the β-basic probability assignment is defined and the Dempster's combination rule for product of two decision tables is constructed. This entire approach is illustrated by examples.

DEMPSTER COMBINATION RULE WITH BOOLEAN-LIKE PROCESSED BELIEF FUNCTIONS

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.

Reinvestigating Dempster's Idea on Evidence Combination

In this paper, we investigate the problem encountered by Dempster's combination rule in view of Dempster's original combination framework. We first show that the root of Dempster's combination rule (defined and named by Shafer) is Dempster's original idea on evidence combination. We then argue that Dempster's original idea on evidence combination is, in fact, richer than what has been formulated in the rule. We conclude that, by strictly following what Dempster has suggested, there should be no counterintuitive results when combining evidence.

Measure-theoretic approach to the inversion problem for belief functions

The problem how to define an inverse operation to the Dempster combination rule for basic probability assignments and belief functions possesses a natural motivation and an intuitive interpretation. Or, if the Dempster rule reflects a modification of one's system of degrees of beliefs when the subject in question becomes familiar with the degrees of beliefs of another subject and accepts the arguments on which these degrees are based, the inverse operation would enable to erase the impact of this modification, and to return back to one's original degrees of beliefs, supposing that the reliability of the second subject is put into doubts. Within the algebraic framework this inversion problem was solved by Ph. Smets in [6], here we suggest an alternative solution based on the apparatus of measure theory and conserving the idea that degrees of beliefs are measures, i.e., numerically quantified sizes, of certain sets of elementary random events defined by set-valued random variables. However, probability measures, used for these purposes when defining classical belief functions, will be replaced by the so-called signed measures, which can also take values outside the unit interval of real numbers including the negative and even infinite ones.

Dempster Combination Rule for Signed Belief Functions

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.

A NOTE ON UNCERTAINTY, DEMPSTER RULE OF COMBINATION, AND CONFLICT

The relationship between uncertainty and the Dempster rule of combination in Dempster-Shafer theory is discussed. In particular, a condition is presented that is sufficient to guarantee that uncertainty does not increase by using the rule. A new definition of conflict of pieces of evidence is proposed and discussed.

Relaxation optimizing operations in extended probabilistic space

The classical probabilistic relaxation method has been widely used to solve optimization problems in various fields, including image processing and pattern recognition. The authors have also been developing handwritten character recognition system using the probabilistic relaxation method. However, we realize that there exist cases in which a probability theoretic model is inadequate, especially where there exists incompleteness in available information by noise such as patchy segments and ink spots. In that case, we must introduce ad hoc rules for one-to-two correspondence and no correspondence in the probabilistic algorithm. As the Dempster-Shafer (DS) theory is a kind of natural extension of the probabilistic theory by reducing additivity on probability measure and it can cope with incomplete data. This paper proposes a relaxation matching method based on the Dempster-Shafer theory. Then the update process in probabilistic relaxation method is derived as a special case of Dempster's combination rule in...

A comprehensive comparison between generalized incidence calculus and the Dempster-Shafer theory of evidence

Dealing with uncertainty problems in intelligent systems has attracted a lot of attention in the AI community. Quite a few techniques have been proposed. Among them, the Dempster-Shafer theory of evidence (DS theory) has been widely appreciated. In DS theory, Dempster's combination rule plays a major role. However, it has been pointed out that the application domains of the rule are rather limited and the application of the theory sometimes gives unexpected results. We have previously explored the problem with Dempster's combination rule and proposed an alternative combination mechanism in generalized incidence calculus. In this paper we give a comprehensive comparison between generalized incidence calculus and the Dempster-Shafer theory of evidence. We first prove that these two theories have the same ability in representing evidence and combining DS-independent evidence. We then show that the new approach can deal with some dependent situations while Dempster's combination rule cannot. Various examples in the paper show the ways of using generalized incidence calculus in expert systems.

A Generalization of Bayes Theorem in Theory of Evidence

A generalized Bayes theorem in the frame of Shafer's theory of evidence is proposed. Smets proposed a procedure for obtaining a posterior basic probability assignment from prior and conditional basic probability assignments using Dempster rule of combination. We show that the uncertainty in the sense of non-specificity of a probability distribution changes by the Smets' procedure. Hence, it contradict with Bayes theorem. We formulate a new combination rule and construct a procedure for obtaining a posterior basic probability assignment without changing the uncertainty. We verify that the proposed rule is reduced to Bayes theorem when each of basic probability assignment is a regular probability assignment. It is also Jeffrey's rule when observation is obtained by a probability distribution and prior distributions are probability distributions.

A mathematical formulation of uncertain information

This paper introduces a mathematical model of uncertain information. Each body of uncertain information is an information quadruplet, consisting of a code space, a message space, an interpretation function, and an evidence space. Each information quadruplet contains prior information as well as possible new evidence which may appear later. The definitions of basic probability and belief function are based on the prior information. Given new evidence, Bayes' rule is used to update the prior information. This paper also introduces an idea of independent information and its combination. A combination formula is derived for combining independent information. Both the conventional Bayesian approach and Dempster-Shafer's approach belong to this mathematical model. A Bayesian prior probability measure is the prior information of a special information quadruplet; Bayesian conditioning is the combination of special independent information. A Dempster's belief function is the belief function of a different information quadruplet; the Dempster combination rule is the combination rule of independent quadruplets. This paper is a mathematical study of handling uncertainty and shows that both the conventional Bayesian approach and Dempster-Shafer's approach originate from the same mathematical theory.

Structural inference and a modification of Dempster's combination rule

In his article Dempster [5] remarks that the method of inference based on the concept of upper (and lower) probabilities (Dempster [3]–[7]), which he proposes has strong similarity to the method of structural inference introduced by Fraser [12]. This paper attempts to clarify this aspect and to prove that under certain circumstances concerning the underlying family of distributions both methods coincide. To do this a major problem has to be handled: Whereas Dempster's theory is built originally for families of discrete distributions, the theory of structural inference typically emphasizes families of absolutely continuous distributions. In what follows, Dempster's theory is modified for families of absolutely continuous distributions. This may be done without remarkable difficulties for sample size one. In cases of larger sample sizes this modification is more complicated on account of the necessity to define conditional distributions on sets whose Lebesgue measures are zero. If nothing else concerning a conditioning sub-σ-field or a conditioning mapping is presupposed, this can be done in different ways. This problem however may be overcome when the inference based on upper probabilities is seen under circumstances of structural inference, i.e. the underlying family of distributions admits a representation by a unitary group of transformations. In those situations a conditioning mapping can be defined in a “natural” way which leads to the same results in both methods of inference.

Two perspectives of the Dempster-Shafer theory ofbelief functions

This study presents two different perspectives of the Dempster-Shafer theory ofbelief functions. The first view of the theory, called the compatibility view , constructs the belief function for a frame of discernment by using the compatibility relationship of the frame with another frame called evidence frame for which the probability function is known. The second view referred to as the probability allocation view provides a generalization of the Bayesian theory by allocating the probability mass to the propositions which may not necessarily be singleton sets of possible answers, based on some vague body of evidence. Both these views are useful in the design and implementation of expert systems. The belief functions constructed using the evidence frames are useful in situations where limited information regarding the relationship between two frames of discernment is available. The belief functions based on the allocation of probability mass are useful when the evidence cannot be explicitly expressed in terms of propositions, but probability allocation based on evidence is possible. A successful implementation of these two views requires a rule for combining evidence. However, the Dempster rule of combination cannot incorporate dependencies among different bodies of evidence. This study illustrates a possible method for incorporating dependencies based on limited information.

Dempster's combination rule and sufficient statistics

In his articles (1966-1968) concerning statistical inference based on lower and upper probabilities, Dempster refers to the connection between Fisher's fiducial argument and his own ideas of statistical inference. Dempster's main concern however focuses on the “Bayesian” aspects of his theory and not on an elaboration of the relation between Fisher's and his ideas. This article attempts to work out the connection between those two approaches and focuses primarily on the question, whether Dempster's combination rule, his upper and lower probabilty based on sufficient statistics and inference based on sufficient statistics in Fisher's sense are consistent. To be adequate to Fisher's reasoning, we deal with absolutely continuous, one parametric families of distributions.This is certainly not the usual assumption in context with Dempster's theory and implies a normative but straightforward definition concerning the underlying conditional distribution; this definition however is done in Dempster's spirit as ca...

Uncertainty management in a pyramid vision system

This paper presents an efficient adaptation and application of the DempsterShafer ( (D-S) theory of evidence, one that can be used effectively in a massively parallel hierarchical system for real-world image analysis. It describes the techniques used and shows some extended examples that use a multilevel set of processes. The knowledge representation technique employs uncertain and incomplete world knowledge. It is modular, flexible, and easy to update. The reasoning mechanism extends the applications of the belief functions and Dempster's combination rule in a relatively efficient and powerful manner. The performance of the proposed D-S evidential reasoning mechanism is compared with other methods based on the Bayesian formalism and a simple weight combination method.

Representation and combination of uncertainty with belief functions and possibility measures

The theory of evidence proposed by G. Shafer is gaining more and more acceptance in the field of artificial intelligence, for the purpose of managing uncertainty in knowledge bases. One of the crucial problems is combining uncertain pieces of evidence stemming from several sources, whether rules or physical sensors. This paper examines the framework of belief functions in terms of expressive power for knowledge representation. It is recalled that probability theory and Zadeh's theory of possibility are mathematically encompassed by the theory of evidence, as far as the evaluation of belief is concerned. Empirical and axiomatic foundations of belief functions and possibility measures are investigated. Then the general problem of combining uncertain evidence is addressed, with focus on Dempster rule of combination. It is pointed out that this rule is not very well adapted to the pooling of conflicting information. Alternative rules are proposed to cope with this problem and deal with specific cases such as nonreliable sources, nonexhaustive sources, inconsistent sources, and dependent sources. It is also indicated that combination rules issued from fuzzy set and possibility theory look more flexible than Dempster rule because many variants exist, and their numerical stability seems to be better.

Probabilistic and Evidential Approaches for Multisource Data Analysis

Two methods for combining the information contents from multiple sources of remote-sensing image data and spatial data in general are described. One is a probabilistic scheme that employs a global membership function (similar to a joint posterior probability) that is derived from all available data sources. The other is an evidential calculus based upon Dempster's orthogonal sum combination rule. A feature of both methods is that uncertainty regarding data analysis can be incorporated into the process. Both schemes are evaluated in terms of their general applicability and certain equivalences are noted. Moreover, both are shown to perform well on mixed multispectral data.

A consideration for measure of information based on the dempster‐shafer theory

AbstractThis paper discusses the measure of information for the event using the probability theory by Dempster and Shafer. By providing various kinds of information to the event containing an ambiguity, the ambiguity is gradually decreased to define the event in a clearer way. The combination rule by Dempster is most suited to represent the combination of information. This paper shows first that several representations can be made for Dempster's combination rule. Then using the upper and lower probabilities defined by the basic probability of the event, the upper and lower information measures are defined, followed by the discussion of their properties. The lower probability of a set (event) is defined as the sum of basic probabilities of all subsets in the set, and the upper probability is defined as the sum of basic probabilities of all sets which contain an element of the set. Thus, the upper and lower information measures can be interpreted as the measures of information concerning the message with ambiguity. Finally, several examples are discussed, determining the information measure for the message containing ambiguity.