Belief Function Framework Research Articles

Learning from partially supervised data using mixture models and belief functions

This paper addresses classification problems in which the class membership of training data are only partially known. Each learning sample is assumed to consist of a feature vector x i ∈ X and an imprecise and/or uncertain “soft” label m i defined as a Dempster–Shafer basic belief assignment over the set of classes. This framework thus generalizes many kinds of learning problems including supervised, unsupervised and semi-supervised learning. Here, it is assumed that the feature vectors are generated from a mixture model. Using the generalized Bayesian theorem, an extension of Bayes’ theorem in the belief function framework, we derive a criterion generalizing the likelihood function. A variant of the expectation maximization (EM) algorithm, dedicated to the optimization of this criterion is proposed, allowing us to compute estimates of model parameters. Experimental results demonstrate the ability of this approach to exploit partial information about class labels.

ECM: An evidential version of the fuzzy c-means algorithm

A new clustering method for object data, called ECM (evidential c-means) is introduced, in the theoretical framework of belief functions. It is based on the concept of credal partition, extending those of hard, fuzzy, and possibilistic ones. To derive such a structure, a suitable objective function is minimized using an FCM-like algorithm. A validity index allowing the determination of the proper number of clusters is also proposed. Experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics.

Multisensor Data Fusion for Spaceborne and Airborne Reduction of Mine Suspected Areas

The problem of mined area reduction is addressed in this paper. Pieces of information collected using airborne multispectral scanners and airborne full polarimetric SAR, together with context information, all integrated in a geographical information system, are classified and combined in order to find indicators of mine presence and mine absence and provide image analysts with adequate tools to interpret mined scenes during the area reduction process. The paper contains a broad description of the whole problem and of the developed method and focuses on classification and data fusion tools based on the belief function framework and fuzzy sets theory.

An Analytical Model for External Auditor Evaluation of the Internal Audit Function Using Belief Functions

The primary purpose of this paper is to advance research in internal audit (IA) evaluation by developing an IA assessment model that considers interrelationships among specific factors used by external auditors when evaluating the strength of the IA function. The model is built on three factors previously identified by auditing standards and prior academic research: Competence, Work Performance, and Objectivity (SAS 65 1991; Messier and Schneider 1988; Krishnamoorthy 2002). Our model generates an overall judgment concerning the strength of the IA function that can aid the external auditor in assessing the reliability of a client's internal control system. Because evaluating the IA function entails a process of gathering items of evidence pertaining to each IA factor and aggregating the evidence to make an overall judgment, we develop the IA assessment model using a conceptual framework based on the prior literature and develop an analytical expression under the belief function framework. By using this framework, we overcome limitations of prior research regarding the modelling of interrelationships among factors and regarding difficulties in application. The results of our analysis revealed that modelling the And relationship is essential for assessing the strength of the IA function. As far as interrelationships are concerned, the analysis revealed that when the three factors have a strong or a perfect relationship, the strength of the IA function remains high even if we have positive or negative evidence about one of the factors. This result holds as long as we have high levels of beliefs about the other two factors.

Refined modeling of sensor reliability in the belief function framework using contextual discounting

In belief functions theory, the discounting operation allows to combine information provided by a source in the form of a belief function with meta-knowledge regarding the reliability of that source, resulting in a “weakened”, less informative belief function. In this article, an extension of the discounting operation is proposed, allowing to use more detailed information regarding the reliability of the source in different contexts, i.e., conditionally on different hypotheses regarding the variable on interest. This results in a contextual discounting operation parameterized with a discount rate vector. Some properties of this contextual discounting operation are studied, and its relationship with classical discounting is explained. A method for learning the discount rates is also presented.

Attribute Sampling: A Belief-Function Approach to Statistical Audit Evidence

The Dempster-Shafer belief function framework has been used to model the aggregation of audit evidence based on subjectively assessed beliefs. This paper shows how statistical evidence obtained by means of attribute sampling may be represented as belief functions, so that it can be incorporated into such models. In particular, the article shows: (1) how to determine the sample size in attribute sampling to obtain a desired level of belief that the true attribute occurrence rate of the population lies in a given interval; (2) what level of belief is obtained for a specified interval, given the sample result. As intuitively expected, we find that the sample size increases as the desired level of belief in the interval increases. In evaluating the sample results, our findings are again intuitively appealing. For example, provided the sample occurrence rate falls in the interval B for a given number of occurrences of the attribute, we find that the belief in B, Bel(B), increases as the sample size increases. However, if the sample occurrence rate falls outside of the interval, then Bel(B) is zero. Note that, in general, both Bel(B) and Bel(notB) are zero when the sample occurrence rate falls at the end points of the interval. These results extend similar results already available for variables sampling. However, the auditor faces an additional problem for attribute sampling: how to convert belief in an interval for control exceptions into belief in an interval for material misstatements in the financial statements, so that it can be combined with evidence from other sources in implementations of the Audit Risk Model.

A belief-function model for the representation and the combination of uncertain audit evidence

This paper describes a belief-function framework for representing and combining evidence in financial statement auditing. The use of belief-function theory for these purposes is not new. A considerable amount of important research in this respect has been performed mainly by Srivastava and is described in numerous publications. One of his major achievements is the construction of a complete framework like the one described in this paper. The framework that we propose in this text takes a different point of view and addresses a number of limitations of Srivastava’s framework. Copyright © 1999 John Wiley & Sons, Ltd.

Integrating statistical and nonstatistical audit evidence using belief functions: A case of variable sampling

The main purpose of this article is to show how one can integrate statistical and nonstatistical items of evidence in the belief function framework. First, we use the properties of consonant belief functions to define the belief that the true mean of a variable lies in a given interval when a statistical test is performed for the variable. Second, we use the above definition to determine the sample size for a statistical test when a desired level of belief is needed from the sample. Third, we determine the level of belief that the true mean lies in a given interval when a statistical test yields certain values for the sample mean and the standard deviation of the mean for the variable. Finally, we use the auditing situation to illustrate the process of integrating statistical and nonstatistical items evidence. © 1994 John Wiley & Sons, Inc.

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.

An alternative approach to the handling of subnormal possibility distributions: — A critical comment on a proposal by Yager

In this short note a definition of the necessity measure in case of a subnormal possibility distribution already proposed by the authors is recalled. This definition both satisfies the axiomatics of necessity measures and preserves the desirable inequality between possibility and necessity measures. The handling of subnormal basic probability assignments in the more general framework of Shafer's belief functions is also discussed.