Classification systems based on rough sets under the belief function framework

In this paper, we present two classification approaches based on Rough Sets (RS) that are able to learn decision rules from uncertain data. We assume that the uncertainty exists only in the decision attribute values of the Decision Table (DT) and is represented by the belief functions. The first technique, named Belief Rough Set Classifier (BRSC), is based only on the basic concepts of the Rough Sets (RS). The second, called Belief Rough Set Classifier, is more sophisticated. It is based on Generalization Distribution Table (BRSC-GDT), which is a hybridization of the Generalization Distribution Table and the Rough Sets (GDT-RS). The two classifiers aim at simplifying the Uncertain Decision Table (UDT) in order to generate significant decision rules for classification process. Furthermore, to improve the time complexity of the construction procedure of the two classifiers, we apply a heuristic method of attribute selection based on rough sets. To evaluate the performance of each classification approach, we carry experiments on a number of standard real-world databases by artificially introducing uncertainty in the decision attribute values. In addition, we test our classifiers on a naturally uncertain web usage database. We compare our belief rough set classifiers with traditional classification methods only for the certain case. Besides, we compare the results relative to the uncertain case with those given by another similar classifier, called the Belief Decision Tree (BDT), which also deals with uncertain decision attribute values.     

Facial expression classification: An approach based on the fusion of facial deformations using the transferable belief model

A method for the classification of facial expressions from the analysis of facial deformations is presented. This classification process is based on the transferable belief model (TBM) framework. Facial expressions are related to the six universal emotions, namely Joy, Surprise, Disgust, Sadness, Anger, Fear, as well as Neutral. The proposed classifier relies on data coming from a contour segmentation technique, which extracts an expression skeleton of facial features (mouth, eyes and eyebrows) and derives simple distance coefficients from every face image of a video sequence. The characteristic distances are fed to a rule-based decision system that relies on the TBM and data fusion in order to assign a facial expression to every face image. In the proposed work, we first demonstrate the feasibility of facial expression classification with simple data (only five facial distances are considered). We also demonstrate the efficiency of TBM for the purpose of emotion classification. The TBM based classifier was compared with a Bayesian classifier working on the same data. Both classifiers were tested on three different databases.     

On a multiplication and a theory of integration for belief and plausibility functions

Belief and plausibility functions have been introduced as generalizations of probability measures, which abandon the axiom of additivity. It turns out that elementwise multiplication is a binary operation on the set of belief functions. If the set functions of the type considered here are defined on a locally compact and separable space X, a theorem by Choquet ensures that they can be represented by a probability measure on the space containing the closed subsets of X, the so-called basic probability assignment. This is basic for defining two new types of integrals. One of them may be used to measure the degree of non-additivity of the belief or plausibility function. The other one is a generalization of the Lebesgue integral. The latter is compared with Choquet's and Sugeno's integrals for non-additive set functions.     

Active classification using belief functions and information gain maximization

Obtaining reliable estimates of the parameters of a probabilistic classification model is often a challenging problem because the amount of available training data is limited. In this paper, we present a classification approach based on belief functions that makes the uncertainty resulting from limited amounts of training data explicit and thereby improves classification performance. In addition, we model classification as an active information acquisition problem where features are sequentially selected by maximizing the expected information gain with respect to the current belief distribution, thus reducing uncertainty as quickly as possible. For this, we consider different measures of uncertainty for belief functions and provide efficient algorithms for computing them. As a result, only a small subset of features need to be extracted without negatively impacting the recognition rate. We evaluate our approach on an object recognition task where we compare different evidential and Bayesian methods for obtaining likelihoods from training data and we investigate the influence of different uncertainty measures on the feature selection process.     

Proper belief revision and rationalizability in dynamic games

In this paper we develop an epistemic model for dynamic games in which players may revise their beliefs about the opponents’ utility functions as the game proceeds. Within this framework, we propose a rationalizability concept that is based upon the following three principles: (1) at every instance of the game, a player should believe that his opponents are carrying out optimal strategies, (2) a player, at information set h, should not change his belief about an opponent’s relative ranking of two strategies s and s′ if both s and s′ could have led to h, and (3) the players’ initial beliefs about the opponents’ utility functions should agree on a given profile u of utility functions. Common belief in these events leads to the concept of persistent rationalizability for the profile u of utility functions. It is shown that for a given game tree with observable deviators and a given profile u of utility functions, every properly point-rationalizable strategy is a persistently rationalizable strategy for u. This result implies that persistently rationalizable strategies always exist for all game trees with observable deviators and all profiles of utility functions. We provide an algorithm that can be used to compute the set of persistently rationalizable strategies for a given profile u of utility functions. For generic games with perfect information, persistent rationalizability uniquely selects the backward induction strategy for every player.     

Inference in directed evidential networks based on the transferable belief model

Inference algorithms in directed evidential networks (DEVN) obtain their efficiency by making use of the represented independencies between variables in the model. This can be done using the disjunctive rule of combination (DRC) and the generalized Bayesian theorem (GBT), both proposed by Smets [Ph. Smets, Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem, International Journal of Approximate Reasoning 9 (1993) 1–35]. These rules make possible the use of conditional belief functions for reasoning in directed evidential networks, avoiding the computations of joint belief function on the product space. In this paper, new algorithms based on these two rules are proposed for the propagation of belief functions in singly and multiply directed evidential networks.     

Belief linear programming

This paper proposes solution approaches to the belief linear programming (BLP). The BLP problem is an uncertain linear program where uncertainty is expressed by belief functions. The theory of belief function provides an uncertainty measure that takes into account the ignorance about the occurrence of single states of nature. This is the case of many decision situations as in medical diagnosis, mechanical design optimization and investigation problems. We extend stochastic programming approaches, namely the chance constrained approach and the recourse approach to obtain a certainty equivalent program. A generic solution strategy for the resulting certainty equivalent is presented.     

A simple logic for reasoning about incomplete knowledge

The semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations.     

Dempster's rule of combination

The theory of belief functions is a generalization of probability theory; a belief function is a set function more general than a probability measure but whose values can still be interpreted as degrees of belief. Dempster's rule of combination is a rule for combining two or more belief functions; when the belief functions combined are based on distinct or “independent” sources of evidence, the rule corresponds intuitively to the pooling of evidence. As a special case, the rule yields a rule of conditioning which generalizes the usual rule for conditioning probability measures. The rule of combination was studied extensively, but only in the case of finite sets of possibilities, in the author's monograph A Mathematical Theory of Evidence. The present paper describes the rule for general, possibly infinite, sets of possibilities. We show that the rule preserves the regularity conditions of continuity and condensability, and we investigate the two distinct generalizations of probabilistic independence which the rule suggests.     

Dominance of capacities by k-additive belief functions

In this paper we deal with the set of k-additive belief functions dominating a given capacity. We follow the line introduced by Chateauneuf and Jaffray for dominating probabilities and continued by Grabisch for general k-additive measures. First, we show that the conditions for the general k-additive case lead to a very wide class of functions and this makes that the properties obtained for probabilities are no longer valid. On the other hand, we show that these conditions cannot be improved. We solve this situation by imposing additional constraints on the dominating functions. Then, we consider the more restrictive case of k-additive belief functions. In this case, a similar result with stronger conditions is proved. Although better, this result is not completely satisfactory and, as before, the conditions cannot be strengthened. However, when the initial capacity is a belief function, we find a subfamily of the set of dominating k-additive belief functions from which it is possible to derive any other dominant k-additive belief function, and such that the conditions are even more restrictive, obtaining the natural extension of the result for probabilities. Finally, we apply these results in the fields of Social Welfare Theory and Decision Under Risk.     

A framework for managing uncertain inputs: An axiomization of rewarding

The success postulate in belief revision ensures that new evidence (input) is always trusted. However, admitting uncertain input has been questioned by many researchers. Darwiche and Pearl argued that strengths of evidence should be introduced to determine the outcome of belief change, and provided a preliminary definition towards this thought. In this paper, we start with Darwiche and Pearl’s idea aiming to develop a framework that can capture the influence of the strengths of inputs with some rational assumptions. To achieve this, we first define epistemic states to represent beliefs attached with strength, and then present a set of postulates to describe the change process on epistemic states that is determined by the strengths of input and establish representation theorems to characterize these postulates. As a result, we obtain a unique rewarding operator which is proved to be a merging operator that is in line with many other works. We also investigate existing postulates on belief merging and compare them with our postulates. In addition, we show that from an epistemic state, a corresponding ordinal conditional function by Spohn can be derived and the result of combining two epistemic states is thus reduced to the result of combining two corresponding ordinal conditional functions proposed by Laverny and Lang. Furthermore, when reduced to the belief revision situation, we prove that our results induce all the Darwiche and Pearl’s postulates as well as the Recalcitrance postulate and the Independence postulate.     

Average optimality for continuous-time Markov decision processes with a policy iteration approach

This paper deals with the average expected reward criterion for continuous-time Markov decision processes in general state and action spaces. The transition rates of underlying continuous-time jump Markov processes are allowed to be unbounded, and the reward rates may have neither upper nor lower bounds. We give conditions on the system's primitive data and under which we prove the existence of the average reward optimality equation and an average optimal stationary policy. Also, under our conditions we ensure the existence of ?-average optimal stationary policies. Moreover, we study some properties of average optimal stationary policies. We not only establish another average optimality equation on an average optimal stationary policy, but also present an interesting “martingale characterization” of such a policy. The approach provided in this paper is based on the policy iteration algorithm. It should be noted that our way is rather different from both the usually “vanishing discounting factor approach” and the “optimality inequality approach” widely used in the previous literature.     

Decision with Dempster-Shafer belief functions: Decision under ignorance and sequential consistency

This paper examines proposals for decision making with Dempster-Shafer belief functions from the perspectives of requirements for rational decision under ignorance and sequential consistency. The focus is on the proposals by Jaffray & Wakker and Giang & Shenoy applied for partially consonant belief functions. We formalize the concept of sequential consistency of an evaluation model and prove results about sequential consistency of Jaffray-Wakker’s model and Giang-Shenoy’s model under various conditions. We demonstrate that the often neglected assumption about two-stage resolution of uncertainty used in Jaffray-Wakker’s model actually disambiguates the foci of a belief function, and therefore, makes it a partially consonant on the extended state space.     

Least committed basic belief density induced by a multivariate Gaussian: Formulation with applications

We consider here the case where our knowledge is partial and based on a betting density function which is n-dimensional Gaussian. The explicit formulation of the least committed basic belief density (bbd) of the multivariate Gaussian pdf is provided in the transferable belief model (TBM) framework. Beliefs are then assigned to hyperspheres and the bbd follows a χ² distribution. Two applications are also presented. The first one deals with model based classification in the joint speed–acceleration feature space. The second is devoted to joint target tracking and classification: the tracking part is performed using a Rao–Blackwellized particle filter, while the classification is carried out within the developed TBM scheme.     

Job-shop local-search move evaluation without direct consideration of the criterion’s value

This article focuses on the evaluation of moves for the local search of the job-shop problem with the makespan criterion. We reason that the omnipresent ranking of moves according to their resulting value of a criterion function makes the local search unnecessarily myopic. Consequently, we introduce an alternative evaluation that relies on a surrogate quantity of the move’s potential, which is related to, but not strongly coupled with, the bare criterion. The approach is confirmed by empirical tests, where the proposed evaluator delivers a new upper bound on the well-known benchmark test yn2. The line of the argumentation also shows that by sacrificing accuracy the established makespan estimators unintentionally improve on the move evaluation in comparison to the exact makespan calculation, in contrast to the belief that the reliance on estimation degrades the optimization results.     

Minimal and maximal operator spaces and operator systems in entanglement theory

We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.     

Nearly optimal convergence result for multigrid with aggressive coarsening and polynomial smoothing

We analyze a general multigrid method with aggressive coarsening and polynomial smoothing. We use a special polynomial smoother that originates in the context of the smoothed aggregation method. Assuming the degree of the smoothing polynomial is, on each level k, at least Ch _k+1/h _k , we prove a convergence result independent of h _k+1/h _k . The suggested smoother is cheaper than the overlapping Schwarz method that allows to prove the same result. Moreover, unlike in the case of the overlapping Schwarz method, analysis of our smoother is completely algebraic and independent of geometry of the problem and prolongators (the geometry of coarse spaces).     

Stake-Invariant Belief

What can rational deliberation indicate about belief? Belief clearly influences deliberation. The principle that rational belief is stake-invariant rules out at least one way that deliberation might influence belief. The principle is widely, if implicitly, held in work on the epistemology of categorical belief, and it is built into the model of choice-guiding degrees of belief that comes to us from Ramsey and de Finetti. Criticisms of subjective probabilism include challenges to the assumption of additive values (the package principle) employed by defenses of probabilism. But the value-interaction phenomena often cited in such challenges are excluded by stake-invariance. A comparison with treatments of categorical belief suggests that the appeal to stake-invariance is not ad hoc. Whether or not to model belief as stake-invariant is a question not settled here.