Credal Sets Research Articles

An expectation operator for belief functions in the Dempster–Shafer theory

ABSTRACTThe main contribution of this paper is a new definition of expected value of belief functions in the Dempster–Shafer (D–S) theory of evidence. Our definition shares many of the properties of the expectation operator in probability theory. Also, for Bayesian belief functions, our definition provides the same expected value as the probabilistic expectation operator. A traditional method of computing expected of real-valued functions is to first transform a D–S belief function to a corresponding probability mass function, and then use the expectation operator for probability mass functions. Transforming a belief function to a probability function involves loss of information. Our expectation operator works directly with D–S belief functions. Another definition is using Choquet integration, which assumes belief functions are credal sets, i.e. convex sets of probability mass functions. Credal sets semantics are incompatible with Dempster's combination rule, the center-piece of the D–S theory. In general, our definition provides different expected values than, e.g. if we use probabilistic expectation using the pignistic transform or the plausibility transform of a belief function. Using our definition of expectation, we provide new definitions of variance, covariance, correlation, and other higher moments and describe their properties.

The contradiction between belief functions: Its description, measurement, and correction based on generalized credal sets

In the paper, we formalize the notion of contradiction between belief functions: we argue that belief functions are not contradictory if they provide non-contradictory models for decision-making. To elaborate on this idea, we take the decision rule from imprecise probabilities and show that sources of information described by belief functions are not contradictory iff the intersection of corresponding credal sets is not empty. We demonstrate that evidential conjunctive and disjunctive rules fit with this idea and they are justified in a probabilistic setting. In the case of contradictory sources of information, we analyze possible conjunctions and show how the result can be described by generalized credal sets. Based on generalized credal sets, we propose a measure of contradiction between information sources and find its axiomatics. We show how the contradiction correction can be produced based on generalized credal sets and how it can be done on sets of surely desirable gambles.

A novel approach to transformer fault diagnosis using IDM and naive credal classifier

Transformer fault diagnosis is important for improving the operational reliability of power systems. Despite great efforts to enhance the accuracy of fault diagnosis, the precise detection of transformer faults remains difficult. The major barriers are attributed to the lack of transformer fault records and related data, which may produce unconvincing diagnostic results, and to the fact that multiple latent faults in a transformer are difficult to detect distinctly. In this paper, an imprecise probability-based approach to transformer fault diagnosis by using the imprecise Dirichlet model and naive credal classifier is proposed. Instead of providing a single-valued probability, the approach calculates the interval probability for each possible type of transformer faults to explicitly indicate the uncertainty of the diagnosis caused by the insufficiency of historical fault records. Meanwhile, according to whether an overlap exists among the probability intervals of different types of faults, the proposed approach can output either one explicit fault or multiple probable latent faults that occurred in the transformer. In the proposed approach, the imprecise Dirichlet model (IDM) is used to evaluate the imprecise probabilistic relationships between each type of transformer fault and its corresponding symptom attributes. A naive credal classifier (NCC) is established to integrate the IDM estimation results and calculate the interval-valued probabilities of all types of possible faults. Then, the transformer fault diagnosis results can be obtained by the classification criteria of the NCC. The approach can compensate for the defects of conventional methods, including their inability to perform reliable diagnosis with insufficient fault samples or distinctly indicate the multiple latent transformer faults. The accuracy of diagnosis and the efficiency of transformer maintenance can be significantly improved by implementing the proposed approach. The effectiveness of the approach is verified through case studies.

Evenly convex credal sets

An evenly convex credal set is a set of probability measures that is evenly convex; that is, a set that is an arbitrary intersection of open halfspaces. An evenly convex credal set can for instance encode preference judgments through strict and non-strict inequalities such as P(A)>1/2 and P(A)≤2/3. This paper presents an axiomatization of evenly convex sets from preferences, where we introduce a new (and very weak) Archimedean condition. We examine the duality between preference orderings and credal sets; we also consider assessments of almost preference and natural extensions. We then discuss regular conditioning, a concept that is closely related to evenly convex sets.

Modeling Deterministic Equations in Discrete Bayesian Networks by Credal Networks

We focus on the problem of embedding deterministic equations in discrete Bayesian networks. This is typically achieved by a discretization of both input and output variables and a degenerate quantification of the corresponding conditional probability table. We note that, generally speaking, this approach based on the classical Bayesian theory of probability cannot properly take into account the information loss induced by the discretization. Consequently, we propose a conservative modeling of such epistemic uncertainty by means of credal sets, i.e., sets of probability mass functions. This approach transforms the original Bayesian network in a credal network, returning interval-valued inferences, that are robust with respect to the discretization. Procedures for optimal discretization in this setup are discussed. The proposed method can be used for both developing knowledge-based expert systems as well as machine learning based on probabilistic graphical models.

Concepts for decision making under severe uncertainty with partial ordinal and partial cardinal preferences

We introduce three different approaches for decision making under uncertainty if (I) there is only partial (both cardinally and ordinally scaled) information on an agent's preferences and (II) the uncertainty about the states of nature is described by a credal set (or some other imprecise probabilistic model). Particularly, situation (I) is modeled by a pair of binary relations, one specifying the partial rank order of the alternatives and the other modeling partial information on the strength of preference. Our first approach relies on decision criteria constructing complete rankings of the available acts that are based on generalized expectation intervals. Subsequently, we introduce different concepts of global admissibility that construct partial orders between the available acts by comparing them all simultaneously. Finally, we define criteria induced by suitable binary relations on the set of acts and, therefore, can be understood as concepts of local admissibility. For certain criteria, we provide linear programming based algorithms for checking optimality/admissibility of acts. Additionally, the paper includes a discussion of a prototypical situation by means of a toy example.

Tackling uncertainty in security assessment of critical infrastructures: Dempster-Shafer Theory vs. Credal Sets Theory

Securing critical infrastructures is a complex task. Required information is usually scarce or inexistent, and experts’ judgments may be inaccurate and biased. In this paper, two methodologies dealing with data scarcity, imprecision, and uncertainty are presented: Evidential network and Credal network. Evidential network is a graphical technique based on Dempster-Shafer Theory to explicitly model the propagation of epistemic uncertainty among variables while Credal network is an extension of Bayesian network to deal with sets of probabilities, known as Credal sets, based on experts’ judgments. Both methodologies constitute robust frameworks to account for high degree of imprecision on data, producing informative results despite the low-informative input. In the present study, the power in expressing uncertainty of these two methodologies have been showed, and their differences have been described through their application to a case study of security vulnerability assessment. Results demonstrate the substantial equivalence of the two methodologies in prognostic analysis, thus, an approximate updating procedure of Evidential network through equivalent Credal network has been proposed, to overcome the lack of possibility to compute updating in the context of Dempster-Shafer Theory.

Applications of Data Mining Techniques in Healthcare and Prediction of Heart Attacks

The healthcare environment is generally perceived as being 'information rich' yet 'knowledge poor'. There is a wealth of data available within the healthcare systems. However, there is a lack of effective analysis tools to discover hidden relationships and trends in data. Knowledge discovery and data mining have found numerous applications in business and scientific domain. Valuable knowledge can be discovered from application of data mining techniques in healthcare system. In this study, we briefly examine the potential use of classification based data mining techniques such as Rule based, Decision tree, Naive Bayes and Artificial Neural Network to massive volume of healthcare data. The healthcare industry collects huge amounts of healthcare data which, unfortunately, are not "mined" to discover hidden information. For data preprocessing and effective decision making One Dependency Augmented Naive Bayes classifier (ODANB) and naive credal classifier 2 (NCC2) are used. This is an extension of naive Bayes to imprecise probabilities that aims at delivering robust classifications also when dealing with small or incomplete data sets. Discovery of hidden patterns and relationships often goes unexploited. Using medical profiles such as age, sex, blood pressure and blood sugar it can predict the likelihood of patients getting a heart disease. It enables significant knowledge, e.g. patterns, relationships between medical factors related to heart disease, to be established.

Modelling uncertainty with generalized credal sets: application to conjunction and decision

To model conflict, non-specificity and contradiction in information, upper and lower generalized credal sets are introduced. Any upper generalized credal set is a convex subset of plausibility measures interpreted as lower probabilities whose bodies of evidence consist of singletons and a certain event. Analogously, contradiction is modelled in the theory of evidence by a belief function that is greater than zero at empty set. Based on generalized credal sets, we extend the conjunctive rule for contradictory sources of information, introduce constructions like natural extension in the theory of imprecise probabilities and show that the model of generalized credal sets coincides with the model of imprecise probabilities if the profile of a generalized credal set consists of probability measures. We give ways how the introduced model can be applied to decision problems.

Compositional models for credal sets

We present the composition operator, already known from probability, possibility, evidence and valuation-based systems theories, for credal sets. We prove that the proposed definition preserves all the properties enabling us to design compositional models in a way analogous to those in the above-mentioned theories. A special kind of compositional models, the so-called perfect sequences of credal sets, is studied in more detail and (among others) its relationship to perfect sequences of probability distributions is revealed. The theoretical results are illustrated by numerous simple examples.

Some Characteristics of Credal Sets and Their Application to Analysis of Polls Results

In the theory of imprecise probabilities there are well established functionals for measuring uncertainty. Meanwhile, when we try to apply them to real problems they have a lack of meaningful interpretation. In this investigation we propose some characteristics of credal sets that seem to be useful for analyzing polls results.

The multilabel naive credal classifier

A credal classifier for multilabel data is presented. This is obtained as an extension of Zaffalon's naive credal classifier to the case of non-exclusive class labels. The dependence relations among the labels are shaped with a tree topology. The classifier, based on a polynomial-time algorithm to compute whether or not a class label is optimal, returns a compact description of the set of optimal sequences of labels. Extensive experiments on real multilabel data show that the classifier gives more robust predictions than its Bayesian counterpart. In practice, when multiple sequences are returned in output, the Bayesian model is more likely to be inaccurate, while the sequences returned by the credal classifier are more likely to include the correct one.

Extreme points of the credal sets generated by comparative probabilities

When using convex probability sets (or, equivalently, lower previsions) as uncertainty models, identifying extreme points can help simplifying various computations or the use of some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixtures or 2-monotone measures may have extreme points easier to compute than generic convex sets. In this paper, we study extreme points of another specific model: comparative probability orderings between the singletons of a finite space. We characterize these extreme points by means of a graphical representation of the comparative model, and use them to study the properties of the lower probability induced by this set. By doing so, we show that 2-monotone capacities are not informative enough to handle this type of comparisons without a loss of information. In addition, we connect comparative probabilities with other uncertainty models, such as imprecise probability masses.

Advanced Line Sampling for efficient robust reliability analysis

A numerical strategy for the efficient estimation of set-valued failure probabilities, coupling Monte Carlo with optimization methods, is presented in this paper. The notion of uncertainty is generalized to include both aleatory and epistemic uncertainties, allowing to capture gaps of knowledge and scarcity of data. The proposed formulation of the generalized uncertainty model allows for sets of probability distribution functions, also known as credal sets, and sets of bounded variables. The developed Advanced Line Sampling method is combined with the generalized uncertainty model, in order to both speed up the reliability analysis, and provide a better estimate for the lower and upper bounds of the failure probability. The proposed strategy knocks down the computational barrier of computing interval failure probabilities, and reduces the cost of a robust reliability analysis by many orders of magnitude. The efficiency and applicability of the developed method is demonstrated via numerical examples. The solution strategy is integrated into the open-source software for uncertainty quantification and risk analysis OpenCossan, allowing its application on large-scale engineering problems as well as broadening its spectrum of applications.

A geometric and game-theoretic study of the conjunction of possibility measures

In this paper, we study the conjunction of possibility measures when they are interpreted as coherent upper probabilities, that is, as upper bounds for some set of probability measures. We identify conditions under which the minimum of two possibility measures remains a possibility measure. We provide graphical way to check these conditions, by means of a zero-sum game formulation of the problem. This also gives us a nice way to adjust the initial possibility measures so their minimum is guaranteed to be a possibility measure. Finally, we identify conditions under which the minimum of two possibility measures is a coherent upper probability, or in other words, conditions under which the minimum of two possibility measures is an exact upper bound for the intersection of the credal sets of those two possibility measures.

Credal model averaging for classification: representing prior ignorance and expert opinions

Bayesian model averaging (BMA) is the state of the art approach for overcoming model uncertainty. Yet, especially on small data sets, the results yielded by BMA might be sensitive to the prior over the models. Credal model averaging (CMA) addresses this problem by substituting the single prior over the models by a set of priors (credal set). Such approach solves the problem of how to choose the prior over the models and automates sensitivity analysis. We discuss various CMA algorithms for building an ensemble of logistic regressors characterized by different sets of covariates. We show how CMA can be appropriately tuned to the case in which one is prior-ignorant and to the case in which instead domain knowledge is available. CMA detects prior-dependent instances, namely instances in which a different class is more probable depending on the prior over the models. On such instances CMA suspends the judgment, returning multiple classes. We thoroughly compare different BMA and CMA variants on a real case study, predicting presence of Alpine marmot burrows in an Alpine valley. We find that BMA is almost a random guesser on the instances recognized as prior-dependent by CMA.

Credal networks under epistemic irrelevance: The sets of desirable gambles approach

We present a new approach to credal networks, which are graphical models that generalise Bayesian networks to deal with imprecise probabilities. Instead of applying the commonly used notion of strong independence, we replace it by the weaker, asymmetrical notion of epistemic irrelevance. We show how assessments of epistemic irrelevance allow us to construct a global model out of given local uncertainty models, leading to an intuitive expression for the so-called irrelevant natural extension of a credal network. In contrast with Cozman [4], who introduced this notion in terms of credal sets, our main results are presented using the language of sets of desirable gambles. This has allowed us to derive some remarkable properties of the irrelevant natural extension, including marginalisation properties and a tight connection with the notion of independent natural extension. Our perhaps most important result is that the irrelevant natural extension satisfies a collection of epistemic irrelevancies that is induced by AD-separation, an asymmetrical adaptation of d-separation. Both AD-separation and the induced collection of irrelevancies are shown to satisfy all graphoid properties except symmetry.

Comments on “Statistical reasoning with set-valued information: Ontic vs. epistemic view” by Inés Couso and Didier Dubois

I discuss some aspects of the distinction between ontic and epistemic views of sets as representation of imprecise or incomplete information. In particular, I consider its implications on imprecise probability representations: credal sets and sets of desirable gambles. It is emphasized that the interpretation of the same mathematical object can be different depending on the point of view from which this element is considered. In the case of a fuzzy information on a random variable, it is possible to define a possibility distribution on the simplex of probability distributions. I add some comments about the properties of this possibility distribution.

Credal-C4.5: Decision tree based on imprecise probabilities to classify noisy data

In the area of classification, C4.5 is a known algorithm widely used to design decision trees. In this algorithm, a pruning process is carried out to solve the problem of the over-fitting. A modification of C4.5, called Credal-C4.5, is presented in this paper. This new procedure uses a mathematical theory based on imprecise probabilities, and uncertainty measures. In this way, Credal-C4.5 estimates the probabilities of the features and the class variable by using imprecise probabilities. Besides it uses a new split criterion, called Imprecise Information Gain Ratio, applying uncertainty measures on convex sets of probability distributions (credal sets). In this manner, Credal-C4.5 builds trees for solving classification problems assuming that the training set is not fully reliable. We carried out several experimental studies comparing this new procedure with other ones and we obtain the following principal conclusion: in domains of class noise, Credal-C4.5 obtains smaller trees and better performance than classic C4.5.

Credal ensembles of classifiers

It is studied how to aggregate the probabilistic predictions generated by different SPODE (Super-Parent-One-Dependence Estimators) classifiers. It is shown that aggregating such predictions via compression-based weights achieves a slight but consistent improvement of performance over previously existing aggregation methods, including Bayesian Model Averaging and simple average (the approach adopted by the AODE algorithm). Then, attention is given to the problem of choosing the prior probability distribution over the models; this is an important issue in any Bayesian ensemble of models. To robustly deal with the choice of the prior, the single prior over the models is substituted by a set of priors over the models (credal set), thus obtaining a credal ensemble of Bayesian classifiers. The credal ensemble recognizes the prior-dependent instances, namely the instances whose most probable class varies when different prior over the models are considered. When faced with prior-dependent instances, the credal ensemble remains reliable by returning a set of classes rather than a single class. Two credal ensembles of SPODEs are developed; the first generalizes the Bayesian Model Averaging and the second the compression-based aggregation. Extensive experiments show that the novel ensembles compare favorably to traditional methods for aggregating SPODEs and also to previous credal classifiers.